Stergios-Konstantinidis/MNLP_M2_rag_dataset_limit_tokens_few

text stringlengths 1 46.8k	source stringclasses 1 value	input_ids sequencelengths 3 512	token_type_ids sequencelengths 3 512	attention_mask sequencelengths 3 512
{	test	[ 101, 1063, 102 ]	[ 0, 0, 0 ]	[ 1, 1, 1 ]
"title": "Prime scattering geodesic theorem",	test	[ 101, 1000, 2516, 1000, 1024, 1000, 3539, 17501, 20248, 6155, 2594, 9872, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"abstract": "Abstract.\nThe modular surface, given by the quotient \u2133=\u210d/PSL\u2062(2,\u2124)\u2133\u210dPSL2\u2124\\mathcal{M}=\\mathbb{H}/\\text{PSL}(2,\\mathbb{Z})caligraphic_M = blackboard_H / PSL ( 2 , blackboard_Z ), can be partitioned into a compact subset \u2133csubscript\u2133\ud835\udc50\\mathcal{M...	test	[ 101, 1000, 10061, 1000, 1024, 1000, 10061, 1012, 1032, 23961, 5369, 19160, 3302, 1010, 2445, 2011, 1996, 22035, 9515, 3372, 1032, 23343, 17134, 2509, 1027, 1032, 23343, 10790, 2094, 1013, 8827, 2140, 1032, 23343, 2692, 2575, 2475, 1006, 101...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"introduction": "The study of geodesics has been a central theme in mathematics due to its connection with different branches of Mathematics and Physics.",	test	[ 101, 1000, 4955, 1000, 1024, 1000, 1996, 2817, 1997, 20248, 6155, 6558, 2038, 2042, 1037, 2430, 4323, 1999, 5597, 2349, 2000, 2049, 4434, 2007, 2367, 5628, 1997, 5597, 1998, 5584, 1012, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"main_content": [	test	[ 101, 1000, 2364, 1035, 4180, 1000, 1024, 1031, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"Inspired by Riemann\u2019s explicit formula, Selberg [21] introduced a zeta function and established his trace formula. Further inspired by the connection of the zeros of the zeta function and prime numbers, Selberg [21] used his mighty trace formula to count the lengths of closed geodesics on a compact Rieman...	test	[ 101, 1000, 4427, 2011, 15544, 17545, 1032, 23343, 24096, 2683, 2015, 13216, 5675, 1010, 7367, 22927, 1031, 2538, 1033, 3107, 1037, 23870, 3853, 1998, 2511, 2010, 7637, 5675, 1012, 2582, 4427, 2011, 1996, 4434, 1997, 1996, 5717, 2015, 1997, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"One of the key characteristics of Selberg\u2019s trace formula is its unique insight into the relationship between the discrete spectrum of the hyperbolic Laplacian and the lengths of closed geodesics on a compact surface. Selberg later extended this formula to the non-compact case, incorporating both the disc...	test	[ 101, 1000, 2028, 1997, 1996, 3145, 6459, 1997, 7367, 22927, 1032, 23343, 24096, 2683, 2015, 7637, 5675, 2003, 2049, 4310, 12369, 2046, 1996, 3276, 2090, 1996, 16246, 8674, 1997, 1996, 23760, 18647, 5001, 2721, 14483, 1998, 1996, 10742, 1997...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"In 1977, Victor Guillemin wrote a paper ([6]) in which he discussed the asymptotic behavior of the scattering matrix arising from the automorphic wave equation, following the work of Lax and Phillips (see [13], [14]). He considered a finite-area hyperbolic surface with cusps given by X=\u210d/\u0393\ud835\udc4...	test	[ 101, 1000, 1999, 3355, 1010, 5125, 26458, 6216, 10020, 2626, 1037, 3259, 1006, 1031, 1020, 1033, 1007, 1999, 2029, 2002, 6936, 1996, 2004, 24335, 13876, 20214, 5248, 1997, 1996, 17501, 8185, 17707, 2013, 1996, 8285, 18078, 4400, 8522, 1010,...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"As part of his framework, Guillemin introduced the notion of a scattering geodesic on such a hyperbolic surface and showed that these geodesics spend only a finite amount of time (called the sojourn time) inside the compact core of the surface. One of the main results in [6] is a trace formula type theorem tha...	test	[ 101, 1000, 2004, 2112, 1997, 2010, 7705, 1010, 26458, 6216, 10020, 3107, 1996, 9366, 1997, 1037, 17501, 20248, 6155, 2594, 2006, 2107, 1037, 23760, 18647, 3302, 1998, 3662, 2008, 2122, 20248, 6155, 6558, 5247, 2069, 1037, 10713, 3815, 1997,...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"We now review the fundamental aspects of scattering geodesics from [6]. Let us assume that X\ud835\udc4bXitalic_X has n\ud835\udc5bnitalic_n number of cusps labeled with \u03ba1,\u03ba2,\u2026,\u03bansubscript\ud835\udf051subscript\ud835\udf052\u2026subscript\ud835\udf05\ud835\udc5b\\kappa_{1},\\kappa_{2},...,...	test	[ 101, 1000, 2057, 2085, 3319, 1996, 8050, 5919, 1997, 17501, 20248, 6155, 6558, 2013, 1031, 1020, 1033, 1012, 2292, 2149, 7868, 2008, 1060, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2549, 2497, 9048, 9080, 2594, 1035, 1060, 2038, 1050, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"The following observation has been made in [cf. Appendix B, Corollary 2 [6]]. \nA scattering geodesic has the property that for large\nnegative and positive times it corresponds to a vertical line in\nthe figure below (i.e. after we have mapped the appropriate cusp neighborhoods onto\nthe standard cusp neighbo...	test	[ 101, 1000, 1996, 2206, 8089, 2038, 2042, 2081, 1999, 1031, 12935, 1012, 22524, 1038, 1010, 2522, 28402, 5649, 1016, 1031, 1020, 1033, 1033, 1012, 1032, 6583, 17501, 20248, 6155, 2594, 2038, 1996, 3200, 2008, 2005, 2312, 1032, 1050, 2638, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"In [6, Theorem B1], Guillemin proved the following result.",	test	[ 101, 1000, 1999, 1031, 1020, 1010, 9872, 29491, 1033, 1010, 26458, 6216, 10020, 4928, 1996, 2206, 2765, 1012, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"There are countably many scattering geodesics that scatter between a given pair of cusps \u03baisubscript\ud835\udf05\ud835\udc56\\kappa_{i}italic_\u03ba start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and \u03bajsubscript\ud835\udf05\ud835\udc57\\kappa_{j}italic_\u03ba start_POSTSUBSCRIPT italic_j end_POSTSUBS...	test	[ 101, 1000, 2045, 2024, 4175, 8231, 2116, 17501, 20248, 6155, 6558, 2008, 8040, 20097, 2090, 1037, 2445, 3940, 1997, 12731, 13102, 2015, 1032, 1057, 2692, 2509, 26068, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2546, 2692, 262...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"In this paper, we prove a Prime number theorem kind result for Scattering geodesics. This is the first such result in the theory of scattering geodesics.\nMore explicitly, we prove the following theorem.",	test	[ 101, 1000, 1999, 2023, 3259, 1010, 2057, 6011, 1037, 3539, 2193, 9872, 2785, 2765, 2005, 17501, 20248, 6155, 6558, 1012, 2023, 2003, 1996, 2034, 2107, 2765, 1999, 1996, 3399, 1997, 17501, 20248, 6155, 6558, 1012, 1032, 13221, 5686, 12045, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"For Y\u226b0much-greater-than\ud835\udc4c0Y\\gg 0italic_Y \u226b 0, let \u03a0\u2062(Y)\u03a0\ud835\udc4c\\Pi(Y)roman_\u03a0 ( italic_Y ) be the number of distinct scattering geodesics in \u2133:=\u210d/PSL\u2062(2,\u2124)assign\u2133\u210dPSL2\u2124\\mathcal{M}:=\\mathbb{H}/\\text{PSL}(2,\\mathbb{Z})caligraph...	test	[ 101, 1000, 2005, 1061, 1032, 23343, 23833, 2497, 2692, 12274, 2818, 1011, 3618, 1011, 2084, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2549, 2278, 2692, 2100, 1032, 1032, 1043, 2290, 1014, 18400, 2594, 1035, 1061, 1032, 23343, 23833, 2497...