Chebyshev's Bias is a phenomenon which notes that the number of primes whose remainder is three when divided by four exceeds the number of primes whose remainder is one after such a division. This can be easily visualized with the following simple graph.

Chebyshev's Bias to n=14. Credit: MakeTheBrainHappy

The x-axis scale denotes the list of the first n primes. So the number 8 in the example would take into account the first eight primes (2, 3, 5, 7, 11, 13, 17, 19). The y-axis measures the difference between the number of primes in this list with remainders of 3 (after dividing each prime by four) and those with remainders if one. For this example, it means that four out of the eight primes have a remainder of three and three out of the eight primes have a remainder of one. 2 is the only number that is not accounted for by this bias because it's remainder when divided by four is neither 1 or 3.





Each time the 4k+1 primes (primes with remainder one when divided by four) equals to number of 4k+3 (primes with remainder three when divided by four). This happens fairly often when the window is this small but it is still evident that there is a skew towards the 4k+3 primes.

Chebyshev's Bias to n=100,000. Credit: MakeTheBrainHappy

Next Question: Are there any sets of primes at which the number of 4k+1 primes is greater than the number of 4k+3 primes? Yes, but the first one is at the set of the first 2946 primes. The graph above shows that the ratio becomes more skewed as larger sets of primes are considered. We can postulate that their are three points on the graph where the 4k+1 primes are either equal or take the lead (near 0 (i.e. the first graph), near 3000, and around 50000).





Here the bias was examined up to the first 100,000,000 primes with some intermediate graphs and new terms. Note: The number of crossings around a particular point increases as the terms increase.





Chebyshev's Bias to n=1,000,000. Credit: MakeTheBrainHappy

Chebyshev's Bias to n=10,000,000. Credit: MakeTheBrainHappy

Chebyshev's Bias to n=100,000,000. Credit: MakeTheBrainHappy

Sets of primes (number n on the x-axis) where the number of 4k+3 primes equals the number of 4k+1 primes: [1, 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, 50375, 50377, 50379, 50381, 50393, 50413, 50423, 50425, 50427, 50429, 50431, 50433, 50435, 50437, 50439, 50445, 50449, 50451, 50503, 50507, 50515, 50517, 50821, 50843, 50853, 50855, 50857, 50859, 50861, 50865, 50893, 50899, 50901, 50903, 50905, 50907, 50909, 50911, 50913, 50915, 50917, 50919, 50921, 50927, 50929, 51119, 51121, 51123, 51127, 51151, 51155, 51157, 51159, 51161, 51163, 51177, 51185, 51187, 51189, 51195, 51227, 51261, 51263, 51285, 51287, 51289, 51291, 51293, 51297, 51299, 51319, 51321, 51389, 51391, 51395, 51397, 51505, 51535, 51537, 51543, 51547, 51551, 51553, 51557, 51559, 51567, 51573, 51575, 51577, 51595, 51599, 51607, 51609, 51611, 51615, 51617, 51619, 51621, 51623, 51627, 806807, 806809, 806817, 806819, 806839, 806841, 806935, 806937, 806963, 806965, 807049, 807051, 807053, 807059, 807071, 807147, 807149, 807151, 807153, 807155, 807157, 807167, 807169, 807181, 807183, 807185, 807199, 807203, 807205, 807209, 807211, 807231, 807233, 807235, 807237, 807239, 807241, 807243, 807245, 807251, 807283, 807285, 807327, 807329, 807331, 807339, 807341, 807353, 807493, 807495, 807501, 807505, 807519, 807573, 807575, 807577, 807579, 807581, 807671, 807673, 807675, 807677, 807679, 807691, 807699, 807701, 807709, 807757, 807759, 807761, 807763, 807765, 807767, 807819, 807821, 807829, 807831, 807833, 807835, 807853, 807855, 807891, 807895, 807897, 807899, 807921, 807927, 807929, 807957, 807963, 807965, 807977, 807979, 807981, 807983, 807985, 807987, 807989, 807991, 807999, 808001, 808007, 808009, 