Double Wieferich pairs and circulant Hadamard matrices Brooke Logan and Michael J. Mossinghoff May 26, 2015 The 4489 values of u <= 10^15 for which n = 4u^2 passes all known requirements for the order of a circulant Hadamard matrix. Each value of u is shown with its prime factorization. 11715 [3,5,11,71] 82005 [3,5,7,11,71] 550605 [3,5,11,47,71] 3854235 [3,5,7,11,47,71] 3877665 [3,5,11,71,331] 5418777 [3,7,13,23,863] 8515221 [3,11,13,23,863] 9031295 [5,7,13,23,863] 9047885 [5,7,11,71,331] 10975393 [11,13,23,47,71] 12663915 [3,5,11,23,47,71] 14192035 [5,11,13,23,863] 16256331 [3,3,7,13,23,863] 27093885 [3,5,7,13,23,863] 27143655 [3,5,7,11,71,331] 29549135 [5,7,11,23,47,71] 32926179 [3,11,13,23,47,71] 42576105 [3,5,11,13,23,863] 54876965 [5,11,13,23,47,71] 59606547 [3,7,11,13,23,863] 76827751 [7,11,13,23,47,71] 81281655 [3,3,5,7,13,23,863] 82880065 [5,41,83,4871] 88647405 [3,5,7,11,23,47,71] 92805449 [11,71,331,359] 99344245 [5,7,11,13,23,863] 120503279 [13,23,467,863] 164630895 [3,5,11,13,23,47,71] 178819641 [3,3,7,11,13,23,863] 182250255 [3,5,11,47,71,331] 230483253 [3,7,11,13,23,47,71] 278416347 [3,11,71,331,359] 298032735 [3,5,7,11,13,23,863] 361509837 [3,13,23,467,863] 384138755 [5,7,11,13,23,47,71] 425250595 [5,7,11,47,71,331] 464027245 [5,11,71,331,359] 556069735 [5,13,23,431,863] 604580691 [3,11,13,23,71,863] 778497629 [7,13,23,431,863] 818187315 [3,5,11,71,211,331] 843522953 [7,13,23,467,863] 894098205 [3,3,5,7,11,13,23,863] 1007634485 [5,11,13,23,71,863] 1152416265 [3,5,7,11,13,23,47,71] 1275751785 [3,5,7,11,47,71,331] 1325536069 [11,13,23,467,863] 1392081735 [3,5,11,71,331,359] 1668209205 [3,5,13,23,431,863] 1948914429 [3,7,11,71,331,359] 2134525327 [11,23,71,331,359] 2232101653 [83,4871,5521] 2335492887 [3,7,13,23,431,863] 2403521885 [5,29,41,83,4871] 2530568859 [3,7,13,23,467,863] 3022903455 [3,5,11,13,23,71,863] 3248190715 [5,7,11,71,331,359] 3494595091 [13,23,29,467,863] 3632855083 [11,13,23,47,71,331] 3892488145 [5,7,13,23,431,863] 3976608207 [3,11,13,23,467,863] 4191755865 [3,5,11,23,47,71,331] 4217614765 [5,7,13,23,467,863] 4232064837 [3,7,11,13,23,71,863] 4361856103 [11,47,71,331,359] 4528485893 [23,83,487,4871] 4736292495 [3,5,11,71,83,4871] 4782199721 [13,23,43,431,863] 5004627615 [3,3,5,13,23,431,863] 5316048657 [3,3,3,83,487,4871] 5508404761 [101,2903,18787] 5727311205 [3,5,7,11,71,211,331] 6116767085 [5,11,13,23,431,863] 6403575981 [3,11,23,71,331,359] 6497392803 [3,11,83,487,4871] 6627680345 [5,11,13,23,467,863] 6630809493 [3,7,11,71,83,4871] 6696304959 [3,83,4871,5521] 7006478661 [3,3,7,13,23,431,863] 7053441395 [5,7,11,13,23,71,863] 7591706577 [3,3,7,13,23,467,863] 8072518331 [41,83,487,4871] 8860081095 [3,3,5,83,487,4871] 9278752483 [7,11,13,23,467,863] 9327263439 [3,13,23,863,12049] 9744572145 [3,5,7,11,71,331,359] 9780763685 [5,7,11,23,47,71,331] 10483785273 [3,13,23,29,467,863] 10672626635 [5,11,23,71,331,359] 10828988005 [5,11,83,487,4871] 10898565249 [3,11,13,23,47,71,331] 11160508265 [5,83,4871,5521] 11677464435 [3,5,7,13,23,431,863] 11899555869 [3,83,4871,9811] 12652844295 [3,5,7,13,23,467,863] 12696194511 [3,3,7,11,13,23,71,863] 13085568309 [3,11,47,71,331,359] 13585457679 [3,23,83,487,4871] 14346599163 [3,13,23,43,431,863] 16525214283 [3,101,2903,18787] 16612175371 [11,71,179,331,359] 16824653195 [5,7,29,41,83,4871] 18164275415 [5,11,13,23,47,71,331] 18350301255 [3,5,11,13,23,431,863] 19492178409 [3,3,11,83,487,4871] 19832593115 [5,83,4871,9811] 19883041035 [3,5,11,13,23,467,863] 21160324185 [3,5,7,11,13,23,71,863] 21763614691 [7,13,23,863,12049] 21809280515 [5,11,47,71,331,359] 22642429465 [5,23,83,487,4871] 23910998605 [5,13,23,43,431,863] 24217554993 [3,41,83,487,4871] 24462165637 [7,13,23,29,467,863] 25429985581 [7,11,13,23,47,71,331] 25690421757 [3,7,11,13,23,431,863] 26580243285 [3,3,3,5,83,487,4871] 27542023805 [5,101,2903,18787] 27748829251 [11,13,23,71,331,359] 27836257449 [3,7,11,13,23,467,863] 28077264007 [13,23,233,467,863] 28415292477 [3,11,13,23,47,71,863] 28491272843 [11,71,307,331,359] 29342291055 [3,5,7,11,23,47,71,331] 30532992721 [7,11,47,71,331,359] 32017879905 [3,5,11,23,71,331,359] 32486964015 [3,5,11,83,487,4871] 33154047465 [3,5,7,11,71,83,4871] 33475398047 [7,13,23,43,431,863] 33481524795 [3,5,83,4871,5521] 34199965943 [11,13,23,863,12049] 34518995081 [37,83,821,13691] 35032393305 [3,3,5,7,13,23,431,863] 37958532885 [3,3,5,7,13,23,467,863] 38440546001 [11,13,23,29,467,863] 38454803805 [3,5,11,47,71,211,331] 38837598459 [3,11,41,71,83,4871] 40362591655 [5,41,83,487,4871] 40756373037 [3,3,23,83,487,4871] 42817369595 [5,7,11,13,23,431,863] 43039797489 [3,3,13,23,43,431,863] 44825031867 [3,7,11,23,71,331,359] 46393762415 [5,7,11,13,23,467,863] 46636317195 [3,5,13,23,863,12049] 47358820795 [5,11,13,23,47,71,863] 49813344823 [11,23,83,487,4871] 49836526113 [3,11,71,179,331,359] 50473959585 [3,5,7,29,41,83,4871] 51338338019 [23,83,4871,5521] 52604196931 [11,13,23,43,431,863] 54492826245 [3,5,11,13,23,47,71,331] 55050903765 [3,3,5,11,13,23,431,863] 58476535227 [3,3,3,11,83,487,4871] 58870316609 [13,23,83,487,4871] 59497779345 [3,5,83,4871,9811] 62242928815 [5,41,83,751,4871] 63480972555 [3,3,5,7,11,13,23,71,863] 65290844073 [3,7,13,23,863,12049] 65427841545 [3,5,11,47,71,331,359] 66705402907 [11,13,23,71,331,863] 67927288395 [3,5,23,83,487,4871] 71732995815 [3,5,13,23,43,431,863] 72652664979 [3,3,41,83,487,4871] 73386496911 [3,7,13,23,29,467,863] 73659354549 [3,11,83,4871,5521] 74708386445 [5,7,11,23,71,331,359] 76289956743 [3,7,11,13,23,47,71,331] 77071265271 [3,3,7,11,13,23,431,863] 82626071415 [3,5,101,2903,18787] 83060876855 [5,11,71,179,331,359] 83246487753 [3,11,13,23,71,331,359] 83508772347 [3,3,7,11,13,23,467,863] 