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"The problem of counting closed geodesics in compact manifolds bears a remarkable resemblance to counting prime numbers (see [5] for more details). In our case, the counting of scattering geodesics on the modular surface is closely linked to the study of positive integers whose prime divisors lie in an arithmet...	test	[ 101, 1000, 1996, 3291, 1997, 10320, 2701, 20248, 6155, 6558, 1999, 9233, 19726, 2015, 6468, 1037, 9487, 14062, 2000, 10320, 3539, 3616, 1006, 2156, 1031, 1019, 1033, 2005, 2062, 4751, 1007, 1012, 1999, 2256, 2553, 1010, 1996, 10320, 1997, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Let \u210d={z=x+i\u2062y\u2223x,y\u2208\u211d,y>0}\u210dconditional-set\ud835\udc67\ud835\udc65\ud835\udc56\ud835\udc66formulae-sequence\ud835\udc65\ud835\udc66\u211d\ud835\udc660\\mathbb{H}=\\{z=x+iy\\mid x,y\\in\\mathbb{R},y>0\\}blackboard_H = { italic_z = italic_x + italic_i italic_y \u2223 italic_x , itali...	test	[ 101, 1000, 2292, 1032, 23343, 10790, 2094, 1027, 1063, 1062, 1027, 1060, 1009, 1045, 1032, 23343, 2692, 2575, 2475, 2100, 1032, 23343, 19317, 2509, 2595, 1010, 1061, 1032, 23343, 11387, 2620, 1032, 23343, 14526, 2094, 1010, 1061, 1028, 1014...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"The modular surface\nis the quotient space \u2133=\u210d/PSL\u2062(2,\u2124)\u2133\u210dPSL2\u2124\\mathcal{M}=\\mathbb{H}/\\text{PSL}(2,\\mathbb{Z})caligraphic_M = blackboard_H / PSL ( 2 , blackboard_Z ), which is equipped with the metric induced from the upper half-plane. We know that the modular surface\nha...	test	[ 101, 1000, 1996, 19160, 3302, 1032, 9152, 2015, 1996, 22035, 9515, 3372, 2686, 1032, 23343, 17134, 2509, 1027, 1032, 23343, 10790, 2094, 1013, 8827, 2140, 1032, 23343, 2692, 2575, 2475, 1006, 1016, 1010, 1032, 23343, 12521, 2549, 1007, 1032...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"(1.3)\n\n\u2133cT0=\u03c0\u2062(\u2131\u2229{\u2111\u2061(z)\u2264T0}),\u2133\u221eT0=\u03c0\u2062(\u2131\u2229{\u2111\u2061(z)>T0}).formulae-sequencesuperscriptsubscript\u2133\ud835\udc50subscript\ud835\udc470\ud835\udf0b\u2131\ud835\udc67subscript\ud835\udc470superscriptsubscript\u2133subscript\ud835\udc470\...	test	[ 101, 1000, 1006, 1015, 1012, 1017, 1007, 1032, 1050, 1032, 1050, 1032, 23343, 17134, 2509, 6593, 2692, 1027, 1032, 1057, 2692, 2509, 2278, 2692, 1032, 23343, 2692, 2575, 2475, 1006, 1032, 23343, 17134, 2487, 1032, 23343, 19317, 2683, 1063, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"We are interested in a class of scattering geodesic in \u2133\u2133\\mathcal{M}caligraphic_M that runs between the cusp, which was first discussed by Victor Guillemin\nin [6].",	test	[ 101, 1000, 2057, 2024, 4699, 1999, 1037, 2465, 1997, 17501, 20248, 6155, 2594, 1999, 1032, 23343, 17134, 2509, 1032, 23343, 17134, 2509, 1032, 1032, 8785, 9289, 1063, 1049, 1065, 10250, 8004, 20721, 1035, 1049, 2008, 3216, 2090, 1996, 12731...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"We recall the following definition of scattering geodesics from [6];",	test	[ 101, 1000, 2057, 9131, 1996, 2206, 6210, 1997, 17501, 20248, 6155, 6558, 2013, 1031, 1020, 1033, 1025, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"A geodesic \u03b3\u2062(t)\ud835\udefe\ud835\udc61\\gamma(t)italic_\u03b3 ( italic_t ) in \u2133\u2133\\mathcal{M}caligraphic_M is called a scattering geodesic if it is contained in \u2133\u2216\u2133cT0\u2133superscriptsubscript\u2133\ud835\udc50subscript\ud835\udc470\\mathcal{M}\\setminus\\mathcal{M}_{c}^{T_...	test	[ 101, 1000, 1037, 20248, 6155, 2594, 1032, 1057, 2692, 2509, 2497, 2509, 1032, 23343, 2692, 2575, 2475, 1006, 1056, 1007, 1032, 20904, 2620, 19481, 1032, 20904, 27235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2575, 2487, 1032, 1032, 1309...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Next, we show that the scattering geodesics are in one-to-one correspondence with a subset of rationals in [0,1)01[0,1)[ 0 , 1 ).",	test	[ 101, 1000, 2279, 1010, 2057, 2265, 2008, 1996, 17501, 20248, 6155, 6558, 2024, 1999, 2028, 1011, 2000, 1011, 2028, 11061, 2007, 1037, 16745, 1997, 11581, 2015, 1999, 1031, 1014, 1010, 1015, 1007, 5890, 1031, 1014, 1010, 1015, 1007, 1031, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"We start with the following result;",	test	[ 101, 1000, 2057, 2707, 2007, 1996, 2206, 2765, 1025, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"For w\u2208[0,1)\u2229\u211a\ud835\udc6401\u211aw\\in[0,1)\\cap\\mathbb{Q}italic_w \u2208 [ 0 , 1 ) \u2229 blackboard_Q, let \u03b3\u00afwsubscript\u00af\ud835\udefe\ud835\udc64\\bar{\\gamma}_{w}over\u00af start_ARG italic_\u03b3 end_ARG start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT be the unique geodesic in ...	test	[ 101, 1000, 2005, 1059, 1032, 23343, 11387, 2620, 1031, 1014, 1010, 1015, 1007, 1032, 23343, 19317, 2683, 1032, 23343, 14526, 2050, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 21084, 24096, 1032, 23343, 14526, 10376, 1032, 1032, 1999, 1031, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"q1=q2=qsubscript\ud835\udc5e1subscript\ud835\udc5e2\ud835\udc5eq_{1}=q_{2}=qitalic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_q.",	test	[ 101, 1000, 1053, 2487, 1027, 1053, 2475, 1027, 1053, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2487, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2475, 1032, 20904, 2620, 1948...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"q\ud835\udc5eqitalic_q divides p1\u2062p2+1subscript\ud835\udc5d1subscript\ud835\udc5d21p_{1}p_{2}+1italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 1.",	test	[ 101, 1000, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 20487, 1052, 2487, 1032, 23343, 2692, 2575, 2475, 2361, 2475, 1009, 1015, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 227...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Fixing a T0subscript\ud835\udc470T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT as before, it follows from [Corollary 2,Theorem B1, [6]] that there are a countable number of non-trivial scattering geodesics in \u2133\u2133\\mathcal{M}caligraphic_M. Our first goal is to show that the set of scattering ge...	test	[ 101, 1000, 15887, 1037, 1056, 16223, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 22610, 2692, 2102, 1035, 1063, 1014, 1065, 2009, 27072, 1035, 1056, 2707, 1035, 8466, 12083, 22483, 1014, 2203, 1035, 8466, 12083, 22483, 2004, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Let \ud835\udcae\ud835\udcae\\mathcal{S}caligraphic_S be the set of scattering geodesics in \u2133\u2133\\mathcal{M}caligraphic_M, then there is a one-to-one correspondence between \ud835\udcae\ud835\udcae\\mathcal{S}caligraphic_S and the set \ud835\udca2\ud835\udca2\\mathcal{G}caligraphic_G defined in (1.8), ...	test	[ 101, 1000, 2292, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2063, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2063, 1032, 1032, 8785, 9289, 1063, 1055, 1065, 10250, 8004, 20721, 1035, 1055, 2022, 1996, 2275, 1997, 17501, 20248, 6155, 655...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Next, we begin the construction of this set, which we denote by \ud835\udca2\ud835\udca2\\mathcal{G}caligraphic_G.",	test	[ 101, 1000, 2279, 1010, 2057, 4088, 1996, 2810, 1997, 2023, 2275, 1010, 2029, 2057, 19090, 2011, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 1032, 8785, 9289, 1063, 1043, 1065, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"For q\u22652\ud835\udc5e2q\\geq 2italic_q \u2265 2, consider the set\n\n\n\nIq:={p\u2208\u2124+\u22231\u2264p<q,gcd\u2061(p,q)=1}.assignsubscript\ud835\udc3c\ud835\udc5econditional-set\ud835\udc5dsuperscript\u2124formulae-sequence1\ud835\udc5d\ud835\udc5e\ud835\udc5d\ud835\udc5e1I_{q}:=\\{p\\in\\mathbb{Z}^{+}\...	