808139, 808141, 808145, 808147, 808157, 808163, 808167, 808169, 808171, 808173, 808175, 808179, 808181, 808183, 808187, 808189, 808285, 808291, 808463, 808465, 808477, 808479, 808481, 808577, 808579, 808587, 808589, 808591, 808593, 808595, 809473, 809479, 809487, 809569, 810139, 810141, 810155, 810157, 810167, 810169, 810171, 810197, 810201, 810203, 810211, 810231, 811209, 811211, 811215, 811223, 811401, 811413, 811415, 811421, 811423, 811425, 811427, 811519, 811523, 811527, 811529, 811537, 811549, 811551, 811557, 811559, 811561, 814061, 814069, 814089, 48517583, 48517585, 48517591, 48517601, 48517603, 48517715, 48517717, 48517719, 48517723, 48517737, 48517739, 48517743, 48517745, 48517747, 48517751, 48517761, 48517765, 48517767, 48517769, 48517773, 48517775, 48517777, 48517783, 48517785, 48517823, 48517825, 48517827, 48517857, 48517861, 48517863, 48517865, 48517867, 48517871, 48517873, 48517879, 48517883, 48517885, 48517887, 48517889, 48517891, 48517903, 48517915, 48517917, 48517921, 48517927, 48517933, 48517935, 48517937, 48517939, 48517941, 48517943, 48517945, 48517947, 48517949, 48517967, 48517975, 48517977, 48518027, 48518029, 48518031, 48518033, 48518055, 48518103, 48518105, 48518107, 48518145, 48518149, 48518151, 48518153, 48518155, 48518163, 48518171, 48518183, 48518185, 48518187, 48518189, 48518191, 48518193, 48518273, 48518275, 48518277, 48518279, 48518281, 48518283, 48518285, 48518541, 48518543, 48518545, 48518577, 48518579, 48518581, 48518583, 48518585, 48518587, 48518589, 48518613, 48518615, 48518617, 48518663, 48518665, 48518667, 48518673, 48518675, 48518677, 48518687, 48518689, 48518691, 48518693, 48518713, 48518715, 48518717, 48518719, 48518721, 48518727, 48518731, 48518733, 48518737, 48518741, 48518743, 48518749, 48518755, 48518757, 48518759, 48518773, 48518775, 48518779, 48518787, 48518789, 48518795, 48518835, 48518837, 48518839, 48518841, 48518889, 48518891, 48518893, 48518929, 48518931, 48518933, 48518939, 48518941, 48518943, 48518963, 48520127, 48520171, 48520177, 48520239, 48520251, 48520253, 48520301, 48520303, 48520325, 48520347, 48520349, 48520353, 48520355, 48520357, 48520527, 48520567, 48520569, 48520573, 48520575, 48520577, 48520579, 48520593, 48520601, 48520607, 48520615, 48522631, 48522633, 48522635, 48522647, 48522651, 48522653, 48522741, 48522753, 48522755, 48522757, 48522761, 48522763, 48522765, 48522767, 48522769, 48522771, 48522773, 48522785, 48522787, 48522789, 48522791, 48522793, 48522795, 48522801, 48522881, 48522889, 48522895, 48522897, 48522899, 48522901, 48522903, 48522907, 48522941, 48522963, 48522965, 48522973, 48522975, 48522989, 48522995, 48522997, 48522999, 48523001, 48523003, 48523009, 48523013, 48523029, 48523031, 48523035, 48523037, 48523039, 48523041, 48523323, 48523325, 48523327, 48523329, 48523331, 48523341, 48523353, 48523417, 48523419, 48523425, 48523427, 48523429, 48523439, 48523441, 48523443, 48523445, 48523471, 48523501, 48523517, 48523523, 48523525, 48523545, 48523547, 48523549, 48523555, 48523557, 48523563, 48523569, 48523809, 48523811, 48523813, 48523835, 48523837, 48523875, 48523895, 48523897, 48523899, 48523901, 48523913, 48523915, 48523921, 48523923, 48523927, 48523931, 48523933, 48524287, 48524293, 48524295, 48524303, 48524305, 48524307, 48524309, 48524311, 48524313, 48524315, 48524317, 48524319, 48524321, 48524323, 48524325, 48524327, 48524331, 48524333, 48524343, 48524345, 48524347, 48524349, 48524351, 48524353, 48524375, 48524377, 48524425, 48524427, 48524429, 48524431, 48524435, 48524437, 48524439, 48524447, 48524457, 48524459, 48524461, 48524463, 48524465, 48524467, 48524469, 48524471, 48524611, 48524761, 48524763, 48524769, 48525159, 48525161, 48537809, 48537811, 48537813, 48537815, 48537817, 48537823, 48537837, 48537839, 48537841, 48537899, 48537951, 48537953, 48537981, 48537983, 48538065, 48538075, 48538077, 48538079, 48538081, 48538083, 48538085, 48538087, 48538091, 48538093, 48538103, 48538105, 48538107, 48538125, 48538145, 48538147, 48538149, 48538151, 48538159, 48538161, 48538163, 48538165, 48538169, 48538173, 48538197, 48538199, 48538207, 48538211, 48538213, 48538217, 48538221, 48538543, 48538555, 48538603, 48538605, 48538607, 48538625, 48538655, 48538931, 48538933, 48538945, 48538949, 48538951, 48538953, 48538955, 48538957, 48538959, 48538969, 48538971]





