 mersenneforum.org Two prime-related conjectures
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sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

31·109 Posts Quote:
 Originally Posted by sweety439 Conjecture 1 (conjecture about square numbers and odd primes): Every number which is not twice a square number (A001105) can be written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 47 known counterexamples, the largest known counterexample is 43358, and I conjectured that all other numbers which is not twice a square number can be written as this form. Code: 1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358 Conjecture 2 (conjecture about triangular numbers and odd primes): Every number which is not twice a triangular number (A002378) can be written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 8 known counterexamples, the largest known counterexample is 432, and I conjectured that all other numbers which is not twice a triangular number can be written as this form. Code: 1, 3, 4, 10, 14, 122, 422, 432
Now OEIS has sequences about these two conjectures!!! A347567 A347568   2022-01-16, 11:48 #13 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 D3316 Posts Related research: perfect power (0 and 1 are not counted as perfect power) + prime (the even prime 2 is allowed) OEIS sequences for this: A119748 A277075 A196228 A253238 A276711 In the past, I conjectured that all integers >24 can be written as perfect power (0 and 1 are not counted as perfect power) + prime, I tested dozens of numbers, and I found many numbers having only one way to write: Code: 6 = 4 + 2 7 = 4 + 3 9 = 4 + 5 10 = 8 + 2 12 = 9 + 3 13 = 8 + 5 14 = 9 + 5 16 = 9 + 7 17 = 4 + 13 18 = 16 + 2 20 = 9 + 11 22 = 9 + 13 25 = 8 + 17 26 = 9 + 17 31 = 8 + 23 36 = 25 + 11 42 = 25 + 17 48 = 25 + 23 58 = 27 + 31 60 = 49 + 11 64 = 27 + 37 74 = 27 + 47 76 = 9 + 67 82 = 9 + 73 85 = 32 + 53 90 = 49 + 41 114 = 25 + 89 120 = 49 + 71 127 = 125 + 2 170 = 81 + 89 193 = 36 + 157 196 = 125 + 71 202 = 9 + 193 214 = 125 + 89 324 = 125 + 199 328 = 225 + 103 331 = 324 + 7 370 = 243 + 127 412 = 81 + 331 505 = 324 + 181 562 = 225 + 337 676 = 243 + 433 706 = 243 + 463 730 = 243 + 487 799 = 576 + 223 841 = 32 + 809 1024 = 27 + 997 1087 = 36 + 1051 1204 = 81 + 1123 1243 = 324 + 919 1681 = 128 + 1553 1849 = 128 + 1721 2146 = 9 + 2137 2293 = 1296 + 997 2986 = 125 + 2861 3319 = 128 + 3191 10404 = 343 + 10061 46656 = 46225 + 431 52900 = 35937 + 16963 112896 = 125 + 112771 122500 = 1331 + 121169 (I checked all numbers <= 2^24) but I don't know why my conjecture fails at the number 1549, also 1771561 is another counterexample, it is known (checked by others), my conjecture works at all numbers <= 10^10 except 1549 and 1771561 (the small numbers cannot be written as this way is 1, 2, 3, 4, 5, 8, 24, thus the set of all numbers which cannot be written as this way is (likely) {1, 2, 3, 4, 5, 8, 24, 1549, 1771561} (if only odd primes are allowed, and the even prime 