test	[ 101, 1000, 2005, 1053, 1032, 23343, 23833, 25746, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2475, 4160, 1032, 1032, 16216, 4160, 1016, 18400, 2594, 1035, 1053, 1032, 23343, 23833, 2629, 1016, 1010, 5136, 1996, 2275, 1032, 105...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"(1.4)\n\nSq:={p\u2208\u2124+\u22231\u2264p<q,p2\u2261\u22121(modq)}.assignsubscript\ud835\udc46\ud835\udc5econditional-set\ud835\udc5dsuperscript\u2124formulae-sequence1\ud835\udc5d\ud835\udc5esuperscript\ud835\udc5d2annotated1pmod\ud835\udc5eS_{q}:=\\{p\\in\\mathbb{Z}^{+}\\mid 1\\leq p<q,\\,p^{2}\\equiv-1\\pm...	test	[ 101, 1000, 1006, 1015, 1012, 1018, 1007, 1032, 1050, 1032, 24978, 4160, 1024, 1027, 1063, 1052, 1032, 23343, 11387, 2620, 1032, 23343, 12521, 2549, 1009, 1032, 23343, 19317, 21486, 1032, 23343, 23833, 2549, 2361, 1026, 1053, 1010, 1052, 247...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"For a given p\u2208Iq\ud835\udc5dsubscript\ud835\udc3c\ud835\udc5ep\\in I_{q}italic_p \u2208 italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, we now examine equation p\u2062p\u2032\u2261\u22121(modq)\ud835\udc5dsuperscript\ud835\udc5d\u2032annotated1pmod\ud835\udc5epp^{\\prime}\\equiv-1\\pmod{q}italic_...	test	[ 101, 1000, 2005, 1037, 2445, 1052, 1032, 23343, 11387, 2620, 18515, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2509, 2278, 1032, 20904, 2620, 19481, 1032, 20904, 2278, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Now, there are two cases to consider:\n\n\n(i)\n\np=yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep=y_{p,q}italic_p = italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT,\n\n\n\n(ii)\n\np\u2260yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep\\neq y_{p,q}italic_p \u22...	test	[ 101, 1000, 2085, 1010, 2045, 2024, 2048, 3572, 2000, 5136, 1024, 1032, 1050, 1032, 1050, 1032, 1050, 1006, 1045, 1007, 1032, 1050, 1032, 27937, 1027, 1061, 2361, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"p=yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep=y_{p,q}italic_p = italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT,",	test	[ 101, 1000, 1052, 1027, 1061, 2361, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, 1032, 20904, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"p\u2260yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep\\neq y_{p,q}italic_p \u2260 italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT.",	test	[ 101, 1000, 1052, 1032, 23343, 23833, 2692, 22571, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"If p=yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep=y_{p,q}italic_p = italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT, then p\u2208Sq\ud835\udc5dsubscript\ud835\udc46\ud835\udc5ep\\in S_{q}italic_p \u2208 italic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, where Sqsubscr...	test	[ 101, 1000, 2065, 1052, 1027, 1061, 2361, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, 1032, 2...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Finally, we have the partition Iq=Cq\u2294Sqsubscript\ud835\udc3c\ud835\udc5esquare-unionsubscript\ud835\udc36\ud835\udc5esubscript\ud835\udc46\ud835\udc5eI_{q}=C_{q}\\sqcup S_{q}italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT \u2294 italic_S st...	test	[ 101, 1000, 2633, 1010, 2057, 2031, 1996, 13571, 26264, 1027, 1039, 4160, 1032, 23343, 24594, 2549, 2015, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2509, 2278, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 22...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Let \ud835\udca21={0}subscript\ud835\udca210\\mathcal{G}_{1}=\\{0\\}caligraphic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { 0 } and for q\u22652,q\u2208\u2124+formulae-sequence\ud835\udc5e2\ud835\udc5esuperscript\u2124q\\geq 2,q\\in\\mathbb{Z}^{+}italic_q \u2265 2 , italic_q \u2208 blackboard_Z start_POSTSUP...	test	[ 101, 1000, 2292, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 17465, 1027, 1063, 1014, 1065, 4942, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 17465, 2692, 1032, 1032, 8785, 9289, 1063, 1043, 1065, 1035, 1063, 1015, 1065, 1027, 1032...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"The set \ud835\udca2qsubscript\ud835\udca2\ud835\udc5e\\mathcal{G}_{q}caligraphic_G start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT is ordered in an obvious way as a subset of rational numbers with standard ordering.",	test	[ 101, 1000, 1996, 2275, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 1032, 1032, 8785, 9289, 1063, 104...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Let \ud835\udca2\u2282[0,1)\ud835\udca201\\mathcal{G}\\subset[0,1)caligraphic_G \u2282 [ 0 , 1 ) be the following subset,\n\n\n(1.8)\n\n\ud835\udca2:=\u2a06q=1\u221e\ud835\udca2q.assign\ud835\udca2superscriptsubscriptsquare-union\ud835\udc5e1subscript\ud835\udca2\ud835\udc5e\\mathcal{G}:=\\displaystyle\\bigsqc...	test	[ 101, 1000, 2292, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 23343, 22407, 2475, 1031, 1014, 1010, 1015, 1007, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 11387, 2487, 1032, 1032, 8785, 9289, 1063, 1043, 1065, 1032, 1032, 1674...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"We order the elements of \ud835\udca2\ud835\udca2\\mathcal{G}caligraphic_G as follows, if x\u2208\ud835\udca2q1,y\u2208\ud835\udca2q2formulae-sequence\ud835\udc65subscript\ud835\udca2subscript\ud835\udc5e1\ud835\udc66subscript\ud835\udca2subscript\ud835\udc5e2x\\in\\mathcal{G}_{q_{1}},y\\in\\mathcal{G}_{q_{2}}...	test	[ 101, 1000, 2057, 2344, 1996, 3787, 1997, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 1032, 8785, 9289, 1063, 1043, 1065, 10250, 8004, 20721, 1035, 1043, 2004, 4076, 1010, 2065,...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"It is clear from the definition that the cardinality of \ud835\udca2qsubscript\ud835\udca2\ud835\udc5e\\mathcal{G}_{q}caligraphic_G start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT is the sum of the cardinalities of the sets Cq\u2217subscriptsuperscript\ud835\udc36\ud835\udc5eC^{*}_{q}italic_C start_POSTSUPERSCR...	test	[ 101, 1000, 2009, 2003, 3154, 2013, 1996, 6210, 2008, 1996, 7185, 3012, 1997, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 2090...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Let nqsubscript\ud835\udc5b\ud835\udc5en_{q}italic_n start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT denote the number of elements in \ud835\udca2qsubscript\ud835\udca2\ud835\udc5e\\mathcal{G}_{q}caligraphic_G start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. Then for q\u22652\ud835\udc5e2q\\geq 2italic_q \u2265 ...	test	[ 101, 1000, 2292, 1050, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2497, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2368, 1035, 1063, 1053, 1065, 2009, 27072, 1035, 1050, 2707, 1035, 8466, 12083, 2248...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"In the next section, we discuss preliminaries results. In the final section, we prove the main results of the paper.",	test	[ 101, 1000, 1999, 1996, 2279, 2930, 1010, 2057, 6848, 3653, 17960, 3981, 5134, 3463, 1012, 1999, 1996, 2345, 2930, 1010, 2057, 6011, 1996, 2364, 3463, 1997, 1996, 3259, 1012, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"In this section, we discuss the results required to prove the main theorems.",	test	[ 101, 1000, 1999, 2023, 2930, 1010, 2057, 6848, 1996, 3463, 3223, 2000, 6011, 1996, 2364, 9872, 2015, 1012, 1000, 1010, 102 ]	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"Recall that for q\u22652\ud835\udc5e2q\\geq 2italic_q \u2265 2,",	test	[ 101, 1000, 9131, 2008, 2005, 1053, 1032, 23343, 23833, 25746, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2475, 4160, 1032, 1032, 16216, 4160, 1016, 18400, 2594, 1035, 1053, 1032, 23343, 23833, 2629, 1016, 1010, 1000, 1010, 102...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