Sets of primes (number n on the x-axis) where the number of 4k+3 primes is less than the number of 4k+1 primes: [2946, 50378, 50380, 50382, 50383, 50384, 50385, 50386, 50387, 50388, 50389, 50390, 50391, 50392, 50414, 50415, 50416, 50417, 50418, 50419, 50420, 50421, 50422, 50424, 50426, 50428, 50430, 50436, 50438, 50446, 50447, 50448, 50450, 50822, 50823, 50824, 50825, 50826, 50827, 50828, 50829, 50830, 50831, 50832, 50833, 50834, 50835, 50836, 50837, 50838, 50839, 50840, 50841, 50842, 50844, 50845, 50846, 50847, 50848, 50849, 50850, 50851, 50852, 50854, 50856, 50858, 50862, 50863, 50864, 50866, 50867, 50868, 50869, 50870, 50871, 50872, 50873, 50874, 50875, 50876, 50877, 50878, 50879, 50880, 50881, 50882, 50883, 50884, 50885, 50886, 50887, 50888, 50889, 50890, 50891, 50892, 50902, 50904, 50906, 50908, 50916, 50918, 50920, 50928, 51120, 51122, 51152, 51153, 51154, 51158, 51160, 51178, 51179, 51180, 51181, 51182, 51183, 51184, 51186, 51188, 51196, 51197, 51198, 51199, 51200, 51201, 51202, 51203, 51204, 51205, 51206, 51207, 51208, 51209, 51210, 51211, 51212, 51213, 51214, 51215, 51216, 51217, 51218, 51219, 51220, 51221, 51222, 51223, 51224, 51225, 51226, 51262, 51264, 51265, 51266, 51267, 51268, 51269, 51270, 51271, 51272, 51273, 51274, 51275, 51276, 51277, 51278, 51279, 51280, 51281, 51282, 51283, 51284, 51286, 51294, 51295, 51296, 51300, 51301, 51302, 51303, 51304, 51305, 51306, 51307, 51308, 51309, 51310, 51311, 51312, 51313, 51314, 51315, 51316, 51317, 51318, 51392, 51393, 51394, 51552, 51558, 51568, 51569, 51570, 51571, 51572, 51574, 51576, 51578, 51579, 51580, 51581, 51582, 51583, 51584, 51585, 51586, 51587, 51588, 51589, 51590, 51591, 51592, 51593, 51594, 51596, 51597, 51598, 51600, 51601, 51602, 51603, 51604, 51605, 51606, 51612, 51613, 51614, 51622, 806808, 806810, 806811, 806812, 806813, 806814, 806815, 806816, 806818, 806820, 806821, 806822, 806823, 806824, 806825, 806826, 806827, 806828, 806829, 806830, 806831, 806832, 806833, 806834, 806835, 806836, 806837, 806838, 806840, 806938, 806939, 806940, 806941, 806942, 806943, 806944, 806945, 806946, 806947, 806948, 806949, 806950, 806951, 806952, 806953, 806954, 806955, 806956, 806957, 806958, 806959, 806960, 806961, 806962, 806964, 807050, 807054, 807055, 807056, 807057, 807058, 807060, 807061, 807062, 807063, 807064, 807065, 807066, 807067, 807068, 807069, 807070, 807072, 807073, 807074, 807075, 807076, 807077, 807078, 807079, 807080, 807081, 807082, 807083, 807084, 807085, 807086, 807087, 807088, 807089, 807090, 807091, 807092, 807093, 807094, 807095, 807096, 807097, 807098, 807099, 807100, 807101, 807102, 807103, 807104, 807105, 807106, 807107, 807108, 807109, 807110, 807111, 807112, 807113, 807114, 807115, 807116, 807117, 807118, 807119, 807120, 807121, 807122, 807123, 807124, 807125, 807126, 807127, 807128, 807129, 807130, 807131, 807132, 807133, 807134, 807135, 807136, 807137, 807138, 807139, 807140, 807141, 807142, 807143, 807144, 807145, 807146, 807148, 807152, 807168, 807182, 807186, 807187, 807188, 807189, 807190, 807191, 807192, 807193, 807194, 807195, 807196, 807197, 807198, 807212, 807213, 807214, 807215, 807216, 807217, 807218, 807219, 807220, 807221, 807222, 807223, 807224, 807225, 807226, 807227, 807228, 807229, 807230, 807232, 