A near Wieferich prime is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

p satisfying the congruence 2^(p−1)/2 ≡ ±1 + Ap (mod p²) with small |A|. Near-Wieferich primes with A = 0 represent Wieferich primes. Recent searches, in addition to their primary search for Wieferich primes, also tried to find near-Wieferich primes. The following table lists all near-Wieferich primes with |A| < 100 up to 3. This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.

p	1 or −1	A
1093	+1	0
3511	+1	0
1222336487	+1	+60
1259662487	+1	−71
1274153897	+1	−86
1494408397	−1	+52
1584392531	−1	−24
1586651309	−1	−24
1662410923	−1	−70
1817972423	+1	−56
1890830857	+1	+69
2062661389	−1	+55
2244893621	−1	+47
2332252547	−1	−33
2416644757	−1	+67
2461090421	−1	+47
2566816313	+1	+52
2570948153	+1	−41
2589186937	+1	−85
2709711233	+1	+50
2760945133	−1	−77
2954547209	+1	+32
3027263587	−1	−95
3133652447	+1	+87
3303616961	+1	−20
3520624567	+1	−6
3606693551	+1	+21
4449676157	−1	−15
5045920247	+1	+76
5409537149	−1	+66
8843450093	−1	−20
10048450537	+1	+82
10329891503	+1	+79
11214704947	−1	+56
20051397221	−1	−46
20366156849	+1	−95
36673326289	+1	−45
46262476201	+1	+5
47004625957	−1	+1
49819566449	+1	+27
53359191887	+1	+50
58481216789	−1	+5
76843523891	−1	+1
82834772291	−1	+82
108058158839	+1	+58
130861186019	−1	−97
138528575509	−1	−23
239398882511	+1	−72
252074060191	+1	+61
252137567497	+1	−31
299948374351	+1	−19
405897532891	−1	+61
443168739911	+1	+64
504568016327	+1	+55
703781283787	−1	−49
955840782881	+1	−84
980377925057	+1	−90
981086885117	−1	+85
1095406033573	−1	+42
1104406423781	−1	+54
1180032105761	+1	−6
1722721869859	−1	−31
1730418792409	+1	−46
1780536689159	+1	+84
2207775149407	+1	−99
2424653846701	−1	−51
2610372685663	+1	−28
3667691800441	+1	+27
3713054321579	−1	−51
3729819224423	+1	+77
4006528141163	−1	+17
4169357937293	−1	−27
5216344035949	−1	+93
5240305919047	+1	−95
7355288787229	−1	−68
7876427903107	−1	−48
8851776421399	+1	−81
11344191252809	+1	−92
12456646902457	+1	+2
15056776355693	−1	−19
23639424831877	−1	−48
24990087401551	+1	+16
28506780213511	+1	−28
28785529445977	+1	+33
29230410915073	+1	+96
30189412701163	−1	−37
30309769394167	+1	+28
63735899194511	+1	−16
63918629031731	−1	+38
67961346537659	−1	−49
68132247624521	+1	−55
92226580839683	−1	−76
118485210646981	−1	−90
134257821895921	+1	+10
153332502585091	−1	+59
181841793213263	+1	+90
205250817470827	−1	−78
259990715684839	+1	−12
339258218134349	−1	+2
342092449620191	+1	+90
346412396858131	−1	−48
362061154308767	+1	−64
694936752678643	−1	+75
696740841781447	+1	+61
734180764265903	+1	+37
739507312099561	+1	−78
765760560131939	−1	+38
1140417231387373	−1	−82
1170553064286511	+1	−84
1302848719581529	+1	+38
1680898792774051	−1	+48
1885825033325021	−1	−79
2276306935816523	−1	−3