"(2.1)\n\nSq={p\u2208\u2124+\u22231\u2264p<q\u2062\u00a0and\u00a0\u2062p2\u2261\u22121(modq)}subscript\ud835\udc46\ud835\udc5econditional-set\ud835\udc5dsuperscript\u21241\ud835\udc5d\ud835\udc5e\u00a0and\u00a0superscript\ud835\udc5d2annotated1pmod\ud835\udc5eS_{q}=\\{p\\in\\mathbb{Z}^{+}\\mid 1\\leq p<q\\hskip...	test	[ 101, 1000, 1006, 1016, 1012, 1015, 1007, 1032, 1050, 1032, 24978, 4160, 1027, 1063, 1052, 1032, 23343, 11387, 2620, 1032, 23343, 12521, 2549, 1009, 1032, 23343, 19317, 21486, 1032, 23343, 23833, 2549, 2361, 1026, 1053, 1032, 23343, 2692, 25...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"and sqsubscript\ud835\udc60\ud835\udc5es_{q}italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT denotes the cardinality of Sqsubscript\ud835\udc46\ud835\udc5eS_{q}italic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. We set s1=1subscript\ud835\udc6011s_{1}=1italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRI...	test	[ 101, 1000, 1998, 5490, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 16086, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2229, 1035, 1063, 1053, 1065, 2009, 27072, 1035, 1055, 2707, 1035, 8466, 12083, 22483, 2009, 27...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"If q\ud835\udc5eqitalic_q is a prime of the form 4\u2062k+34\ud835\udc5834k+34 italic_k + 3, then sq=0subscript\ud835\udc60\ud835\udc5e0s_{q}=0italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = 0.",	test	[ 101, 1000, 2065, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 2003, 1037, 3539, 1997, 1996, 2433, 1018, 1032, 23343, 2692, 2575, 2475, 2243, 1009, 4090, 1032, 20904, 2620, 19481, 1032, 20904,...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"This follows from the fact the value of the Legendre symbol (\u22121p)1\ud835\udc5d\\left(\\frac{-1}{p}\\right)( divide start_ARG - 1 end_ARG start_ARG italic_p end_ARG ) is \u221211-1- 1 for a prime p\ud835\udc5dpitalic_p of the form 4\u2062k+34\ud835\udc5834k+34 italic_k + 3.\n\u220e",	test	[ 101, 1000, 2023, 4076, 2013, 1996, 2755, 1996, 3643, 1997, 1996, 5722, 2890, 6454, 1006, 1032, 23343, 17465, 17465, 2361, 1007, 1015, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, 1032, 1032, 2187, 1006, 1032, 1032, 25312, 2278, ...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"If q\ud835\udc5eqitalic_q is a prime of the form 4\u2062k+34\ud835\udc5834k+34 italic_k + 3, and m=qk\ud835\udc5asuperscript\ud835\udc5e\ud835\udc58m=q^{k}italic_m = italic_q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT with k\u22652\ud835\udc582k\\geq 2italic_k \u2265 2, then\nsm=0subscript\ud835\udc60\...	test	[ 101, 1000, 2065, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 2003, 1037, 3539, 1997, 1996, 2433, 1018, 1032, 23343, 2692, 2575, 2475, 2243, 1009, 4090, 1032, 20904, 2620, 19481, 1032, 20904,...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Suppose we have a solution y\ud835\udc66yitalic_y that satisfies y2\u2261\u22121(modm)superscript\ud835\udc662annotated1pmod\ud835\udc5ay^{2}\\equiv-1\\pmod{m}italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT \u2261 - 1 start_MODIFIER ( roman_mod start_ARG italic_m end_ARG ) end_MODIFIER. This implies y2\u2...	test	[ 101, 1000, 6814, 2057, 2031, 1037, 5576, 1061, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 10139, 9080, 2594, 1035, 1061, 2008, 2938, 2483, 14213, 1061, 2475, 1032, 23343, 23833, 2487, 1032, 23343, 17465, 17465, 1006, 16913, 2213, 1...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Suppose q\ud835\udc5eqitalic_q is a prime of the form 4\u2062k+14\ud835\udc5814k+14 italic_k + 1. Then for any m\u22651\ud835\udc5a1m\\geq 1italic_m \u2265 1, the equation p2\u2261\u22121(modqm)superscript\ud835\udc5d2annotated1\ud835\udc5d\ud835\udc5a\ud835\udc5c\ud835\udc51superscript\ud835\udc5e\ud835\udc5a...	test	[ 101, 1000, 6814, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 2003, 1037, 3539, 1997, 1996, 2433, 1018, 1032, 23343, 2692, 2575, 2475, 2243, 1009, 2403, 1032, 20904, 2620, 19481, 1032, 20904,...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Note that if qmsuperscript\ud835\udc5e\ud835\udc5aq^{m}italic_q start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT divides p2+1superscript\ud835\udc5d21p^{2}+1italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1, then since q\ud835\udc5eqitalic_q is a prime, we must have q\ud835\udc5eqitalic_q dividing p2+...	test	[ 101, 1000, 3602, 2008, 2065, 1053, 5244, 6279, 2545, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 20784, 1034, 1063, 1049, 1065, 2009, 27072, 1035, 1053, 2707, 1035, 8466...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"We perform this lifting inductively. Suppose we start with a solution x0subscript\ud835\udc650x_{0}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT modulo qjsuperscript\ud835\udc5e\ud835\udc57q^{j}italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT satisfying x02\u2261\u22121(modqj)superscriptsubscript...	test	[ 101, 1000, 2057, 4685, 2023, 8783, 27427, 14194, 25499, 1012, 6814, 2057, 2707, 2007, 1037, 5576, 1060, 16223, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 26187, 2692, 2595, 1035, 1063, 1014, 1065, 2009, 27072, 1035, 1060, 27...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"We begin by substituting the expression x1=x0+qj\u2062y0subscript\ud835\udc651subscript\ud835\udc650superscript\ud835\udc5e\ud835\udc57subscript\ud835\udc660x_{1}=x_{0}+q^{j}y_{0}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_q start_POSTSUPERSCRIPT...	test	[ 101, 1000, 2057, 4088, 2011, 4942, 21532, 1996, 3670, 1060, 2487, 1027, 1060, 2692, 1009, 1053, 3501, 1032, 23343, 2692, 2575, 2475, 2100, 16223, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 26187, 2487, 6342, 5910, 23235, 103...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
"Since gcd\u2061(2\u2062x0,q)=12subscript\ud835\udc650\ud835\udc5e1\\gcd(2x_{0},q)=1roman_gcd ( 2 italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q ) = 1, the above equation yields a unique solution y0subscript\ud835\udc660y_{0}italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT modulo q\ud835\udc5eqital...	test	[ 101, 1000, 2144, 1043, 19797, 1032, 23343, 2692, 2575, 2487, 1006, 1016, 1032, 23343, 2692, 2575, 2475, 2595, 2692, 1010, 1053, 1007, 1027, 2260, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 26187, 2692, 1032, 20904, 2620...	[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

