Thema: Doppelpaar Wieferich-Primzahlen mit der Basis B und der Primzahl P (double Wieferich prime pairs) B^(P-1) == 1 (mod P^2) und P^(B-1) == 1 (mod B^2) Main table of content: http://www.fermatquotient.com/ Basis Primzahl^2 B*P 2 1093 2186 3 1006003 3018009 5 1645333507 8226667535 5 188748146801 943740734005 83 4871 404293 911 318917 290533387 2903 18787 54538661 Bisher entdeckte Nahe-Doppel-Wieferich-Paare mit kleinen Resten Abs(R) <= 10*P+1 Basis B ; Primzahl P>100 ; B^[(P-1)/2] == R (mod P^2) (2 5 R = P-1) 2 569 R = 9*P+1 2 1733 R = -4*P-1 2 2633 R = 2*P+1 2 15329 R = 6*P+1 2 546097 R = 5*P+1 2 2549177 R = 6*P+1 2 110057537 R = -P+1 2 209227901 R = 8*P-1 2 671499313 R = -5*P+1 2 46262476201 R = 5*P+1 2 47004625957 R = P-1 2 58481216789 R = 5*P-1 2 1180032105761 R = -6*P+1 2 12456646902457 R = 2*P+1 2 134257821895921 R = 10*P+1 2 339258218134349 R = 2*P-1 (2 3723113065138349 R = 18*P-1) (2 4150209531584437 R = -24*P-1) (2 5131427559624857 R = -18*P+1) (2 6517506365514181 R = -29*P-1) (2 20903942824901597 R = -48*P-1) 2 82687771042557349 R = -10*P-1 (2 115341564749415161 R = 34*P+1) (2 145172216558999221 R = 26*P-1) (2 480637248454682153 R = -41*P+1) (2 550129374002911793 R = -251*P+1) 3 109 R = -4*P+1 3 269 R = 5*P-1 3 523 R = 3*P-1 3 1097 R = 7*P-1 3 2357 R = -7*P-1 3 2593 R = -10*P+1 3 2789 R = -2*P-1 3 2861 R = -P-1 3 29789 R = -8*P-1 3 163061 R = 7*P-1 3 187559 R = 7*P+1 3 308899 R = 5*P-1 3 1386773 R = 2*P-1 3 9501101 R = 9*P-1 3 15978511 R = P-1 3 90614762099 R = 8*P+1 3 39526424220761 R = 10*P-1 3 61629351935149 R = P+1 (3 183022968916817 R = 49*P-1) (3 229551010019207 R = 44*P+1) (3 408980446804529 R = 38*P-1) (3 615112232160347 R = 45*P+1) (3 1012090335713603 R = 115*P+1) (3 2079977499026767 R = 51*P-1) (3 2402852024873467 R = -91*P-1) (3 3321907273300807 R = 122*P-1) 5 199 R = -2*P+1 5 349 R = -3*P+1 5 457 R = -9*P-1 5 2957 R = -4*P-1 5 2503720796399 R = -8*P+1 (5 492829739161393 R = -73*P-1) (5 571956190367207 R = 75*P-1) (5 1032351331230607 R = 34*P-1) (5 1411237230610049 R = -69*P+1) 7 15551 R = -9*P-1 7 119297 R = -8*P-1 7 775639 R = 3*P-1 7 8726803 R = -10*P-1 7 3018906857 R = 7*P+1 (7 754753733881711 R = -58*P-1) (11 3 R = P-1) 11 239 R = -3*P+1 11 3631 R = 8*P-1 11 3863 R = -8*P+1 11 4826814791 R = -10*P+1 11 14664628387 R = -3*P+1 11 47090999897 R = -2*P-1 11 168377484659 R = 2*P+1 (11 129631798750637 R = -44*P-1) (11 244936278919987 R = 30*P+1) (11 755870372079973 R = -81*P-1) (11 813642522621493 R = -54*P+1) 13 8597 R = -6*P+1 13 74761397 R = 5*P+1 (13 290662065849773 R = -64*P-1) 17 251 R = -9*P+1 17 3371399 R = 2*P-1 19 127 R = -9*P+1 19 945391 R = -3*P+1 23 7583 R = -6*P-1 23 16391339 R = 5*P+1 (23 207061886077819 R = -35*P+1) 29 2939 R = -5*P-1 31 10454806891 R = -8*P+1 (37 69605729752387 R = -41*P+1) 41 3049 R = -8*P-1 (41 174876270954911 R = -47*P-1) (41 376756147986209 R = 97*P+1) (43 19 R = 2*P+1) 47 22968479591 R = 5*P-1 53 212790645247 R = -P-1 61 580477 R = -5*P+1 (79 52637650750123 R = -15*P+1) 83 293 R = 5*P+1 83 127110119 R = 4*P+1 83 192034559 R = -7*P+1 89 4903 R = 9*P+1 89 7529177 R = 5*P+1 (101 83101912174733 R = 81*P-1) 107 1125950115061 R = -7*P+1 149 3538990931981 R = 3*P+1 151 388070033147 R = 10*P-1 167 94360929809777 R = 2*P-1 181 826939 R = 6*P-1 191 5867 R = -10*P+1 199 8981205221 R = 9*P-1 241 8681 R = 6*P+1 293 18593 R = -8*P-1 313 669271 R = -10*P-1 (359 53217502874509 R = -28*P-1) 401 2131 R = 3*P+1 (479 