2 is not allowed, then the set of all numbers is (likely) {1, 2, 3, 4, 5, 6, 8, 10, 18, 24, 127, 1549, 1771561}, the number 127 is interesting as it is the first odd number >3 which is not the sum of power of 2 and a prime) Last fiddled with by sweety439 on 2022-01-26 at 20:31   2022-01-16, 16:55 #14 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 64638 Posts Compare with the numbers with only one way to write as: "twice a positive square number" + "odd prime or twice an odd prime" or "twice a positive triangular number" + "odd prime or twice an odd prime" For the former (square number): Code: 5 = 1*3 + 2*1^2 7 = 1*5 + 2*1^2 9 = 1*7 + 2*1^2 11 = 1*3 + 2*2^2 12 = 2*5 + 2*1^2 14 = 2*3 + 2*2^2 16 = 2*7 + 2*1^2 22 = 2*7 + 2*2^2 23 = 1*5 + 2*3^2 27 = 1*19 + 2*2^2 29 = 1*11 + 2*3^2 30 = 2*11 + 2*2^2 33 = 1*31 + 2*1^2 34 = 2*13 + 2*2^2 36 = 2*17 + 2*1^2 38 = 2*3 + 2*4^2 41 = 1*23 + 2*3^2 44 = 2*13 + 2*3^2 47 = 1*29 + 2*3^2 48 = 2*23 + 2*1^2 52 = 2*17 + 2*3^2 53 = 1*3 + 2*5^2 57 = 1*7 + 2*5^2 58 = 2*13 + 2*4^2 59 = 1*41 + 2*3^2 65 = 1*47 + 2*3^2 71 = 1*53 + 2*3^2 80 = 2*31 + 2*3^2 83 = 1*11 + 2*6^2 86 = 2*7 + 2*6^2 92 = 2*37 + 2*3^2 95 = 1*23 + 2*6^2 100 = 2*41 + 2*3^2 102 = 2*47 + 2*2^2 107 = 1*89 + 2*3^2 110 = 2*19 + 2*6^2 113 = 1*41 + 2*6^2 123 = 1*73 + 2*5^2 140 = 2*61 + 2*3^2 143 = 1*71 + 2*6^2 146 = 2*37 + 2*6^2 148 = 2*73 + 2*1^2 149 = 1*131 + 2*3^2 152 = 2*67 + 2*3^2 158 = 2*43 + 2*6^2 161 = 1*89 + 2*6^2 164 = 2*73 + 2*3^2 188 = 2*13 + 2*9^2 194 = 2*61 + 2*6^2 197 = 1*179 + 2*3^2 198 = 2*83 + 2*4^2 212 = 2*97 + 2*3^2 218 = 2*73 + 2*6^2 230 = 2*79 + 2*6^2 233 = 1*71 + 2*9^2 239 = 1*167 + 2*6^2 240 = 2*71 + 2*7^2 257 = 1*239 + 2*3^2 266 = 2*97 + 2*6^2 272 = 2*127 + 2*3^2 278 = 2*103 + 2*6^2 281 = 1*263 + 2*3^2 284 = 2*61 + 2*9^2 287 = 1*269 + 2*3^2 290 = 2*109 + 2*6^2 302 = 2*7 + 2*12^2 308 = 2*73 + 2*9^2 314 = 2*13 + 2*12^2 317 = 1*29 + 2*12^2 318 = 2*59 + 2*10^2 323 = 1*251 + 2*6^2 332 = 2*157 + 2*3^2 340 = 2*89 + 2*9^2 347 = 1*59 + 2*12^2 356 = 2*97 + 2*9^2 362 = 2*37 + 2*12^2 368 = 2*103 + 2*9^2 383 = 1*311 + 2*6^2 386 = 2*157 + 2*6^2 404 = 2*193 + 2*3^2 407 = 1*389 + 2*3^2 410 = 2*61 + 2*12^2 413 = 1*251 + 2*9^2 422 = 2*67 + 2*12^2 438 = 2*23 + 2*14^2 442 = 2*157 + 2*8^2 443 = 1*281 + 2*9^2 446 = 2*79 + 2*12^2 449 = 1*431 + 2*3^2 458 = 2*193 + 2*6^2 470 = 2*199 + 2*6^2 482 = 2*97 + 2*12^2 492 = 2*197 + 2*7^2 500 = 2*241 + 2*3^2 506 = 2*109 + 2*12^2 530 = 2*229 + 2*6^2 536 = 2*43 + 2*15^2 542 = 2*127 + 2*12^2 548 = 2*193 + 2*9^2 554 = 2*241 + 2*6^2 566 = 2*139 + 2*12^2 569 = 1*281 + 2*12^2 590 = 2*151 + 2*12^2 596 = 2*73 + 2*15^2 602 = 2*157 + 2*12^2 620 = 2*229 + 2*9^2 626 = 2*277 + 2*6^2 632 = 2*307 + 2*3^2 638 = 2*283 + 2*6^2 650 = 2*181 + 2*12^2 656 = 2*103 + 2*15^2 662 = 2*7 + 2*18^2 668 = 2*109 + 2*15^2 680 = 2*331 + 2*3^2 692 = 2*337 + 2*3^2 698 = 2*313 + 2*6^2 743 = 1*293 + 2*15^2 773 = 1*701 + 2*6^2 782 = 2*67 + 2*18^2 785 = 1*137 + 2*18^2 788 = 2*313 + 2*9^2 794 = 2*73 + 2*18^2 798 = 2*383 + 2*4^2 818 = 2*373 + 2*6^2 824 = 2*331 + 2*9^2 848 = 2*199 + 2*15^2 863 = 1*701 + 2*9^2 872 = 2*211 + 2*15^2 884 = 2*433 + 2*3^2 890 = 2*409 + 2*6^2 926 = 2*139 + 2*18^2 938 = 2*433 + 2*6^2 980 = 2*409 + 2*9^2 998 = 2*463 + 2*6^2 1010 = 2*181 + 2*18^2 1022 = 2*367 + 2*12^2 1082 = 2*397 + 2*12^2 1094 = 2*223 + 2*18^2 1118 = 2*523 + 2*6^2 1124 = 2*337 + 2*15^2 1148 = 2*349 + 2*15^2 1172 = 2*577 + 2*3^2 1178 = 2*13 + 2*24^2 1220 = 2*601 + 2*3^2 1227 = 1*1129 + 2*7^2 1232 = 2*607 + 2*3^2 1238 = 2*43 + 2*24^2 1292 = 2*421 + 2*15^2 1322 = 2*337 + 2*18^2 1367 = 1*719 + 2*18^2 1388 = 2*613 + 2*9^2 1415 = 1*263 + 2*24^2 1418 = 2*673 + 2*6^2 1478 = 2*163 + 2*24^2 1502 = 2*607 + 2*12^2 1562 = 2*457 + 2*18^2 1586 = 2*757 + 2*6^2 1598 = 2*223 + 2*24^2 1622 = 2*487 + 2*18^2 1668 = 2*809 + 2*5^2 1670 = 2*691 + 2*12^2 1703 = 1*821 + 2*21^2 1748 = 2*433 + 2*21^2 1754 = 2*733 + 2*12^2 1787 = 1*1499 + 2*12^2 1828 = 2*73 + 2*29^2 1844 = 2*193 + 2*27^2 1892 = 2*937 + 2*3^2 1898 = 2*373 + 2*24^2 1940 = 2*241 + 2*27^2 1958 = 2*79 + 2*30^2 1988 = 2*769 + 2*15^2 2042 = 2*877 + 2*12^2 2060 = 2*1021 + 2*3^2 2090 = 2*1009 + 2*6^2 2123 = 1*971 + 2*24^2 2132 = 2*337 + 2*27^2 2138 = 2*1033 + 2*6^2 2174 = 2*1051 + 2*6^2 2180 = 2*1009 + 2*9^2 2210 = 2*1069 + 2*6^2 2234 = 2*541 + 2*24^2 2328 = 2*1163 + 2*1^2 2342 = 2*271 + 2*30^2 2402 = 2*877 + 2*18^2 2408 = 2*1123 + 2*9^2 2438 = 2*643 + 2*24^2 2486 = 2*919 + 2*18^2 2507 = 1*1049 + 2*27^2 2558 = 2*379 + 2*30^2 2582 = 2*967 + 2*18^2 2648 = 2*883 + 2*21^2 2708 = 2*1129 + 2*15^2 2732 = 2*277 + 2*33^2 2762 = 2*1237 + 2*12^2 2768 = 2*1303 + 2*9^2 2822 = 2*1087 + 2*18^2 2858 = 2*853 + 2*24^2 2900 = 2*1009 + 2*21^2 2933 = 1*2861 + 2*6^2 3002 = 2*601 + 2*30^2 3062 = 2*631 + 2*30^2 3110 = 2*1231 + 2*18^2 3242 = 2*1297 + 2*18^2 3284 = 2*1201 + 2*21^2 3317 = 1*3299 + 2*3^2 3434 = 2*421 + 2*36^2 3452 = 2*997 + 2*27^2 3482 = 2*1597 + 2*12^2 3515 = 1*2633 + 2*21^2 3530 = 2*1621 + 2*12^2 3572 = 2*1777 + 2*3^2 3620 = 2*1801 + 2*3^2 3662 = 2*67 + 2*42^2 3713 = 1*1913 + 2*30^2 3722 = 2*97 + 2*42^2 3758 = 2*1303 + 2*24^2 3770 = 2*1741 + 2*12^2 3962 = 2*1657 + 2*18^2 3980 = 2*1549 + 2*21^2 3998 = 2*1423 + 2*24^2 4022 = 2*1867 + 2*12^2 4082 = 2*277 + 2*42^2 4118 = 2*1483 + 2*24^2 4148 = 2*1993 + 2*9^2 4178 = 2*2053 + 2*6^2 4292 = 2*2137 + 2*3^2 4334 = 2*2131 + 2*6^2 4490 = 2*1669 + 2*24^2 4502 = 2*487 + 2*42^2 4532 = 2*241 + 2*45^2 4538 = 2*1693 + 2*24^2 4568 = 2*2203 + 2*9^2 4586 = 2*997 + 2*36^2 4673 = 1*2081 + 2*36^2 4688 = 2*823 + 2*39^2 4820 = 2*1321 + 2*33^2 4832 = 2*1327 + 2*33^2 4958 = 2*1579 + 2*30^2 5078 = 2*2503 + 2*6^2 5102 = 2*787 + 2*42^2 5300 = 2*1129 + 2*39^2 5612 = 2*2797 + 2*3^2 5642 = 2*2677 + 2*12^2 5708 = 2*829 + 2*45^2 5798 = 2*1999 + 2*30^2 5852 = 2*2917 + 2*3^2 5942 = 2*2647 + 2*18^2 5987 = 1*2459 + 2*42^2 6008 = 2*1483 + 2*39^2 6188 = 2*1069 + 2*45^2 6218 = 2*193 + 2*54^2 6302 = 2*2251 + 2*30^2 6332 = 2*2437 + 2*27^2 6368 = 2*1663 + 2*39^2 6518 = 2*2683 + 2*24^2 6602 = 2*997 + 2*48^2 6797 = 1*6779 + 2*3^2 