{

test

[ 101, 1063, 102 ]

[ 0, 0, 0 ]

[ 1, 1, 1 ]

"title": "Prime scattering geodesic theorem",

test

[ 101, 1000, 2516, 1000, 1024, 1000, 3539, 17501, 20248, 6155, 2594, 9872, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"abstract": "Abstract.\nThe modular surface, given by the quotient \u2133=\u210d/PSL\u2062(2,\u2124)\u2133\u210dPSL2\u2124\\mathcal{M}=\\mathbb{H}/\\text{PSL}(2,\\mathbb{Z})caligraphic_M = blackboard_H / PSL ( 2 , blackboard_Z ), can be partitioned into a compact subset \u2133csubscript\u2133\ud835\udc50\\mathcal{M...

test

[ 101, 1000, 10061, 1000, 1024, 1000, 10061, 1012, 1032, 23961, 5369, 19160, 3302, 1010, 2445, 2011, 1996, 22035, 9515, 3372, 1032, 23343, 17134, 2509, 1027, 1032, 23343, 10790, 2094, 1013, 8827, 2140, 1032, 23343, 2692, 2575, 2475, 1006, 101...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"introduction": "The study of geodesics has been a central theme in mathematics due to its connection with different branches of Mathematics and Physics.",

test

[ 101, 1000, 4955, 1000, 1024, 1000, 1996, 2817, 1997, 20248, 6155, 6558, 2038, 2042, 1037, 2430, 4323, 1999, 5597, 2349, 2000, 2049, 4434, 2007, 2367, 5628, 1997, 5597, 1998, 5584, 1012, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"main_content": [

test

[ 101, 1000, 2364, 1035, 4180, 1000, 1024, 1031, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"Inspired by Riemann\u2019s explicit formula, Selberg [21] introduced a zeta function and established his trace formula. Further inspired by the connection of the zeros of the zeta function and prime numbers, Selberg [21] used his mighty trace formula to count the lengths of closed geodesics on a compact Rieman...

test

[ 101, 1000, 4427, 2011, 15544, 17545, 1032, 23343, 24096, 2683, 2015, 13216, 5675, 1010, 7367, 22927, 1031, 2538, 1033, 3107, 1037, 23870, 3853, 1998, 2511, 2010, 7637, 5675, 1012, 2582, 4427, 2011, 1996, 4434, 1997, 1996, 5717, 2015, 1997, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"One of the key characteristics of Selberg\u2019s trace formula is its unique insight into the relationship between the discrete spectrum of the hyperbolic Laplacian and the lengths of closed geodesics on a compact surface. Selberg later extended this formula to the non-compact case, incorporating both the disc...

test

[ 101, 1000, 2028, 1997, 1996, 3145, 6459, 1997, 7367, 22927, 1032, 23343, 24096, 2683, 2015, 7637, 5675, 2003, 2049, 4310, 12369, 2046, 1996, 3276, 2090, 1996, 16246, 8674, 1997, 1996, 23760, 18647, 5001, 2721, 14483, 1998, 1996, 10742, 1997...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"In 1977, Victor Guillemin wrote a paper ([6]) in which he discussed the asymptotic behavior of the scattering matrix arising from the automorphic wave equation, following the work of Lax and Phillips (see [13], [14]). He considered a finite-area hyperbolic surface with cusps given by X=\u210d/\u0393\ud835\udc4...

test

[ 101, 1000, 1999, 3355, 1010, 5125, 26458, 6216, 10020, 2626, 1037, 3259, 1006, 1031, 1020, 1033, 1007, 1999, 2029, 2002, 6936, 1996, 2004, 24335, 13876, 20214, 5248, 1997, 1996, 17501, 8185, 17707, 2013, 1996, 8285, 18078, 4400, 8522, 1010,...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"As part of his framework, Guillemin introduced the notion of a scattering geodesic on such a hyperbolic surface and showed that these geodesics spend only a finite amount of time (called the sojourn time) inside the compact core of the surface. One of the main results in [6] is a trace formula type theorem tha...

test

[ 101, 1000, 2004, 2112, 1997, 2010, 7705, 1010, 26458, 6216, 10020, 3107, 1996, 9366, 1997, 1037, 17501, 20248, 6155, 2594, 2006, 2107, 1037, 23760, 18647, 3302, 1998, 3662, 2008, 2122, 20248, 6155, 6558, 5247, 2069, 1037, 10713, 3815, 1997,...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"We now review the fundamental aspects of scattering geodesics from [6]. Let us assume that X\ud835\udc4bXitalic_X has n\ud835\udc5bnitalic_n number of cusps labeled with \u03ba1,\u03ba2,\u2026,\u03bansubscript\ud835\udf051subscript\ud835\udf052\u2026subscript\ud835\udf05\ud835\udc5b\\kappa_{1},\\kappa_{2},...,...

test

[ 101, 1000, 2057, 2085, 3319, 1996, 8050, 5919, 1997, 17501, 20248, 6155, 6558, 2013, 1031, 1020, 1033, 1012, 2292, 2149, 7868, 2008, 1060, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2549, 2497, 9048, 9080, 2594, 1035, 1060, 2038, 1050, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"The following observation has been made in [cf. Appendix B, Corollary 2 [6]]. \nA scattering geodesic has the property that for large\nnegative and positive times it corresponds to a vertical line in\nthe figure below (i.e. after we have mapped the appropriate cusp neighborhoods onto\nthe standard cusp neighbo...

test

[ 101, 1000, 1996, 2206, 8089, 2038, 2042, 2081, 1999, 1031, 12935, 1012, 22524, 1038, 1010, 2522, 28402, 5649, 1016, 1031, 1020, 1033, 1033, 1012, 1032, 6583, 17501, 20248, 6155, 2594, 2038, 1996, 3200, 2008, 2005, 2312, 1032, 1050, 2638, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"In [6, Theorem B1], Guillemin proved the following result.",

test

[ 101, 1000, 1999, 1031, 1020, 1010, 9872, 29491, 1033, 1010, 26458, 6216, 10020, 4928, 1996, 2206, 2765, 1012, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"There are countably many scattering geodesics that scatter between a given pair of cusps \u03baisubscript\ud835\udf05\ud835\udc56\\kappa_{i}italic_\u03ba start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and \u03bajsubscript\ud835\udf05\ud835\udc57\\kappa_{j}italic_\u03ba start_POSTSUBSCRIPT italic_j end_POSTSUBS...

test

[ 101, 1000, 2045, 2024, 4175, 8231, 2116, 17501, 20248, 6155, 6558, 2008, 8040, 20097, 2090, 1037, 2445, 3940, 1997, 12731, 13102, 2015, 1032, 1057, 2692, 2509, 26068, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2546, 2692, 262...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"In this paper, we prove a Prime number theorem kind result for Scattering geodesics. This is the first such result in the theory of scattering geodesics.\nMore explicitly, we prove the following theorem.",

test

[ 101, 1000, 1999, 2023, 3259, 1010, 2057, 6011, 1037, 3539, 2193, 9872, 2785, 2765, 2005, 17501, 20248, 6155, 6558, 1012, 2023, 2003, 1996, 2034, 2107, 2765, 1999, 1996, 3399, 1997, 17501, 20248, 6155, 6558, 1012, 1032, 13221, 5686, 12045, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"For Y\u226b0much-greater-than\ud835\udc4c0Y\\gg 0italic_Y \u226b 0, let \u03a0\u2062(Y)\u03a0\ud835\udc4c\\Pi(Y)roman_\u03a0 ( italic_Y ) be the number of distinct scattering geodesics in \u2133:=\u210d/PSL\u2062(2,\u2124)assign\u2133\u210dPSL2\u2124\\mathcal{M}:=\\mathbb{H}/\\text{PSL}(2,\\mathbb{Z})caligraph...

test

[ 101, 1000, 2005, 1061, 1032, 23343, 23833, 2497, 2692, 12274, 2818, 1011, 3618, 1011, 2084, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2549, 2278, 2692, 2100, 1032, 1032, 1043, 2290, 1014, 18400, 2594, 1035, 1061, 1032, 23343, 23833, 2497...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"The problem of counting closed geodesics in compact manifolds bears a remarkable resemblance to counting prime numbers (see [5] for more details). In our case, the counting of scattering geodesics on the modular surface is closely linked to the study of positive integers whose prime divisors lie in an arithmet...