807236, 807238, 807240, 807244, 807246, 807247, 807248, 807249, 807250, 807284, 807286, 807287, 807288, 807289, 807290, 807291, 807292, 807293, 807294, 807295, 807296, 807297, 807298, 807299, 807300, 807301, 807302, 807303, 807304, 807305, 807306, 807307, 807308, 807309, 807310, 807311, 807312, 807313, 807314, 807315, 807316, 807317, 807318, 807319, 807320, 807321, 807322, 807323, 807324, 807325, 807326, 807340, 807342, 807343, 807344, 807345, 807346, 807347, 807348, 807349, 807350, 807351, 807352, 807354, 807355, 807356, 807357, 807358, 807359, 807360, 807361, 807362, 807363, 807364, 807365, 807366, 807367, 807368, 807369, 807370, 807371, 807372, 807373, 807374, 807375, 807376, 807377, 807378, 807379, 807380, 807381, 807382, 807383, 807384, 807385, 807386, 807387, 807388, 807389, 807390, 807391, 807392, 807393, 807394, 807395, 807396, 807397, 807398, 807399, 807400, 807401, 807402, 807403, 807404, 807405, 807406, 807407, 807408, 807409, 807410, 807411, 807412, 807413, 807414, 807415, 807416, 807417, 807418, 807419, 807420, 807421, 807422, 807423, 807424, 807425, 807426, 807427, 807428, 807429, 807430, 807431, 807432, 807433, 807434, 807435, 807436, 807437, 807438, 807439, 807440, 807441, 807442, 807443, 807444, 807445, 807446, 807447, 807448, 807449, 807450, 807451, 807452, 807453, 807454, 807455, 807456, 807457, 807458, 807459, 807460, 807461, 807462, 807463, 807464, 807465, 807466, 807467, 807468, 807469, 807470, 807471, 807472, 807473, 807474, 807475, 807476, 807477, 807478, 807479, 807480, 807481, 807482, 807483, 807484, 807485, 807486, 807487, 807488, 807489, 807490, 807491, 807492, 807494, 807496, 807497, 807498, 807499, 807500, 807502, 807503, 807504, 807520, 807521, 807522, 807523, 807524, 807525, 807526, 807527, 807528, 807529, 807530, 807531, 807532, 807533, 807534, 807535, 807536, 807537, 807538, 807539, 807540, 807541, 807542, 807543, 807544, 807545, 807546, 807547, 807548, 807549, 807550, 807551, 807552, 807553, 807554, 807555, 807556, 807557, 807558, 807559, 807560, 807561, 807562, 807563, 807564, 807565, 807566, 807567, 807568, 807569, 807570, 807571, 807572, 807580, 807672, 807674, 807676, 807762, 807764, 807820, 807832, 807834, 807836, 807837, 807838, 807839, 807840, 807841, 807842, 807843, 807844, 807845, 807846, 807847, 807848, 807849, 807850, 807851, 807852, 807854, 807898, 807928, 807930, 807931, 807932, 807933, 807934, 807935, 807936, 807937, 807938, 807939, 807940, 807941, 807942, 807943, 807944, 807945, 807946, 807947, 807948, 807949, 807950, 807951, 807952, 807953, 807954, 807955, 807956, 807958, 807959, 807960, 807961, 807962, 807966, 807967, 807968, 807969, 807970, 807971, 807972, 807973, 807974, 807975, 807976, 807982, 807984, 808000, 808002, 808003, 808004, 808005, 808006, 808008, 808164, 808165, 808166, 808168, 808172, 808174, 808184, 808185, 808186, 808466, 808467, 808468, 808469, 808470, 808471, 808472, 808473, 808474, 808475, 808476, 808478, 808578, 808580, 808581, 808582, 808583, 808584, 808585, 808586, 808594, 809488, 809489, 809490, 809491, 809492, 809493, 809494, 809495, 809496, 809497, 809498, 809499, 809500, 809501, 809502, 809503, 809504, 809505, 809506, 809507, 809508, 809509, 809510, 809511, 809512, 809513, 809514, 809515, 809516, 809517, 809518, 809519, 809520, 809521, 809522, 809523, 809524, 809525, 809526, 809527, 809528, 809529, 809530, 809531, 809532, 809533, 809534, 809535, 809536, 809537, 809538, 809539, 809540, 809541, 809542, 809543, 809544, 809545, 809546, 809547, 809548, 809549, 809550, 809551, 809552, 809553, 809554, 809555, 809556, 809557, 809558, 809559, 809560, 809561, 809562, 809563, 809564, 809565, 809566, 809567, 809568, 810142, 810143, 810144, 810145, 810146, 810147, 810148, 810149, 810150, 810151, 810152, 810153, 810154, 810168, 810172, 810173, 810174, 810175, 