84231792021 [3,13,23,233,467,863] 85473818529 [3,11,71,307,331,359] 88629852235 [5,19,83,821,13691] 88797701641 [11,41,83,487,4871] 89533936297 [13,23,467,743,863] 89727875545 [5,7,11,47,71,211,331] 91516167773 [41,83,4871,5521] 91598978163 [3,7,11,47,71,331,359] 94113060899 [11,13,23,71,467,863] 97460892045 [3,3,5,11,83,487,4871] 100322690369 [11,23,47,71,331,359] 100426194141 [3,7,13,23,43,431,863] 102599897829 [3,11,13,23,863,12049] 103556985243 [3,37,83,821,13691] 107288768495 [5,23,83,821,13691] 108818073455 [5,7,13,23,863,12049] 111042711087 [3,7,11,29,41,83,4871] 115321638003 [3,11,13,23,29,467,863] 121087774965 [3,5,41,83,487,4871] 122269119111 [3,3,3,23,83,487,4871] 122310828185 [5,7,13,23,29,467,863] 124081793129 [7,19,83,821,13691] 124182233587 [83,409,751,4871] 127149927905 [5,7,11,13,23,47,71,331] 128452108785 [3,5,7,11,13,23,431,863] 130895114559 [3,11,83,4871,9811] 137541691479 [3,83,151,751,4871] 138744146255 [5,11,13,23,71,331,359] 139181287245 [3,5,7,11,13,23,467,863] 142076462385 [3,5,11,13,23,47,71,863] 142456364215 [5,11,71,307,331,359] 149440034469 [3,11,23,83,487,4871] 152664963605 [5,7,11,47,71,331,359] 154015014057 [3,23,83,4871,5521] 157812590793 [3,11,13,23,43,431,863] 162627263543 [41,83,4871,9811] 167376990235 [5,7,13,23,43,431,863] 170760547591 [31,101,2903,18787] 170999829715 [5,11,13,23,863,12049] 172594975405 [5,37,83,821,13691] 176610949827 [3,13,23,83,487,4871] 181777357113 [3,11,101,2903,18787] 185667921613 [23,41,83,487,4871] 186728786445 [3,5,41,83,751,4871] 191253891665 [5,41,83,821,13691] 192202730005 [5,11,13,23,29,467,863] 194187992295 [3,5,11,41,71,83,4871] 194241804757 [7,11,13,23,71,331,359] 195872532219 [3,3,7,13,23,863,12049] 196540848049 [7,13,23,233,467,863] 198907047339 [3,7,11,13,23,47,71,863] 199438909901 [7,11,71,307,331,359] 200116208721 [3,11,13,23,71,331,863] 203781865185 [3,3,5,23,83,487,4871] 206317331467 [13,53,467,743,863] 207920510787 [3,11,1657,1667,2281] 215198987445 [3,3,5,13,23,43,431,863] 217376374429 [83,233,821,13691] 217957994937 [3,3,3,41,83,487,4871] 220159490733 [3,3,7,13,23,29,467,863] 222605747265 [3,5,11,47,71,83,4871] 224125159335 [3,5,7,11,23,71,331,359] 229236152465 [5,83,151,751,4871] 230437615811 [13,19,83,821,13691] 234103031599 [29,41,83,487,4871] 236035290689 [11,23,83,821,13691] 239399761601 [7,11,13,23,863,12049] 249066724115 [5,11,23,83,487,4871] 249182630565 [3,5,11,71,179,331,359] 250489835475 [3,5,5,11,83,751,4871] 255806986799 [7,7,13,19,43,491531] 256421455587 [3,3,11,71,307,331,359] 256691690095 [5,23,83,4871,5521] 259684566245 [5,13,23,431,467,863] 260574277821 [3,11,13,23,71,431,863] 263020984655 [5,11,13,23,43,431,863] 265889556705 [3,5,19,83,821,13691] 266134975171 [13,23,431,863,2393] 266393104923 [3,11,41,83,487,4871] 268601808891 [3,13,23,467,743,863] 269083822007 [7,11,13,23,29,467,863] 