6836 = 2*2689 + 2*27^2 6938 = 2*3433 + 2*6^2 7004 = 2*3061 + 2*21^2 7142 = 2*2671 + 2*30^2 7622 = 2*211 + 2*60^2 7718 = 2*3823 + 2*6^2 7730 = 2*3541 + 2*18^2 7928 = 2*3739 + 2*15^2 7982 = 2*3847 + 2*12^2 8432 = 2*967 + 2*57^2 8444 = 2*1621 + 2*51^2 8558 = 2*4243 + 2*6^2 8660 = 2*3889 + 2*21^2 8828 = 2*2389 + 2*45^2 9008 = 2*4423 + 2*9^2 9020 = 2*541 + 2*63^2 9122 = 2*2797 + 2*42^2 9290 = 2*2341 + 2*48^2 9308 = 2*2053 + 2*51^2 9422 = 2*4567 + 2*12^2 9722 = 2*2557 + 2*48^2 9860 = 2*4201 + 2*27^2 10964 = 2*3457 + 2*45^2 11012 = 2*5281 + 2*15^2 11090 = 2*4969 + 2*24^2 11498 = 2*2833 + 2*54^2 11972 = 2*2017 + 2*63^2 12062 = 2*3727 + 2*48^2 12098 = 2*1693 + 2*66^2 12548 = 2*3673 + 2*51^2 12602 = 2*1117 + 2*72^2 12878 = 2*2083 + 2*66^2 14018 = 2*4093 + 2*54^2 14162 = 2*997 + 2*78^2 14882 = 2*7297 + 2*12^2 15758 = 2*823 + 2*84^2 15908 = 2*7873 + 2*9^2 16172 = 2*6997 + 2*33^2 16838 = 2*5503 + 2*54^2 17168 = 2*3823 + 2*69^2 17648 = 2*8599 + 2*15^2 18428 = 2*9133 + 2*9^2 19142 = 2*1471 + 2*90^2 20330 = 2*3109 + 2*84^2 20918 = 2*9883 + 2*24^2 21548 = 2*10333 + 2*21^2 21722 = 2*457 + 2*102^2 23018 = 2*2293 + 2*96^2 23612 = 2*9781 + 2*45^2 25022 = 2*6427 + 2*78^2 27668 = 2*8209 + 2*75^2 30212 = 2*7537 + 2*87^2 30668 = 2*13309 + 2*45^2 31130 = 2*15241 + 2*18^2 32162 = 2*15937 + 2*12^2 32372 = 2*7537 + 2*93^2 47702 = 2*6427 + 2*132^2 63758 = 2*28279 + 2*60^2 66410 = 2*6961 + 2*162^2 89072 = 2*11047 + 2*183^2 For the latter (triangular number): Code: 5 = 1*3 + 1*2 7 = 1*5 + 1*2 8 = 2*3 + 1*2 11 = 1*5 + 2*3 18 = 2*3 + 3*4 21 = 1*19 + 1*2 22 = 2*5 + 3*4 24 = 2*11 + 1*2 27 = 1*7 + 4*5 32 = 2*13 + 2*3 38 = 2*13 + 3*4 50 = 2*19 + 3*4 51 = 1*31 + 4*5 54 = 2*17 + 4*5 57 = 1*37 + 4*5 60 = 2*29 + 1*2 62 = 2*3 + 7*8 74 = 2*31 + 3*4 84 = 2*41 + 1*2 105 = 1*103 + 1*2 108 = 2*53 + 1*2 111 = 1*109 + 1*2 126 = 2*53 + 4*5 140 = 2*67 + 2*3 150 = 2*47 + 7*8 174 = 2*59 + 7*8 180 = 2*89 + 1*2 186 = 2*83 + 4*5 242 = 2*43 + 12*13 252 = 2*71 + 10*11 258 = 2*101 + 7*8 270 = 2*107 + 7*8 357 = 1*337 + 4*5 372 = 2*131 + 10*11 471 = 1*199 + 16*17 492 = 2*191 + 10*11 510 = 2*227 + 7*8 630 = 2*179 + 16*17 666 = 2*197 + 16*17 690 = 2*317 + 7*8 765 = 1*709 + 7*8 792 = 2*71 + 25*26 810 = 2*269 + 16*17 1080 = 2*449 + 13*14 1112 = 2*541 + 5*6 1380 = 2*599 + 13*14 1434 = 2*311 + 28*29 1602 = 2*773 + 7*8 1848 = 2*599 + 25*26 1920 = 2*257 + 37*38 2160 = 2*827 + 22*23 3726 = 2*1367 + 31*32 4752 = 2*1151 + 49*50 5397 = 1*1237 + 64*65 5652 = 2*1601 + 49*50 7800 = 2*2819 + 46*47 12420 = 2*5507 + 37*38 16632 = 2*3851 + 94*95 Like "perfect power (not including 0 and 1) + prime", it is conjectured that 89072 and 16632 are the largest examples of them, respectively.   2022-01-16, 17:05 #15 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 D3316 Posts There are complement of A065377 and A065397 of the primes A000040 in OEIS: A065376 and A065396, respectively. Last fiddled with by sweety439 on 2022-01-16 at 17:06   2022-01-20, 22:18   #16
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