810176, 810177, 810178, 810179, 810180, 810181, 810182, 810183, 810184, 810185, 810186, 810187, 810188, 810189, 810190, 810191, 810192, 810193, 810194, 810195, 810196, 810198, 810199, 810200, 810204, 810205, 810206, 810207, 810208, 810209, 810210, 810212, 810213, 810214, 810215, 810216, 810217, 810218, 810219, 810220, 810221, 810222, 810223, 810224, 810225, 810226, 810227, 810228, 810229, 810230, 810232, 810233, 810234, 810235, 810236, 810237, 810238, 810239, 810240, 810241, 810242, 810243, 810244, 810245, 810246, 810247, 810248, 810249, 810250, 810251, 810252, 810253, 810254, 810255, 810256, 810257, 810258, 810259, 810260, 810261, 810262, 810263, 810264, 810265, 810266, 810267, 810268, 810269, 810270, 810271, 810272, 810273, 810274, 810275, 810276, 810277, 810278, 810279, 810280, 810281, 810282, 810283, 810284, 810285, 810286, 810287, 810288, 810289, 810290, 810291, 810292, 810293, 810294, 810295, 810296, 810297, 810298, 810299, 810300, 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48523991, 48523992, 48523993, 48523994, 48523995, 48523996, 48523997, 48523998, 48523999, 48524000, 48524001, 48524002, 48524003, 48524004, 48524005, 48524006, 48524007, 48524008, 48524009, 48524010, 48524011, 48524012, 48524013, 48524014, 48524015, 48524016, 48524017, 48524018, 48524019, 48524020, 48524021, 48524022, 48524023, 48524024, 48524025, 48524026, 48524027, 48524028, 48524029, 48524030, 48524031, 48524032, 48524033, 48524034, 48524035, 48524036, 48524037, 48524038, 48524039, 48524040, 48524041, 48524042, 48524043, 48524044, 48524045, 48524046, 48524047, 48524048, 48524049, 48524050, 48524051, 48524052, 48524053, 48524054, 48524055, 48524056, 48524057, 48524058, 48524059, 48524060, 48524061, 48524062, 48524063, 48524064, 48524065, 48524066, 48524067, 48524068, 48524069, 48524070, 48524071, 48524072, 48524073, 48524074, 48524075, 48524076, 48524077, 48524078, 48524079, 48524080, 48524081, 48524082, 48524083, 48524084, 48524085, 48524086, 48524087, 48524088, 48524089, 48524090, 48524091, 48524092, 48524093, 48524094, 48524095, 48524096, 48524097, 48524098, 48524099, 48524100, 48524101, 48524102, 48524103, 48524104, 48524105, 48524106, 48524107, 48524108, 48524109, 48524110, 48524111, 48524112, 48524113, 48524114, 48524115, 48524116, 48524117, 48524118, 48524119, 48524120, 48524121, 48524122, 48524123, 48524124, 48524125, 48524126, 48524127, 48524128, 48524129, 48524130, 48524131, 48524132, 48524133, 48524134, 48524135, 48524136, 48524137, 48524138, 48524139, 48524140, 48524141, 48524142, 48524143, 48524144, 48524145, 48524146, 48524147, 48524148, 48524149, 48524150, 48524151, 48524152, 48524153, 48524154, 48524155, 48524156, 48524157, 48524158, 48524159, 48524160, 48524161, 48524162, 48524163, 48524164, 48524165, 48524166, 48524167, 48524168, 48524169, 48524170, 48524171, 48524172, 48524173, 48524174, 48524175, 48524176, 48524177, 48524178, 48524179, 48524180, 48524181, 48524182, 48524183, 48524184, 48524185, 48524186, 48524187, 48524188, 48524189, 48524190, 48524191, 48524192, 48524193, 48524194, 48524195, 48524196, 48524197, 48524198, 48524199, 48524200, 48524201, 48524202, 48524203, 48524204, 48524205, 48524206, 48524207, 48524208, 48524209, 48524210, 48524211, 48524212, 48524213, 48524214, 48524215, 48524216, 48524217, 48524218, 48524219, 48524220, 48524221, 48524222, 48524223, 48524224, 48524225, 48524226, 48524227, 48524228, 48524229, 48524230, 48524231, 48524232, 48524233, 48524234, 48524235, 48524236, 48524237, 48524238, 48524239, 48524240, 48524241, 48524242, 48524243, 48524244, 48524245, 48524246, 48524247, 48524248, 48524249, 48524250, 48524251, 48524252, 48524253, 48524254, 48524255, 48524256, 48524257, 48524258, 48524259, 48524260, 48524261, 48524262, 48524263, 48524264, 48524265, 48524266, 48524267, 48524268, 