test

[ 101, 1000, 1996, 3291, 1997, 10320, 2701, 20248, 6155, 6558, 1999, 9233, 19726, 2015, 6468, 1037, 9487, 14062, 2000, 10320, 3539, 3616, 1006, 2156, 1031, 1019, 1033, 2005, 2062, 4751, 1007, 1012, 1999, 2256, 2553, 1010, 1996, 10320, 1997, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Let \u210d={z=x+i\u2062y\u2223x,y\u2208\u211d,y>0}\u210dconditional-set\ud835\udc67\ud835\udc65\ud835\udc56\ud835\udc66formulae-sequence\ud835\udc65\ud835\udc66\u211d\ud835\udc660\\mathbb{H}=\\{z=x+iy\\mid x,y\\in\\mathbb{R},y>0\\}blackboard_H = { italic_z = italic_x + italic_i italic_y \u2223 italic_x , itali...

test

[ 101, 1000, 2292, 1032, 23343, 10790, 2094, 1027, 1063, 1062, 1027, 1060, 1009, 1045, 1032, 23343, 2692, 2575, 2475, 2100, 1032, 23343, 19317, 2509, 2595, 1010, 1061, 1032, 23343, 11387, 2620, 1032, 23343, 14526, 2094, 1010, 1061, 1028, 1014...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"The modular surface\nis the quotient space \u2133=\u210d/PSL\u2062(2,\u2124)\u2133\u210dPSL2\u2124\\mathcal{M}=\\mathbb{H}/\\text{PSL}(2,\\mathbb{Z})caligraphic_M = blackboard_H / PSL ( 2 , blackboard_Z ), which is equipped with the metric induced from the upper half-plane. We know that the modular surface\nha...

test

[ 101, 1000, 1996, 19160, 3302, 1032, 9152, 2015, 1996, 22035, 9515, 3372, 2686, 1032, 23343, 17134, 2509, 1027, 1032, 23343, 10790, 2094, 1013, 8827, 2140, 1032, 23343, 2692, 2575, 2475, 1006, 1016, 1010, 1032, 23343, 12521, 2549, 1007, 1032...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"(1.3)\n\n\u2133cT0=\u03c0\u2062(\u2131\u2229{\u2111\u2061(z)\u2264T0}),\u2133\u221eT0=\u03c0\u2062(\u2131\u2229{\u2111\u2061(z)>T0}).formulae-sequencesuperscriptsubscript\u2133\ud835\udc50subscript\ud835\udc470\ud835\udf0b\u2131\ud835\udc67subscript\ud835\udc470superscriptsubscript\u2133subscript\ud835\udc470\...

test

[ 101, 1000, 1006, 1015, 1012, 1017, 1007, 1032, 1050, 1032, 1050, 1032, 23343, 17134, 2509, 6593, 2692, 1027, 1032, 1057, 2692, 2509, 2278, 2692, 1032, 23343, 2692, 2575, 2475, 1006, 1032, 23343, 17134, 2487, 1032, 23343, 19317, 2683, 1063, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"We are interested in a class of scattering geodesic in \u2133\u2133\\mathcal{M}caligraphic_M that runs between the cusp, which was first discussed by Victor Guillemin\nin [6].",

test

[ 101, 1000, 2057, 2024, 4699, 1999, 1037, 2465, 1997, 17501, 20248, 6155, 2594, 1999, 1032, 23343, 17134, 2509, 1032, 23343, 17134, 2509, 1032, 1032, 8785, 9289, 1063, 1049, 1065, 10250, 8004, 20721, 1035, 1049, 2008, 3216, 2090, 1996, 12731...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"We recall the following definition of scattering geodesics from [6];",

test

[ 101, 1000, 2057, 9131, 1996, 2206, 6210, 1997, 17501, 20248, 6155, 6558, 2013, 1031, 1020, 1033, 1025, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"A geodesic \u03b3\u2062(t)\ud835\udefe\ud835\udc61\\gamma(t)italic_\u03b3 ( italic_t ) in \u2133\u2133\\mathcal{M}caligraphic_M is called a scattering geodesic if it is contained in \u2133\u2216\u2133cT0\u2133superscriptsubscript\u2133\ud835\udc50subscript\ud835\udc470\\mathcal{M}\\setminus\\mathcal{M}_{c}^{T_...

test

[ 101, 1000, 1037, 20248, 6155, 2594, 1032, 1057, 2692, 2509, 2497, 2509, 1032, 23343, 2692, 2575, 2475, 1006, 1056, 1007, 1032, 20904, 2620, 19481, 1032, 20904, 27235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2575, 2487, 1032, 1032, 1309...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Next, we show that the scattering geodesics are in one-to-one correspondence with a subset of rationals in [0,1)01[0,1)[ 0 , 1 ).",

test

[ 101, 1000, 2279, 1010, 2057, 2265, 2008, 1996, 17501, 20248, 6155, 6558, 2024, 1999, 2028, 1011, 2000, 1011, 2028, 11061, 2007, 1037, 16745, 1997, 11581, 2015, 1999, 1031, 1014, 1010, 1015, 1007, 5890, 1031, 1014, 1010, 1015, 1007, 1031, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"We start with the following result;",

test

[ 101, 1000, 2057, 2707, 2007, 1996, 2206, 2765, 1025, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"For w\u2208[0,1)\u2229\u211a\ud835\udc6401\u211aw\\in[0,1)\\cap\\mathbb{Q}italic_w \u2208 [ 0 , 1 ) \u2229 blackboard_Q, let \u03b3\u00afwsubscript\u00af\ud835\udefe\ud835\udc64\\bar{\\gamma}_{w}over\u00af start_ARG italic_\u03b3 end_ARG start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT be the unique geodesic in ...

test

[ 101, 1000, 2005, 1059, 1032, 23343, 11387, 2620, 1031, 1014, 1010, 1015, 1007, 1032, 23343, 19317, 2683, 1032, 23343, 14526, 2050, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 21084, 24096, 1032, 23343, 14526, 10376, 1032, 1032, 1999, 1031, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"q1=q2=qsubscript\ud835\udc5e1subscript\ud835\udc5e2\ud835\udc5eq_{1}=q_{2}=qitalic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_q.",

test

[ 101, 1000, 1053, 2487, 1027, 1053, 2475, 1027, 1053, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2487, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2475, 1032, 20904, 2620, 1948...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"q\ud835\udc5eqitalic_q divides p1\u2062p2+1subscript\ud835\udc5d1subscript\ud835\udc5d21p_{1}p_{2}+1italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 1.",

test

[ 101, 1000, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 20487, 1052, 2487, 1032, 23343, 2692, 2575, 2475, 2361, 2475, 1009, 1015, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 227...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Fixing a T0subscript\ud835\udc470T_{0}italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT as before, it follows from [Corollary 2,Theorem B1, [6]] that there are a countable number of non-trivial scattering geodesics in \u2133\u2133\\mathcal{M}caligraphic_M. Our first goal is to show that the set of scattering ge...

test

[ 101, 1000, 15887, 1037, 1056, 16223, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 22610, 2692, 2102, 1035, 1063, 1014, 1065, 2009, 27072, 1035, 1056, 2707, 1035, 8466, 12083, 22483, 1014, 2203, 1035, 8466, 12083, 22483, 2004, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Let \ud835\udcae\ud835\udcae\\mathcal{S}caligraphic_S be the set of scattering geodesics in \u2133\u2133\\mathcal{M}caligraphic_M, then there is a one-to-one correspondence between \ud835\udcae\ud835\udcae\\mathcal{S}caligraphic_S and the set \ud835\udca2\ud835\udca2\\mathcal{G}caligraphic_G defined in (1.8), ...