31·109 Posts The conjecture 2 in post #1 (i.e. A347568 is full with 8 terms: {1, 3, 4, 10, 14, 122, 422, 432}):

Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers

(page 3 has "furthermore, any positive integer n (not∈) {2, 5, 7, 61, 211, 216} can be written in the form p + Tx with x ∈ Z+, where p is an odd prime or zero", which is equivalent to the only even numbers in A347568 are {4, 10, 14, 122, 422, 432}) and (page 5 has "Any odd integer n > 3 can be written in the form p + x(x + 1) with p a prime and and x a positive integer", which is equivalent to the only odd numbers in A347568 are {1, 3})

The conjecture 1 in post #1 (i.e. A347567 is full with 47 terms: {1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}):

G. H. Hardy, J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes (see the attached pdf file)

(page 49 has "Conjecture H. Every large number n is either a square or the sum of a prime and a square.", which is called "Hardy & Littlewood's Conjecture H" (not to be confused with Hardy–Littlewood conjecture about prime k-tuples), and is equivalent to A347567 contains only finitely many even numbers (or finitely many even numbers which are twice composite numbers, if 0 is counted as square in that article), also "probably because, the idea that every number is a square, or the sum of a prime and a square, is refuted (even if I is counted as a prime) by such immediate examples as 34 and 58. But the problem of the representation of an odd number in the form t+2m^2 has received some attention; and it has been verified that the only odd numbers less than 9000, and not of the form desired, are 5777 and 5993", which is related to the even numbers in A347567 and the odd numbers in A347567, respectively)
Attached Files BF02403921.pdf (2.47 MB, 21 views)

Last fiddled with by sweety439 on 2022-01-21 at 21:20   2022-01-21, 21:32 #17 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 31·109 Posts For A347567, 2*square+p, where p is either odd prime or twice an odd prime, the set is: {1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358} If p = 4 (twice the even prime 2) is allowed, then the set become: {1, 3, 4, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358} (the same set except the number 6) If p = 1, 2, 4 are all allowed, then the set become: {1, 17, 26, 62, 68, 116, 122, 137, 170, 182, 227, 254, 260, 428, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358} If square = 0 is allowed, then the set become: {1, 4, 20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5777, 5972, 5993, 7352, 19268, 43358} If square = 0 and p = 4 (twice the even prime 2) are both allowed, then the set become: {1, 4, 20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5777, 5972, 5993, 7352, 19268, 43358} (the same set as the previous) If square = 0 and p = 1, 2, 4 are all allowed, then the set become: {68, 116, 170, 182, 260, 428, 740, 1052, 1412, 1542, 2510, 2702, 2828, 3812, 5777, 5972, 