Problems & Puzzles: Conjectures

Conjecture 92. For any integer m there is at least one set of consecutive primes...

Paola Lava sent on March 2022, the following conjecture:

Conjecture

For any integer m there is at least one set of n>1, consecutive primes p_1, p_2, …, p_n such that the sum of every two consecutive primes in this set is divisible by m.

E.g. m=5

The first least sets for n from 2 to 10 are:

2+3=5 and 5/5=1

3   [43, 47, 53]

43+47=90 and 90/5=18; 47+53=100 and 100/5=20.

4   [157, 163, 167, 173]

157+163=320 and 320/5=64; 163+167=330 and 330/5=66; 167+173=340 and 340/5=68.

And so on …

5   [3593, 3607, 3613, 3617, 3623]

6   [6037, 6043, 6047, 6053, 6067, 6073]

7   [32633, 32647, 32653, 32687, 32693, 32707, 32713]

8   [32633, 32647, 32653, 32687, 32693, 32707, 32713, 32717]

9   [833843, 833857, 833873, 833887, 833893, 833897, 833923, 833927, 833933]

10   [833843, 833857, 833873, 833887, 833893, 833897, 833923, 833927, 833933, 833947]

Here below the list of the least sets for n from 2 to 5 for m from 1 to 50 (Giovanni Resta helped me in finding values for n = 5 and  m = 29, 31, 35, 37, 41, 43, 45, 47, 49):

