An interesting problem is finding primes p such that 2p-1 = 1 (mod p2). The only known such primes are 1093 and 3511. It is now known, from the contributions of several researchers, there are no other solutions less than 5 * 1014. That is a lot of calculation!
Update: New search underway by other researches, see http://borg.cs.dal.ca/~knauer/wieferich/
Wieferich primes are named after a brilliant mathematician A. Wieferich and owe a majority of their fame to a connection with FLT (Fermat's Last Theorem) which Wieferich demonstrated in 1909. If a counter-example exponent p were to exist for FLT, it must be a Wieferich prime, satisfying the criteria ap-1 = 1 (mod p2) for a=2. Mirimanoff in 1910 showed it must also be true for a=3. Searching for a prime to satisfy both of these was probably a subject of much interest, until Granville-Monagan demonstrated in 1988 FLT is true for all exponents below 714,591,416,091,389 which is even today pretty much out of reach without an impressive parallel effort. Of course today, since Andrew Wiles and modern elliptic curve theory, it is generally accepted that there is indeed no solution to FLT. However, there is not much known about the curious Wieferich primes. Are there only two?
The only known Wieferich primes are 1093 and 3511. Because of recent search efforts, the next Wieferich prime (if one exists) must be greater than 5 * 1014. Here is the collection of "close calls" with |A|<100 (where 2(p-1)/2 mod p2 = +/-1 + Ap) and p>109:
1222336487 +1 + 60 p 1259662487 +1 - 71 p 1274153897 +1 - 86 p 1494408397 -1 + 52 p 1584392531 -1 - 24 p 1586651309 -1 - 24 p 1662410923 -1 - 70 p 1817972423 +1 - 56 p 1890830857 +1 + 69 p 2062661389 -1 + 55 p 2244893621 -1 + 47 p 2332252547 -1 - 33 p 2416644757 -1 + 67 p 2461090421 -1 + 47 p 2566816313 +1 + 52 p 2570948153 +1 - 41 p 2589186937 +1 - 85 p 2709711233 +1 + 50 p 2760945133 -1 - 77 p 2954547209 +1 + 32 p 3027263587 -1 - 95 p 3133652447 +1 + 87 p 3303616961 +1 - 20 p 3520624567 +1 - 6 p 3606693551 +1 + 21 p 4449676157 -1 - 15 p 5045920247 +1 + 76 p 5409537149 -1 + 66 p 8843450093 -1 - 20 p 10048450537 +1 + 82 p 10329891503 +1 + 79 p 11214704947 -1 + 56 p 20051397221 -1 - 46 p 20366156849 +1 - 95 p 36673326289 +1 - 45 p 46262476201 +1 + 5 p 47004625957 -1 + 1 p 49819566449 +1 + 27 p 53359191887 +1 + 50 p 58481216789 -1 + 5 p 76843523891 -1 + 1 p 82834772291 -1 + 82 p 108058158839 +1 + 58 p 130861186019 -1 - 97 p 138528575509 -1 - 23 p 239398882511 +1 - 72 p 252137567497 +1 - 31 p 252074060191 +1 + 61 p 299948374351 +1 - 19 p 405897532891 -1 + 61 p 443168739911 +1 + 64 p 504568016327 +1 + 55 p 703781283787 -1 - 49 p 955840782881 +1 - 84 p 980377925057 +1 - 90 p 981086885117 -1 + 85 p 1095406033573 -1 + 42 p 1104406423781 -1 + 54 p 1180032105761 +1 - 6 p 1722721869859 -1 - 31 p 1730418792409 +1 - 46 p 1780536689159 +1 + 84 p 2207775149407 +1 - 99 p 2424653846701 -1 - 51 p 2610372685663 +1 - 28 p 3667691800441 +1 + 27 p 3713054321579 -1 - 54 p 3729819224423 +1 + 77 p 4006528141163 -1 + 17 p 4169357937293 -1 - 27 p 5216344035949 -1 + 93 p 5240305919047 +1 - 95 p 7355288787229 -1 - 68 p 7876427903107 -1 - 48 p 8851776421399 +1 - 81 p 11344191252809 +1 - 92 p 12456646902457 +1 + 2 p 15056776355693 -1 - 19 p 23639424831877 -1 - 48 p 24990087401551 +1 + 16 p 28506780213511 +1 - 28 p 28785529445977 +1 + 33 p 29230410915073 +1 + 96 p 30189412701163 -1 - 37 p 30309769394167 +1 + 28 p 63735899194511 +1 - 16 p 63918629031731 -1 + 38 p 67961346537659 -1 - 49 p 68132247624521 +1 - 55 p 92226580839683 -1 - 76 p 118485210646981 -1 - 90 p 134257821895921 +1 + 10 p 153332502585091 -1 + 59 p 181841793213263 +1 + 90 p
Click here to read an older update.