5993, 7352, 19268, 43358}   2022-01-21, 21:51 #18 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 31·109 Posts The numbers in A347567: {1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358} Categories: odd unit: {1} odd primes: {3, 17, 137, 227, 977, 1187, 1493} (A042978 Stern primes, except the "oddest" prime 2) odd composites: {5777, 5993} (counterexamples of the less-known Goldbach conjecture, composites in A060003) even number whose half is unit: {} (the only such number is 2, but 2 is twice a square thus not considered) even numbers whose half are primes: {4, 10, 26, 62, 74, 122, 254, 758, 878, 1142, 1658, 1982, 3098, 6638, 15098} (2*A065377) even numbers whose half are composites: {20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5972, 7352, 19268, 43358} (2*A020495) Factorizations: Code: 1 = unit 3 = 3 4 = 2^2 6 = 2 * 3 10 = 2 * 5 17 = 17 20 = 2^2 * 5 26 = 2 * 13 62 = 2 * 31 68 = 2^2 * 17 74 = 2 * 37 116 = 2^2 * 29 122 = 2 * 61 137 = 137 170 = 2 * 5 * 17 182 = 2 * 7 * 13 227 = 227 254 = 2 * 127 260 = 2^2 * 5 * 13 428 = 2^2 * 107 452 = 2^2 * 113 740 = 2^2 * 5 * 37 758 = 2 * 379 878 = 2 * 439 977 = 977 1052 = 2^2 * 263 1142 = 2 * 571 1187 = 1187 1412 = 2^2 * 353 1460 = 2^2 * 5 * 73 1493 = 1493 1542 = 2 * 3 * 257 1658 = 2 * 829 1982 = 2 * 991 2510 = 2 * 5 * 251 2702 = 2 * 7 * 193 2828 = 2^2 * 7 * 101 3098 = 2 * 1549 3812 = 2^2 * 953 5777 = 53 * 109 5972 = 2^2 * 1493 5993 = 13 * 461 6638 = 2 * 3319 7352 = 2^3 * 919 15098 = 2 * 7549 19268 = 2^2 * 4817 43358 = 2 * 7 * 19 * 163 Modulos: Mod 2: {1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0} (nothing interesting, the only interesting thing is that "0" is much more than "1") Mod 3: {1, 0, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2} (all are "2" after 10, with only an exception of 1542, which is "0", see the reference in Prime Curios!) Mod 4: {1, 3, 0, 2, 2, 1, 0, 2, 2, 0, 2, 0, 2, 1, 2, 2, 3, 2, 0, 0, 0, 0, 2, 2, 1, 0, 2, 3, 0, 0, 1, 2, 2, 2, 2, 2, 0, 2, 0, 1, 0, 1, 2, 0, 2, 0, 2} (the same as mod 2, most are "0" or "2", and "0" is more than "2" (most "2" occurs for twice primes, see A308516), "1" is more than "3" (the only "3" are 3, 227, 1187)) Mod 5: {1, 3, 4, 1, 0, 2, 0, 1, 2, 3, 4, 1, 2, 2, 0, 2, 2, 4, 0, 3, 2, 0, 3, 3, 2, 2, 2, 2, 2, 0, 3, 2, 3, 2, 0, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 3, 3} (all are "2" or "3" after 2510, and all are "0", "2", or "3" after 254) Mod 6: {1, 3, 4, 0, 4, 5, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 5, 2, 2, 5, 0, 2, 2, 2, 2, 2, 2, 2, 5, 2, 5, 2, 2, 2, 2, 2} (most are "2", and the interesting thing the same as mod 3, all are "2" or "5" after 10, with only an exception of 1542, which is "0", the only "3" is for the number 3, see the Graham comment in A060003) Mod 7: {1, 3, 4, 6, 3, 3, 6, 5, 6, 5, 4, 4, 3, 4, 2, 0, 3, 2, 1, 1, 4, 5, 2, 3, 4, 2, 1, 4, 5, 4, 2, 2, 6, 1, 4, 0, 0, 4, 4, 2, 1, 1, 2, 2, 6, 4, 0} (nothing interesting) Mod 8: {1, 3, 4, 6, 2, 1, 4, 2, 6, 4, 2, 4, 