 m n = 2 n = 3 n = 4 n = 5 1 2, 3 2, 3, 5 2, 3, 5, 7 2, 3, 5, 7, 11 2 3, 5 3, 5, 7 3, 5, 7, 11 3, 5, 7, 11, 13 3 5, 7 5, 7, 11 5, 7, 11, 13 5, 7, 11, 13, 17 4 3, 5 3, 5, 7 23, 29, 31, 37 47, 53, 59, 61, 67 5 2, 3 43, 47, 53 157, 163, 167, 173 3593, 3607, 3613, 3617, 3623 6 5, 7 5, 7, 11 5, 7, 11, 13 [5, 7, 11, 13, 17 7 19, 23 977, 983, 991 977, 983, 991, 997 10487, 10499, 10501, 10513, 10529 8 3, 5 53, 59, 61 53, 59, 61, 67 523, 541, 547, 557, 563] 9 7, 11 313, 317, 331 5171, 5179, 5189, 5197 38377, 38393, 38431, 38447, 38449 10 13, 17 43, 47, 53 157, 163, 167, 173 3593, 3607, 3613, 3617, 3623 11 97, 101 787, 797, 809 33871, 33889, 33893, 33911 1796671, 1796677, 1796693, 1796699, 1796759 12 5, 7 137, 139, 149 137, 139, 149, 151 409, 419, 421, 431, 433 13 23, 29 9587, 9601, 9613 159293, 159311, 159319, 159337 947423, 947431, 947449, 947483, 947501 14 19, 23 977, 983, 991 977, 983, 991, 997 10487, 10499, 10501, 10513, 10529 15 13, 17 2473, 2477, 2503 2969, 2971, 2999, 3001 60383, 60397, 60413, 60427, 60443 16 53, 59 541, 547, 557 541, 547, 557, 563 62501, 62507, 62533, 62539, 62549 17 31, 37 3967, 3989, 4001 406873, 406883, 406907, 406951 18164651, 18164689, 18164719, 18164723, 18164753 18 7, 11 313, 317, 331 5171, 5179, 5189, 5197 38377, 38393, 38431, 38447, 38449 19 73, 79 28979, 29009, 29017 471313, 471353, 471389, 471391 15095579, 15095611, 15095617, 15095687, 15095693 20 29, 31 947, 953, 967 6047, 6053, 6067, 6073 32633, 32647, 32653, 32687, 32693 21 19, 23 3121, 3137, 3163 166739, 166741, 166781, 166783 3272567, 3272587, 3272609, 3272629, 3272651 22 97, 101 787, 797, 809 33871, 33889, 33893, 33911 1796671, 1796677, 1796693, 1796699, 1796759 23 67, 71 72823, 72858, 72869 2112193, 2112217, 2112239, 2112263 116863451, 116863469, 116863543, 116863561, 116863589 24 11, 13 283, 293, 307 5309, 5323, 5333, 5347 67819, 67829, 67843, 67853, 67867 25 47, 53 47441, 47459, 47491 520763, 520787, 520813, 520837 65835479, 65835521, 65835529, 65835571, 65835629 26 23, 29 9587, 9601, 9613 159293, 159311, 159319, 159337 947423, 947431, 947449, 947483, 947501 27 79, 83 81463, 81509, 81517 207869, 207877, 207923, 207931 7005239, 7005277, 7005293, 7005331, 7005347 28 41, 43 4363, 4373, 4391 5443, 5449, 5471, 5477 1165217, 1165223, 1165273, 1165279, 1165301 29 347, 349 61153, 61169, 61211 2404471, 2404483, 2404529, 2404541 1154953243, 1154953249, 1154953301, 1154953307, 1154953417 30 13, 17 2473, 2477, 2503 2969, 2971, 2999, 3001 60383, 60397, 60413, 60427, 60443 31 89, 97 478001, 478039, 478063 1531487, 1531499, 1531549, 1531561 800037461, 800037521, 800037523, 800037583, 800037647 32 61, 67 21617, 21647, 21649 88919, 88937, 88951, 88969 7442557, 7442563, 7442621, 7442627, 7442653 33 97, 101 160243, 160253, 160309 2673791, 2673793, 2673857, 2673859 15442121, 15442183, 15442187, 15442249, 15442253 34 31, 37 3967, 3989, 4001 406873, 406883, 406907, 406951 18164651, 18164689, 18164719, 18164723, 18164753 35 103, 107 132763, 132817, 132833 6056569, 6056581, 6056639, 6056651 771405431, 771405559, 771405571, 771405629, 771405641 36 17, 19 8017, 8039, 8053 95737, 95747, 95773, 95783 558791, 558793, 558827, 558829, 558863 37 109, 113 227873, 227893, 227947 8480357, 8480369, 8480431, 8480443 1956833159, 1956833183, 1956833233, 1956833257, 1956833381 38 73, 79 28979, 29009, 29017 471313, 471353, 471389, 471391 15095579, 15095611, 15095617, 15095687, 15095693 39 37, 41 218279, 218287, 218357 561829, 561839, 561907, 561917 128805491, 128805499, 128805569, 128805577, 128805647 40 59, 61 12163, 12197, 12203 73477, 73483, 73517, 73523 24653231, 24653249, 24653311, 24653329, 24653351 41 199, 211 1772119, 1772167, 1772201 8756689, 8756707, 8756771, 8756789 46267575553, 46267575593, 46267575799, 46267575839, 46267575881 42 19, 23 3121, 3137, 3163 166739, 166741, 166781, 166783 3272567, 3272587, 3272609, 3272629, 3272651 43 83, 89 3070187, 3070213, 3070273 15409001, 15409013, 15409087, 15409099 9523396021, 9523396187, 9523396193, 9523396273, 9523396279 44 151, 157 57413, 57427, 57457 839087, 839117, 839131, 839161 30716599, 30716641, 30716687, 30716729, 30716731 45 43, 47 841459, 841541, 841549 2517289, 2517311, 2517379, 2517401 1036205567, 1036205623, 1036205657, 1036205713, 1036205747 46 67, 71 72823, 72859, 72869 2112193, 2112217, 2112239, 2112263 116863451, 116863469, 116863543, 116863561, 116863589 47 281, 283 6135979, 6136003, 6136073 36726643, 36726649, 36726737, 36726743 72019040869, 72019040917, 72019040963, 72019041011, 72019041151 48 71, 73 39451, 39461, 39499 60811, 60821, 60859, 60869 17426083, 17426093, 17426131, 17426141, 17426179 49 439, 443 1194059, 1194103, 1194157 67473239, 67473251, 67473337, 67473349 23035373797, 23035373863, 23035373993, 23035374059, 23035374091 50 47, 53 47441, 47459, 47491 520763, 520787, 520813, 520837 65835479, 65835521, 65835529, 65835571, 65835629

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