(a) - 1/20/2001: Wieferich primes also satisfy 2p2 = 2 (mod p2).
(b) - 1/20/2001: Similarly, Let 2d = 1 (mod p), then 2d = 1 (mod p2).
(c) - 1/20/2001: Binary representations of (p-1) for known Wieferich primes have a repeating pattern, although this is most likely coincedence.
(d) - 1/20/2001: Factors of (p-1) of known Wieferich primes are all small (<= 13), but this is also most likely coincedence given that the Wieferich primes themselves are small.
(e) - 1/20/2001: Factorization of 2p-1-1 for which p2 divides...
2^1092-1 = 32 * 5 * 72 * 132 * 29 * 43 * 53 * 79 * 113 * 127 * 157 * 313 * 337 * 547 * 911 * (((10932))) * 1249 * 1429 * 1613 * 2731 * 3121 * 4733 * 5419 * 8191 * 14449 * 21841 * 121369 * 224771 * 503413 * 1210483 * 1948129 * 22366891 * 108749551 * 112901153 * 23140471537 * 25829691707 * 105310750819 * 467811806281 * 4093204977277417 * 8861085190774909 * 556338525912325157 * 86977595801949844993 * 275700717951546566946854497 * 292653113147157205779127526827 * 3194753987813988499397428643895659569
2^3510-1 = 34 * 7 * 11 * 19 * 31 * 73 * 79 * 131 * 151 * 271 * 331 * 631 * 811 * 937 * 1171 * 2731 * (((35112))) * 6553 * 8191 * 10531 * 15121 * 23311 * 65521 * 86113 * 87211 * 107251 * 121369 * 262657 * 348031 * 409891 * 446473 * 1024921 * 1969111 * 4633201 * 7623851 * 18837001 * 22366891 * 29121769 * 409251961 * 7830118297 * 2400314671 * 26959262851 * 21497866557571 * 49971617830801 * 145295143558111 * 385838642647891 * 571890896913727 * 93715008807883087 * 194900834792501371 * 339175003117573351 * 4242734772486358591 * 85488365519409100951 * 150832426800173710177 * 255375215316698521591 * 1439538040790707946401 * 5302306226370307681801 * 571403921126076957182161 * 4247713303224552237738169 * 43725552532343303477113703251 * 134304196845099262572814573351 * 2728334536034592865339299805712535332071 * 4897406518564079146139572699835240681611 * 24841125429051585062538961751269988364169 * (two composites)
Composite factor 1 = 994499868210136010981361559557803621721545129593921219614629055332298029738879358633257877455053636767566800834052466239516617923218534966326533248229553609904541805972325270693726491
Composite factor 2 = 13998469549388339590468017073657389503840227715447303027205450636561786721059586640554982761800971743769332153370421665868047306056094647597984314626970971405894583302525615193026605012021992879604120513454431
Note: If you find a factor for one of these, please send it to me and I'll credit you
with its discovery. Be aware thouh, I've run 1120 ECM curves at B1=1e6 for composite 1, and 340 curves
with B1=1e6 for composite 2. Rich Brown
extended this by running 5100 curves with B1=3e6 on both composites, and 10600 curves with B1=11e6 on composite 2.
I plan on starting another Wieferich search before long. I believe I can extend my sieve method from the 2n = c (mod n) problem to Wieferich primes, by analyzing the Wieferich quotient.
Another interesting thing about the Wieferich quotient is that GCD(Q(1093), Q(3511))=278-1, surely due to both 1093 and 3511 being equal to 1 mod 78. This information may prove helpful in writing up an attack to find a new Wieferich prime.
If you have any new facts or tidbits regarding Wieferich primes, please send me an email so I can add it to the site. Let's find another Wieferich prime!
Here are some miscellaneous graphs I've made regarding the distribution of A's.