2, 1, 2, 6, 3, 6, 4, 4, 4, 4, 6, 6, 1, 4, 6, 3, 4, 4, 5, 6, 2, 6, 6, 6, 4, 2, 4, 1, 4, 1, 6, 0, 2, 4, 6} (most are "2", "4", "6", there is only one "0", also there is no "7", see A308516 and A317966) Mod 9: {1, 3, 4, 6, 1, 8, 2, 8, 8, 5, 2, 8, 5, 2, 8, 2, 2, 2, 8, 5, 2, 2, 2, 5, 5, 8, 8, 8, 8, 2, 8, 3, 2, 2, 8, 2, 2, 2, 5, 8, 5, 8, 5, 8, 5, 8, 5} (the same as mod 3, most are "2", "5", "8", 9 is the smallest mod such that there is no "0", the next such mod is 11) Mod 10: {1, 3, 4, 6, 0, 7, 0, 6, 2, 8, 4, 6, 2, 7, 0, 2, 7, 4, 0, 8, 2, 0, 8, 8, 7, 2, 2, 7, 2, 0, 3, 2, 8, 2, 0, 2, 8, 8, 2, 7, 2, 3, 8, 2, 8, 8, 8} (the same as mod 5, all are "2", "3", "7", "8" after 2510, and all are "0", "2", "3", "7", "8" after 254, there is no "5") Mod 11: {1, 3, 4, 6, 10, 6, 9, 4, 7, 2, 8, 6, 1, 5, 5, 6, 7, 1, 7, 10, 1, 3, 10, 9, 9, 7, 9, 10, 4, 8, 8, 2, 8, 2, 2, 7, 1, 7, 6, 2, 10, 9, 5, 4, 6, 7, 7} (nothing interesting) Mod 12: {1, 3, 4, 6, 10, 5, 8, 2, 2, 8, 2, 8, 2, 5, 2, 2, 11, 2, 8, 8, 8, 8, 2, 2, 5, 8, 2, 11, 8, 8, 5, 6, 2, 2, 2, 2, 8, 2, 8, 5, 8, 5, 2, 8, 2, 8, 2} (the same as mod 4 and mod 6, all are "2", "5", "8", "11" after 10, with only an exception of 1542, which is "6") (In general, possible residues of 2*m^2 mod n are more than impossible residues of 2*m^2 mod n, to the list of mod n) Last fiddled with by sweety439 on 2022-01-28 at 23:49   2022-01-22, 10:47 #19 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 337910 Posts Related conjectures: 11 is the only non-generalized pentagonal number which cannot be written as sum of a prime and a positive generalized pentagonal number. besides 1 and 3, 79 is the only odd number which cannot be written as sum of a prime and twice a positive generalized pentagonal number. and the only non-generalized pentagonal numbers can be written as sum of a prime and a positive generalized pentagonal number in only one way are {3, 6, 13, 16, 21, 23, 27, 47, 50, 61, 67, 127, 211} Last fiddled with by sweety439 on 2022-01-22 at 10:49   2022-03-14, 10:31 #20 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 31×109 Posts Conjecture 1: Every natural number n > 21679 is a square or the sum of a square and a prime number. Conjecture 2: Every prime number p > 7549 is the sum of a nonzero square and a prime number. Conjecture 3: Every odd natural number n > 5993 is the sum of twice a square and a prime number. Conjecture 4: Every odd prime number p > 1493 is the sum of twice a nonzero square and a prime number. Conjecture 5: Every natural number n > 216 is a triangular number or the sum of a triangular number and a prime number. Conjecture 6: Every prime number p > 211 is the sum of a nonzero triangular number and a prime number. Conjecture 7: Every odd natural number n > 1 is the sum of twice a triangular number and a prime number. Conjecture 8: Every odd prime number p > 3 is the sum of twice a nonzero triangular number and a prime number. 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