test

[ 101, 1000, 2292, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2063, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2063, 1032, 1032, 8785, 9289, 1063, 1055, 1065, 10250, 8004, 20721, 1035, 1055, 2022, 1996, 2275, 1997, 17501, 20248, 6155, 655...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Next, we begin the construction of this set, which we denote by \ud835\udca2\ud835\udca2\\mathcal{G}caligraphic_G.",

test

[ 101, 1000, 2279, 1010, 2057, 4088, 1996, 2810, 1997, 2023, 2275, 1010, 2029, 2057, 19090, 2011, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 1032, 8785, 9289, 1063, 1043, 1065, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"For q\u22652\ud835\udc5e2q\\geq 2italic_q \u2265 2, consider the set\n\n\n\nIq:={p\u2208\u2124+\u22231\u2264p<q,gcd\u2061(p,q)=1}.assignsubscript\ud835\udc3c\ud835\udc5econditional-set\ud835\udc5dsuperscript\u2124formulae-sequence1\ud835\udc5d\ud835\udc5e\ud835\udc5d\ud835\udc5e1I_{q}:=\\{p\\in\\mathbb{Z}^{+}\...

test

[ 101, 1000, 2005, 1053, 1032, 23343, 23833, 25746, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2475, 4160, 1032, 1032, 16216, 4160, 1016, 18400, 2594, 1035, 1053, 1032, 23343, 23833, 2629, 1016, 1010, 5136, 1996, 2275, 1032, 105...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"(1.4)\n\nSq:={p\u2208\u2124+\u22231\u2264p<q,p2\u2261\u22121(modq)}.assignsubscript\ud835\udc46\ud835\udc5econditional-set\ud835\udc5dsuperscript\u2124formulae-sequence1\ud835\udc5d\ud835\udc5esuperscript\ud835\udc5d2annotated1pmod\ud835\udc5eS_{q}:=\\{p\\in\\mathbb{Z}^{+}\\mid 1\\leq p<q,\\,p^{2}\\equiv-1\\pm...

test

[ 101, 1000, 1006, 1015, 1012, 1018, 1007, 1032, 1050, 1032, 24978, 4160, 1024, 1027, 1063, 1052, 1032, 23343, 11387, 2620, 1032, 23343, 12521, 2549, 1009, 1032, 23343, 19317, 21486, 1032, 23343, 23833, 2549, 2361, 1026, 1053, 1010, 1052, 247...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"For a given p\u2208Iq\ud835\udc5dsubscript\ud835\udc3c\ud835\udc5ep\\in I_{q}italic_p \u2208 italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, we now examine equation p\u2062p\u2032\u2261\u22121(modq)\ud835\udc5dsuperscript\ud835\udc5d\u2032annotated1pmod\ud835\udc5epp^{\\prime}\\equiv-1\\pmod{q}italic_...

test

[ 101, 1000, 2005, 1037, 2445, 1052, 1032, 23343, 11387, 2620, 18515, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2509, 2278, 1032, 20904, 2620, 19481, 1032, 20904, 2278, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Now, there are two cases to consider:\n\n\n(i)\n\np=yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep=y_{p,q}italic_p = italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT,\n\n\n\n(ii)\n\np\u2260yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep\\neq y_{p,q}italic_p \u22...

test

[ 101, 1000, 2085, 1010, 2045, 2024, 2048, 3572, 2000, 5136, 1024, 1032, 1050, 1032, 1050, 1032, 1050, 1006, 1045, 1007, 1032, 1050, 1032, 27937, 1027, 1061, 2361, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"p=yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep=y_{p,q}italic_p = italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT,",

test

[ 101, 1000, 1052, 1027, 1061, 2361, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, 1032, 20904, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"p\u2260yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep\\neq y_{p,q}italic_p \u2260 italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT.",

test

[ 101, 1000, 1052, 1032, 23343, 23833, 2692, 22571, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"If p=yp,q\ud835\udc5dsubscript\ud835\udc66\ud835\udc5d\ud835\udc5ep=y_{p,q}italic_p = italic_y start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT, then p\u2208Sq\ud835\udc5dsubscript\ud835\udc46\ud835\udc5ep\\in S_{q}italic_p \u2208 italic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, where Sqsubscr...

test

[ 101, 1000, 2065, 1052, 1027, 1061, 2361, 1010, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 5104, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, 1032, 2...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Finally, we have the partition Iq=Cq\u2294Sqsubscript\ud835\udc3c\ud835\udc5esquare-unionsubscript\ud835\udc36\ud835\udc5esubscript\ud835\udc46\ud835\udc5eI_{q}=C_{q}\\sqcup S_{q}italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT \u2294 italic_S st...

test

[ 101, 1000, 2633, 1010, 2057, 2031, 1996, 13571, 26264, 1027, 1039, 4160, 1032, 23343, 24594, 2549, 2015, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2509, 2278, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 22...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Let \ud835\udca21={0}subscript\ud835\udca210\\mathcal{G}_{1}=\\{0\\}caligraphic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { 0 } and for q\u22652,q\u2208\u2124+formulae-sequence\ud835\udc5e2\ud835\udc5esuperscript\u2124q\\geq 2,q\\in\\mathbb{Z}^{+}italic_q \u2265 2 , italic_q \u2208 blackboard_Z start_POSTSUP...

test

[ 101, 1000, 2292, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 17465, 1027, 1063, 1014, 1065, 4942, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 17465, 2692, 1032, 1032, 8785, 9289, 1063, 1043, 1065, 1035, 1063, 1015, 1065, 1027, 1032...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"The set \ud835\udca2qsubscript\ud835\udca2\ud835\udc5e\\mathcal{G}_{q}caligraphic_G start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT is ordered in an obvious way as a subset of rational numbers with standard ordering.",

test

[ 101, 1000, 1996, 2275, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 1032, 1032, 8785, 9289, 1063, 104...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Let \ud835\udca2\u2282[0,1)\ud835\udca201\\mathcal{G}\\subset[0,1)caligraphic_G \u2282 [ 0 , 1 ) be the following subset,\n\n\n(1.8)\n\n\ud835\udca2:=\u2a06q=1\u221e\ud835\udca2q.assign\ud835\udca2superscriptsubscriptsquare-union\ud835\udc5e1subscript\ud835\udca2\ud835\udc5e\\mathcal{G}:=\\displaystyle\\bigsqc...

test

[ 101, 1000, 2292, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 23343, 22407, 2475, 1031, 1014, 1010, 1015, 1007, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 11387, 2487, 1032, 1032, 8785, 9289, 1063, 1043, 1065, 1032, 1032, 1674...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"We order the elements of \ud835\udca2\ud835\udca2\\mathcal{G}caligraphic_G as follows, if x\u2208\ud835\udca2q1,y\u2208\ud835\udca2q2formulae-sequence\ud835\udc65subscript\ud835\udca2subscript\ud835\udc5e1\ud835\udc66subscript\ud835\udca2subscript\ud835\udc5e2x\\in\\mathcal{G}_{q_{1}},y\\in\\mathcal{G}_{q_{2}}...

test

[ 101, 1000, 2057, 2344, 1996, 3787, 1997, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 1032, 8785, 9289, 1063, 1043, 1065, 10250, 8004, 20721, 1035, 1043, 2004, 4076, 1010, 2065,...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"It is clear from the definition that the cardinality of \ud835\udca2qsubscript\ud835\udca2\ud835\udc5e\\mathcal{G}_{q}caligraphic_G start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT is the sum of the cardinalities of the sets Cq\u2217subscriptsuperscript\ud835\udc36\ud835\udc5eC^{*}_{q}italic_C start_POSTSUPERSCR...

test

[ 101, 1000, 2009, 2003, 3154, 2013, 1996, 6210, 2008, 1996, 7185, 3012, 1997, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 3540, 2475, 1032, 20904, 2620, 19481, 1032, 2090...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Let nqsubscript\ud835\udc5b\ud835\udc5en_{q}italic_n start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT denote the number of elements in \ud835\udca2qsubscript\ud835\udca2\ud835\udc5e\\mathcal{G}_{q}caligraphic_G start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. Then for q\u22652\ud835\udc5e2q\\geq 2italic_q \u2265 ...