48524269, 48524270, 48524271, 48524272, 48524273, 48524274, 48524275, 48524276, 48524277, 48524278, 48524279, 48524280, 48524281, 48524282, 48524283, 48524284, 48524285, 48524286, 48524296, 48524297, 48524298, 48524299, 48524300, 48524301, 48524302, 48524304, 48524308, 48524310, 48524320, 48524326, 48524332, 48524334, 48524335, 48524336, 48524337, 48524338, 48524339, 48524340, 48524341, 48524342, 48524344, 48524348, 48524354, 48524355, 48524356, 48524357, 48524358, 48524359, 48524360, 48524361, 48524362, 48524363, 48524364, 48524365, 48524366, 48524367, 48524368, 48524369, 48524370, 48524371, 48524372, 48524373, 48524374, 48524430, 48524432, 48524433, 48524434, 48524436, 48524440, 48524441, 48524442, 48524443, 48524444, 48524445, 48524446, 48524458, 48524460, 48524462, 48524464, 48524466, 48524468, 48524470, 48524472, 48524473, 48524474, 48524475, 48524476, 48524477, 48524478, 48524479, 48524480, 48524481, 48524482, 48524483, 48524484, 48524485, 48524486, 48524487, 48524488, 48524489, 48524490, 48524491, 48524492, 48524493, 48524494, 48524495, 48524496, 48524497, 48524498, 48524499, 48524500, 48524501, 48524502, 48524503, 48524504, 48524505, 48524506, 48524507, 48524508, 48524509, 48524510, 48524511, 48524512, 48524513, 48524514, 48524515, 48524516, 48524517, 48524518, 48524519, 48524520, 48524521, 48524522, 48524523, 48524524, 48524525, 48524526, 48524527, 48524528, 48524529, 48524530, 48524531, 48524532, 48524533, 48524534, 48524535, 48524536, 48524537, 48524538, 48524539, 48524540, 48524541, 48524542, 48524543, 48524544, 48524545, 48524546, 48524547, 48524548, 48524549, 48524550, 48524551, 48524552, 48524553, 48524554, 48524555, 48524556, 48524557, 48524558, 48524559, 48524560, 48524561, 48524562, 48524563, 48524564, 48524565, 48524566, 48524567, 48524568, 48524569, 48524570, 48524571, 48524572, 48524573, 48524574, 48524575, 48524576, 48524577, 48524578, 48524579, 48524580, 48524581, 48524582, 48524583, 48524584, 48524585, 48524586, 48524587, 48524588, 48524589, 48524590, 48524591, 48524592, 48524593, 48524594, 48524595, 48524596, 48524597, 48524598, 48524599, 48524600, 48524601, 48524602, 48524603, 48524604, 48524605, 48524606, 48524607, 48524608, 48524609, 48524610, 48524612, 48524613, 48524614, 48524615, 48524616, 48524617, 48524618, 48524619, 48524620, 48524621, 48524622, 48524623, 48524624, 48524625, 48524626, 48524627, 48524628, 48524629, 48524630, 48524631, 48524632, 48524633, 48524634, 48524635, 48524636, 48524637, 48524638, 48524639, 48524640, 48524641, 48524642, 48524643, 48524644, 48524645, 48524646, 48524647, 48524648, 48524649, 48524650, 48524651, 48524652, 48524653, 48524654, 48524655, 48524656, 48524657, 48524658, 48524659, 48524660, 48524661, 48524662, 48524663, 48524664, 48524665, 48524666, 48524667, 48524668, 48524669, 48524670, 48524671, 48524672, 48524673, 48524674, 48524675, 48524676, 48524677, 48524678, 48524679, 48524680, 48524681, 48524682, 48524683, 48524684, 48524685, 48524686, 48524687, 48524688, 48524689, 48524690, 48524691, 48524692, 48524693, 48524694, 48524695, 48524696, 48524697, 48524698, 48524699, 48524700, 48524701, 48524702, 48524703, 48524704, 48524705, 48524706, 48524707, 48524708, 48524709, 48524710, 48524711, 48524712, 48524713, 48524714, 48524715, 48524716, 48524717, 48524718, 48524719, 48524720, 48524721, 48524722, 48524723, 48524724, 48524725, 48524726, 48524727, 48524728, 48524729, 48524730, 48524731, 48524732, 48524733, 48524734, 48524735, 48524736, 48524737, 48524738, 48524739, 48524740, 48524741, 48524742, 48524743, 48524744, 48524745, 48524746, 48524747, 48524748, 48524749, 48524750, 48524751, 48524752, 48524753, 48524754, 48524755, 48524756, 48524757, 48524758, 48524759, 48524760, 48524764, 48524765, 48524766, 48524767, 48524768, 48537810, 48537814, 48537824, 48537825, 