269183626635 [3,5,7,11,47,71,211,331] 271863189213 [3,7,11,41,71,83,4871] 273353123209 [83,293,821,13691] 274548503319 [3,41,83,4871,5521] 278950798087 [13,23,83,821,13691] 278965251015 [3,5,11,31,2903,18787] 282339182697 [3,11,13,23,71,467,863] 282406981307 [11,17,71,179,331,359] 292382676135 [3,3,3,5,11,83,487,4871] 293729246085 [3,5,11,71,211,331,359] 294351583045 [5,13,23,83,487,4871] 294820162505 [5,11,23,47,71,211,331] 300968071107 [3,11,23,47,71,331,359] 301278582423 [3,3,7,13,23,43,431,863] 302962261855 [5,11,101,2903,18787] 308849904077 [11,13,23,233,467,863] 311214644075 [5,5,41,83,751,4871] 311648046171 [3,7,11,47,71,83,4871] 313542563169 [3,11,71,83,331,4871] 321331698489 [3,17,1657,1667,2281] 326454220365 [3,5,7,13,23,863,12049] 331511745565 [5,7,11,13,23,47,71,863] 333527014535 [5,11,13,23,71,331,863] 346534184645 [5,11,1657,1667,2281] 348855682791 [3,7,11,71,179,331,359] 359135427723 [3,19,1657,1667,2281] 363263324895 [3,3,5,41,83,487,4871] 363558392743 [7,13,23,431,467,863] 366932484555 [3,5,7,13,23,29,467,863] 368229378517 [7,11,13,23,43,431,863] 368296772745 [3,5,11,83,4871,5521] 372245379387 [3,7,19,83,821,13691] 372546700761 [3,83,409,751,4871] 381449783715 [3,5,7,11,13,23,47,71,331] 382080033533 [11,23,71,179,331,359] 385356326355 [3,3,5,7,11,13,23,431,863] 407697320281 [19,23,83,821,13691] 410803330179 [3,11,41,83,751,4871] 416232438765 [3,5,11,13,23,71,331,359] 417543861735 [3,3,5,7,11,13,23,467,863] 427369092645 [3,5,11,71,307,331,359] 432350529907 [83,109,4871,9811] 434290463035 [5,11,13,23,71,431,863] 434742886191 [3,23,1657,1667,2281] 443988508205 [5,11,41,83,487,4871] 447669681485 [5,13,23,467,743,863] 448320103407 [3,3,11,23,83,487,4871] 450384843997 [11,23,71,211,331,359] 457580838865 [5,41,83,4871,5521] 457994890815 [3,5,7,11,47,71,331,359] 459945372347 [73,1657,1667,2281] 461314889013 [3,11,71,83,487,4871] 466937820349 [7,11,13,23,71,331,863] 470565304495 [5,11,13,23,71,467,863] 473437772379 [3,3,11,13,23,43,431,863] 484351638331 [11,17,71,307,331,359] 485147858503 [7,11,1657,1667,2281] 487881790629 [3,41,83,4871,9811] 492566571263 [13,23,43,103,431,863] 497556692711 [13,43,431,863,2393] 501613451845 [5,11,23,47,71,331,359] 502130970705 [3,5,7,13,23,43,431,863] 512281642773 [3,31,101,2903,18787] 512999489145 [3,5,11,13,23,863,12049] 517784926215 [3,5,37,83,821,13691] 520698864465 [3,3,5,13,431,863,2393] 529832849481 [3,3,13,23,83,487,4871] 535552830815 [5,17,1657,1667,2281] 541334184017 [11,19,71,307,331,359] 555213555435 [3,5,7,11,29,41,83,4871] 557003764839 [3,23,41,83,487,4871] 564721718209 [11,23,83,4871,5521] 576608190015 [3,5,11,13,23,29,467,863] 581426137985 [5,7,11,71,179,331,359] 582725414271 [3,7,11,13,23,71,331,359] 589622544147 [3,7,13,23,233,467,863] 596721142017 [3,3,7,11,13,23,47,71,863] 598316729703 [3,7,11,71,307,331,359] 598559046205 [5,19,1657,1667,2281] 600348626163 [3,3,11,13,23,71,331,863] 611160289283 [97,1657,1667,2281] 611345595555 [3,3,3,5,23,83,487,4871] 618292157881 [83,277,4871,5521] 618951994401 [3,13,53,467,743,863] 620408965645 [5,7,19,83,821,13691] 620911167935 [5,83,409,751,4871] 626737554079 [7,13,23,467,743,863] 636545178245 [5,23,83,4871,13691] 647573482699 [11,13,23,83,487,4871] 652129123287 [3,83,233,821,13691] 654475572795 [3,5,11,83,4871,9811] 655299275389 [11,23,71,307,331,359] 655860906539 [19,37,83,821,13691] 658791426293 [7,11,13,23,71,467,863] 661565261595 [3,5,7,1657,1667,2281] 667398394247 [13,23,83,4871,5521] 667825903945 [5,31,79,2903,18787] 687708457395 [3,5,83,151,751,4871] 691312847433 [3,13,19,83,821,13691] 702258832583 [7,11,23,47,71,331,359] 702309094797 [3,29,41,83,487,4871] 711391132749 [3,11,71,83,751,4871] 712281821075 [5,5,11,71,307,331,359] 718199284803 [3,7,11,13,23,863,12049] 724571476985 [5,23,1657,1667,2281] 726764788327 [19,41,83,821,13691] 728874379241 [83,4871,1802837] 747200172345 [3,5,11,23,83,487,4871] 749773963141 [7,17,1657,1667,2281] 762216729221 [19,43,83,821,13691] 765668261779 [101,139,2903,18787] 766532422513 [11,13,23,47,71,211,331] 767420960397 [3,7,7,13,19,43,491531] 768858148355 [5,11,71,83,487,4871] 770075070285 [3,5,23,83,4871,5521] 779053698735 [3,5,13,23,431,467,863] 780381041063 [13,23,251,863,12049] 780772242437 [11,47,71,179,331,359] 781722833463 [3,3,11,13,23,71,431,863] 789062953965 [3,5,11,13,23,43,431,863] 793936886863 [23,37,83,821,13691] 798404925513 [3,13,23,431,863,2393] 799179314769 [3,3,11,41,83,487,4871] 807251466021 [3,7,11,13,23,29,467,863] 813136317715 [5,41,83,4871,9811] 814240656203 [13,23,29,233,467,863] 836852394261 [3,13,23,83,821,13691] 837982664687 [7,19,1657,1667,2281] 847220943921 [3,11,17,71,179,331,359] 847595706315 [3,5,431,2393,54787] 853802737955 [5,31,101,2903,18787] 879767901659 [23,41,83,821,13691] 883054749135 [3,5,13,23,83,487,4871] 884460487515 [3,5,11,23,47,71,211,331] 890880741647 [13,23,467,863,7393] 900988880077 [11,13,1657,1667,2281] 908886785565 [3,5,11,101,2903,18787] 926549712231 [3,11,13,23,233,467,863] 928339608065 [5,23,41,83,487,4871] 933643932225 [3,5,5,41,83,751,4871] 938792609311 [149,1657,1667,2281] 963995095467 [3,3,17,1657,1667,2281] 971209023785 [5,7,11,13,23,71,331,359] 979362661095 [3,3,5,7,13,23,863,12049] 982704240245 [5,7,13,23,233,467,863] 984873299267 [11,13,23,467,743,863] 994535236695 [3,5,7,11,13,23,47,71,863] 997194549505 [5,7,11,71,307,331,359] 1000581043605 [3,5,11,13,23,71,331,863] 1014400067779 [7,23,1657,1667,2281] 1024007460655 [5,37,83,4871,13691] 1031586657335 [5,13,53,467,743,863] 1039602553935 [3,5,11,1657,1667,2281] 1076401407697 [7,11,71,83,487,4871] 1077406283169 [3,3,19,1657,1667,2281] 1086881872145 [5,83,233,821,13691] 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