test

[ 101, 1000, 2292, 1050, 4160, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2497, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2368, 1035, 1063, 1053, 1065, 2009, 27072, 1035, 1050, 2707, 1035, 8466, 12083, 2248...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"In the next section, we discuss preliminaries results. In the final section, we prove the main results of the paper.",

test

[ 101, 1000, 1999, 1996, 2279, 2930, 1010, 2057, 6848, 3653, 17960, 3981, 5134, 3463, 1012, 1999, 1996, 2345, 2930, 1010, 2057, 6011, 1996, 2364, 3463, 1997, 1996, 3259, 1012, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"In this section, we discuss the results required to prove the main theorems.",

test

[ 101, 1000, 1999, 2023, 2930, 1010, 2057, 6848, 1996, 3463, 3223, 2000, 6011, 1996, 2364, 9872, 2015, 1012, 1000, 1010, 102 ]

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"Recall that for q\u22652\ud835\udc5e2q\\geq 2italic_q \u2265 2,",

test

[ 101, 1000, 9131, 2008, 2005, 1053, 1032, 23343, 23833, 25746, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 2475, 4160, 1032, 1032, 16216, 4160, 1016, 18400, 2594, 1035, 1053, 1032, 23343, 23833, 2629, 1016, 1010, 1000, 1010, 102...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

"(2.1)\n\nSq={p\u2208\u2124+\u22231\u2264p<q\u2062\u00a0and\u00a0\u2062p2\u2261\u22121(modq)}subscript\ud835\udc46\ud835\udc5econditional-set\ud835\udc5dsuperscript\u21241\ud835\udc5d\ud835\udc5e\u00a0and\u00a0superscript\ud835\udc5d2annotated1pmod\ud835\udc5eS_{q}=\\{p\\in\\mathbb{Z}^{+}\\mid 1\\leq p<q\\hskip...

test

[ 101, 1000, 1006, 1016, 1012, 1015, 1007, 1032, 1050, 1032, 24978, 4160, 1027, 1063, 1052, 1032, 23343, 11387, 2620, 1032, 23343, 12521, 2549, 1009, 1032, 23343, 19317, 21486, 1032, 23343, 23833, 2549, 2361, 1026, 1053, 1032, 23343, 2692, 25...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"and sqsubscript\ud835\udc60\ud835\udc5es_{q}italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT denotes the cardinality of Sqsubscript\ud835\udc46\ud835\udc5eS_{q}italic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. We set s1=1subscript\ud835\udc6011s_{1}=1italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRI...

test

[ 101, 1000, 1998, 5490, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 16086, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2229, 1035, 1063, 1053, 1065, 2009, 27072, 1035, 1055, 2707, 1035, 8466, 12083, 22483, 2009, 27...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"If q\ud835\udc5eqitalic_q is a prime of the form 4\u2062k+34\ud835\udc5834k+34 italic_k + 3, then sq=0subscript\ud835\udc60\ud835\udc5e0s_{q}=0italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = 0.",

test

[ 101, 1000, 2065, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 2003, 1037, 3539, 1997, 1996, 2433, 1018, 1032, 23343, 2692, 2575, 2475, 2243, 1009, 4090, 1032, 20904, 2620, 19481, 1032, 20904,...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"This follows from the fact the value of the Legendre symbol (\u22121p)1\ud835\udc5d\\left(\\frac{-1}{p}\\right)( divide start_ARG - 1 end_ARG start_ARG italic_p end_ARG ) is \u221211-1- 1 for a prime p\ud835\udc5dpitalic_p of the form 4\u2062k+34\ud835\udc5834k+34 italic_k + 3.\n\u220e",

test

[ 101, 1000, 2023, 4076, 2013, 1996, 2755, 1996, 3643, 1997, 1996, 5722, 2890, 6454, 1006, 1032, 23343, 17465, 17465, 2361, 1007, 1015, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2094, 1032, 1032, 2187, 1006, 1032, 1032, 25312, 2278, ...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"If q\ud835\udc5eqitalic_q is a prime of the form 4\u2062k+34\ud835\udc5834k+34 italic_k + 3, and m=qk\ud835\udc5asuperscript\ud835\udc5e\ud835\udc58m=q^{k}italic_m = italic_q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT with k\u22652\ud835\udc582k\\geq 2italic_k \u2265 2, then\nsm=0subscript\ud835\udc60\...

test

[ 101, 1000, 2065, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 2003, 1037, 3539, 1997, 1996, 2433, 1018, 1032, 23343, 2692, 2575, 2475, 2243, 1009, 4090, 1032, 20904, 2620, 19481, 1032, 20904,...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Suppose we have a solution y\ud835\udc66yitalic_y that satisfies y2\u2261\u22121(modm)superscript\ud835\udc662annotated1pmod\ud835\udc5ay^{2}\\equiv-1\\pmod{m}italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT \u2261 - 1 start_MODIFIER ( roman_mod start_ARG italic_m end_ARG ) end_MODIFIER. This implies y2\u2...

test

[ 101, 1000, 6814, 2057, 2031, 1037, 5576, 1061, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 28756, 10139, 9080, 2594, 1035, 1061, 2008, 2938, 2483, 14213, 1061, 2475, 1032, 23343, 23833, 2487, 1032, 23343, 17465, 17465, 1006, 16913, 2213, 1...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Suppose q\ud835\udc5eqitalic_q is a prime of the form 4\u2062k+14\ud835\udc5814k+14 italic_k + 1. Then for any m\u22651\ud835\udc5a1m\\geq 1italic_m \u2265 1, the equation p2\u2261\u22121(modqm)superscript\ud835\udc5d2annotated1\ud835\udc5d\ud835\udc5a\ud835\udc5c\ud835\udc51superscript\ud835\udc5e\ud835\udc5a...

test

[ 101, 1000, 6814, 1053, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 14702, 9080, 2594, 1035, 1053, 2003, 1037, 3539, 1997, 1996, 2433, 1018, 1032, 23343, 2692, 2575, 2475, 2243, 1009, 2403, 1032, 20904, 2620, 19481, 1032, 20904,...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Note that if qmsuperscript\ud835\udc5e\ud835\udc5aq^{m}italic_q start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT divides p2+1superscript\ud835\udc5d21p^{2}+1italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1, then since q\ud835\udc5eqitalic_q is a prime, we must have q\ud835\udc5eqitalic_q dividing p2+...

test

[ 101, 1000, 3602, 2008, 2065, 1053, 5244, 6279, 2545, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 2063, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 2629, 20784, 1034, 1063, 1049, 1065, 2009, 27072, 1035, 1053, 2707, 1035, 8466...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"We perform this lifting inductively. Suppose we start with a solution x0subscript\ud835\udc650x_{0}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT modulo qjsuperscript\ud835\udc5e\ud835\udc57q^{j}italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT satisfying x02\u2261\u22121(modqj)superscriptsubscript...

test

[ 101, 1000, 2057, 4685, 2023, 8783, 27427, 14194, 25499, 1012, 6814, 2057, 2707, 2007, 1037, 5576, 1060, 16223, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 26187, 2692, 2595, 1035, 1063, 1014, 1065, 2009, 27072, 1035, 1060, 27...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"We begin by substituting the expression x1=x0+qj\u2062y0subscript\ud835\udc651subscript\ud835\udc650superscript\ud835\udc5e\ud835\udc57subscript\ud835\udc660x_{1}=x_{0}+q^{j}y_{0}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_q start_POSTSUPERSCRIPT...

test

[ 101, 1000, 2057, 4088, 2011, 4942, 21532, 1996, 3670, 1060, 2487, 1027, 1060, 2692, 1009, 1053, 3501, 1032, 23343, 2692, 2575, 2475, 2100, 16223, 12083, 22483, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 26187, 2487, 6342, 5910, 23235, 103...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...

"Since gcd\u2061(2\u2062x0,q)=12subscript\ud835\udc650\ud835\udc5e1\\gcd(2x_{0},q)=1roman_gcd ( 2 italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q ) = 1, the above equation yields a unique solution y0subscript\ud835\udc660y_{0}italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT modulo q\ud835\udc5eqital...

test

[ 101, 1000, 2144, 1043, 19797, 1032, 23343, 2692, 2575, 2487, 1006, 1016, 1032, 23343, 2692, 2575, 2475, 2595, 2692, 1010, 1053, 1007, 1027, 2260, 6342, 5910, 23235, 1032, 20904, 2620, 19481, 1032, 20904, 2278, 26187, 2692, 1032, 20904, 2620...

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...