48537826, 48537827, 48537828, 48537829, 48537830, 48537831, 48537832, 48537833, 48537834, 48537835, 48537836, 48537838, 48537952, 48537954, 48537955, 48537956, 48537957, 48537958, 48537959, 48537960, 48537961, 48537962, 48537963, 48537964, 48537965, 48537966, 48537967, 48537968, 48537969, 48537970, 48537971, 48537972, 48537973, 48537974, 48537975, 48537976, 48537977, 48537978, 48537979, 48537980, 48538078, 48538082, 48538086, 48538088, 48538089, 48538090, 48538092, 48538104, 48538126, 48538127, 48538128, 48538129, 48538130, 48538131, 48538132, 48538133, 48538134, 48538135, 48538136, 48538137, 48538138, 48538139, 48538140, 48538141, 48538142, 48538143, 48538144, 48538146, 48538150, 48538160, 48538162, 48538166, 48538167, 48538168, 48538198, 48538200, 48538201, 48538202, 48538203, 48538204, 48538205, 48538206, 48538212, 48538222, 48538223, 48538224, 48538225, 48538226, 48538227, 48538228, 48538229, 48538230, 48538231, 48538232, 48538233, 48538234, 48538235, 48538236, 48538237, 48538238, 48538239, 48538240, 48538241, 48538242, 48538243, 48538244, 48538245, 48538246, 48538247, 48538248, 48538249, 48538250, 48538251, 48538252, 48538253, 48538254, 48538255, 48538256, 48538257, 48538258, 48538259, 48538260, 48538261, 48538262, 48538263, 48538264, 48538265, 48538266, 48538267, 48538268, 48538269, 48538270, 48538271, 48538272, 48538273, 48538274, 48538275, 48538276, 48538277, 48538278, 48538279, 48538280, 48538281, 48538282, 48538283, 48538284, 48538285, 48538286, 48538287, 48538288, 48538289, 48538290, 48538291, 48538292, 48538293, 48538294, 48538295, 48538296, 48538297, 48538298, 48538299, 48538300, 48538301, 48538302, 48538303, 48538304, 48538305, 48538306, 48538307, 48538308, 48538309, 48538310, 48538311, 48538312, 48538313, 48538314, 48538315, 48538316, 48538317, 48538318, 48538319, 48538320, 48538321, 48538322, 48538323, 48538324, 48538325, 48538326, 48538327, 48538328, 48538329, 48538330, 48538331, 48538332, 48538333, 48538334, 48538335, 48538336, 48538337, 48538338, 48538339, 48538340, 48538341, 48538342, 48538343, 48538344, 48538345, 48538346, 48538347, 48538348, 48538349, 48538350, 48538351, 48538352, 48538353, 48538354, 48538355, 48538356, 48538357, 48538358, 48538359, 48538360, 48538361, 48538362, 48538363, 48538364, 48538365, 48538366, 48538367, 48538368, 48538369, 48538370, 48538371, 48538372, 48538373, 48538374, 48538375, 48538376, 48538377, 48538378, 48538379, 48538380, 48538381, 48538382, 48538383, 48538384, 48538385, 48538386, 48538387, 48538388, 48538389, 48538390, 48538391, 48538392, 48538393, 48538394, 48538395, 48538396, 48538397, 48538398, 48538399, 48538400, 48538401, 48538402, 48538403, 48538404, 48538405, 48538406, 48538407, 48538408, 48538409, 48538410, 48538411, 48538412, 48538413, 48538414, 48538415, 48538416, 48538417, 48538418, 48538419, 48538420, 48538421, 48538422, 48538423, 48538424, 48538425, 48538426, 48538427, 48538428, 48538429, 48538430, 48538431, 48538432, 48538433, 48538434, 48538435, 48538436, 48538437, 48538438, 48538439, 48538440, 48538441, 48538442, 48538443, 48538444, 48538445, 48538446, 48538447, 48538448, 48538449, 48538450, 48538451, 48538452, 48538453, 48538454, 48538455, 48538456, 48538457, 48538458, 48538459, 48538460, 48538461, 48538462, 48538463, 48538464, 48538465, 48538466, 48538467, 48538468, 48538469, 48538470, 48538471, 48538472, 48538473, 48538474, 48538475, 48538476, 48538477, 48538478, 48538479, 48538480, 48538481, 48538482, 48538483, 48538484, 48538485, 48538486, 48538487, 48538488, 48538489, 48538490, 48538491, 48538492, 48538493, 48538494, 48538495, 48538496, 48538497, 48538498, 48538499, 48538500, 48538501, 48538502, 48538503, 48538504, 48538505, 48538506, 48538507, 48538508, 48538509, 48538510, 48538511, 48538512, 48538513, 48538514, 48538515, 48538516, 48538517, 