40079294340583 R = -32*P+1) 523 2971 R = -8*P-1 541 244005992723 R = 8*P-1 673 455145767 R = 7*P+1 773 3001 R = 9*P+1 1151 5295343 R = 7*P-1 1279 683 R = 9*P-1 (1301 165761878266251 R = 42*P+1) 1583 2269 R = -5*P-1 1619 2021251 R = -10*P+1 1709 97821393408467 R = -10*P+1 2221 659 R = -7*P+1 2803 2131 R = 2*P+1 (2879 44752556049379 R = 35*P-1) 2953 656634313 R = -6*P+1 3911 3187 R = 7*P-1 5827 270506645441 R = P+1 6133 4134311373869 R = 8*P-1 6733 137 R = -7*P-1 11059 256469 R = 2*P+1 13049 14326961641 R = 8*P-1 17351 57338793569 R = P-1 (19559 127923899282227 R = -84*P+1) (20887 31276549111967 R = -57*P-1) 23549 151133681 R = -3*P+1 30773 434377 R = -5*P-1 (37441 127239991056863 R = -68*P+1) (40487 5 R = -P-1) 46861 2928463 R = -P-1 50957 160639 R = 7*P+1 51673 138007 R = -7*P-1 86197 1801 R = 7*P-1 94873 766574317 R = 6*P-1 (107507 143699237788787 R = 63*P-1) 143537 4973 R = 5*P+1 (192103 24181638413587 R = -68*P-1) 208729 42884747 R = 2*P+1 231323 657983 R = 2*P-1 (266291 13473643983643 R = -28*P-1) (345601 26282496590083 R = 49*P+1) 379133 191 R = 4*P-1 530701 187335405683317 R = -5*P-1 537221 509413549 R = 9*P+1 750077 7159 R = 7*P-1 (1012573 79 R = P-1) (1209379 43299039237551 R = 35*P+1) 1610251 691193 R = 2*P+1 (1975163 148169567669687 R = 90*P+1) 2163971 51503 R = -10*P-1 (2481757 23 R = -P-1) 3924731 4357 R = -3*P-1 4242923 157 R = -7*P-1 5168309 30820477 R = -8*P+1 6149761 23429215331 R = 9*P-1 (9744481 99602590528093 R = 65*P+1) 11027333 13978673 R = 3*P+1 12588133 2872033819 R = -9*P+1 (14438887 31296163893061 R = -65*P+1) 29843371 1547245666933 R = -7*P+1 32017411 12473 R = -2*P-1 50871187 27708041 R = 4*P-1 (53471161 5 R = -P+1) 58517909 7607 R = 9*P-1 68788403 20021 R = 5*P-1 69310961 42372293239993 R = -9*P+1 (89377243 161890761699391 R = -74*P+1) 94457777 10256419 R = P+1 (108986699 74856481275661 R = 39*P-1) (131871013 5359581605767 R = -43*P-1) (138246593 43215198318769 R = 92*P+1) (188720783 18807203836727 R = 98*P-1) (203638433 1410328356529 R = -80*P+1) 221448961 223019471 R = -7*P+1 (274516927 2288703249299 R = 39*P-1) (344154523 62198713145977 R = -17*P+1) (382112833 29945869 R = -39*P+1) (386109631 75803113 R = -56*P-1) (470105899 2260673971 R = -74*P+1) 501070837 1045649123 R = 2*P-1 (520941667 19747901 R = -46*P-1) (548984147 54842060798143 R = 49*P+1) (564954157 10643484113653 R = -58*P+1) (576005341 6588257692673 R = 49*P+1) (625492547 2721869 R = -66*P-1) (666504107 2404190355227 R = -22*P-1) (777105521 139175998573121 R = -196*P+1) (846866039 33315254741129 R = -79*P+1) (935839703 113226661294651 R = -179*P-1) ... 3160333793 32633 R = 4*P+1 (6692367337 5 R = -P-1) 7215975149 509 R = -6*P+1 40325433091 1051 R = -8*P+1 243547988443 277 R = -3*P+1 275318049829 157 R = 9*P+1 988813725773 3037 R = -9*P-1 Berechnung der Erwartungswerte (expectation values): Näherung bis zu einen Produkt X = B*P Erwartungswert = 0.53*[ln(ln X)]^2 - 0.424 Gegenwärtig durchsucht bis X = 1E+16, liefert einen Erwartungswert von 6.47 Näherung bis zu einem Rechteck B*P und P>=B S = Summe[1/P(i)^2] = 0.452247420041065... ; P(i) entspricht der i-Primzahl M = 0.261497212847642... (Mertenskonstante) Erwartungswert = {[ln(ln P) + M]^2 - [ln(ln P) - ln(ln B)]^2 - S} / 2 Gegenwärtig durchsucht bis B = 1000000007 und P = 2E+14, liefert einen Erwartungswert von 6.72 Dies ist nicht gerade erfolgsversprechend, ein weiteres Doppel-Wieferich-Paar zu finden. 13.05.2022 Richard Fischer