48538518, 48538519, 48538520, 48538521, 48538522, 48538523, 48538524, 48538525, 48538526, 48538527, 48538528, 48538529, 48538530, 48538531, 48538532, 48538533, 48538534, 48538535, 48538536, 48538537, 48538538, 48538539, 48538540, 48538541, 48538542, 48538606, 48538608, 48538609, 48538610, 48538611, 48538612, 48538613, 48538614, 48538615, 48538616, 48538617, 48538618, 48538619, 48538620, 48538621, 48538622, 48538623, 48538624, 48538626, 48538627, 48538628, 48538629, 48538630, 48538631, 48538632, 48538633, 48538634, 48538635, 48538636, 48538637, 48538638, 48538639, 48538640, 48538641, 48538642, 48538643, 48538644, 48538645, 48538646, 48538647, 48538648, 48538649, 48538650, 48538651, 48538652, 48538653, 48538654, 48538656, 48538657, 48538658, 48538659, 48538660, 48538661, 48538662, 48538663, 48538664, 48538665, 48538666, 48538667, 48538668, 48538669, 48538670, 48538671, 48538672, 48538673, 48538674, 48538675, 48538676, 48538677, 48538678, 48538679, 48538680, 48538681, 48538682, 48538683, 48538684, 48538685, 48538686, 48538687, 48538688, 48538689, 48538690, 48538691, 48538692, 48538693, 48538694, 48538695, 48538696, 48538697, 48538698, 48538699, 48538700, 48538701, 48538702, 48538703, 48538704, 48538705, 48538706, 48538707, 48538708, 48538709, 48538710, 48538711, 48538712, 48538713, 48538714, 48538715, 48538716, 48538717, 48538718, 48538719, 48538720, 48538721, 48538722, 48538723, 48538724, 48538725, 48538726, 48538727, 48538728, 48538729, 48538730, 48538731, 48538732, 48538733, 48538734, 48538735, 48538736, 48538737, 48538738, 48538739, 48538740, 48538741, 48538742, 48538743, 48538744, 48538745, 48538746, 48538747, 48538748, 48538749, 48538750, 48538751, 48538752, 48538753, 48538754, 48538755, 48538756, 48538757, 48538758, 48538759, 48538760, 48538761, 48538762, 48538763, 48538764, 48538765, 48538766, 48538767, 48538768, 48538769, 48538770, 48538771, 48538772, 48538773, 48538774, 48538775, 48538776, 48538777, 48538778, 48538779, 48538780, 48538781, 48538782, 48538783, 48538784, 48538785, 48538786, 48538787, 48538788, 48538789, 48538790, 48538791, 48538792, 48538793, 48538794, 48538795, 48538796, 48538797, 48538798, 48538799, 48538800, 48538801, 48538802, 48538803, 48538804, 48538805, 48538806, 48538807, 48538808, 48538809, 48538810, 48538811, 48538812, 48538813, 48538814, 48538815, 48538816, 48538817, 48538818, 48538819, 48538820, 48538821, 48538822, 48538823, 48538824, 48538825, 48538826, 48538827, 48538828, 48538829, 48538830, 48538831, 48538832, 48538833, 48538834, 48538835, 48538836, 48538837, 48538838, 48538839, 48538840, 48538841, 48538842, 48538843, 48538844, 48538845, 48538846, 48538847, 48538848, 48538849, 48538850, 48538851, 48538852, 48538853, 48538854, 48538855, 48538856, 48538857, 48538858, 48538859, 48538860, 48538861, 48538862, 48538863, 48538864, 48538865, 48538866, 48538867, 48538868, 48538869, 48538870, 48538871, 48538872, 48538873, 48538874, 48538875, 48538876, 48538877, 48538878, 48538879, 48538880, 48538881, 48538882, 48538883, 48538884, 48538885, 48538886, 48538887, 48538888, 48538889, 48538890, 48538891, 48538892, 48538893, 48538894, 48538895, 48538896, 48538897, 48538898, 48538899, 48538900, 48538901, 48538902, 48538903, 48538904, 48538905, 48538906, 48538907, 48538908, 48538909, 48538910, 48538911, 48538912, 48538913, 48538914, 48538915, 48538916, 48538917, 48538918, 48538919, 48538920, 48538921, 48538922, 48538923, 48538924, 48538925, 48538926, 48538927, 48538928, 48538929, 48538930, 48538932, 48538934, 48538935, 48538936, 48538937, 48538938, 48538939, 48538940, 48538941, 48538942, 48538943, 48538944, 48538946, 48538947, 48538948, 48538950, 48538952, 48538954, 48538956, 48538958, 48538970]





The following program code was used to test the different values up to the first 100,000,000 primes.



