Named after | Arthur Wieferich |
---|---|

Publication year | 1909 |

Author of publication | Wieferich, A. |

No. of known terms | 2 |

Conjectured no. of terms | Infinite |

Subsequence of | Crandall numbers^{[1]}Wieferich numbers ^{[2]}Lucas–Wieferich primes ^{[3]}near-Wieferich primes |

First terms | 1093, 3511 |

Largest known term | 3511 |

OEIS index | A001220 |

In number theory, a **Wieferich prime** is a prime number *p* such that *p*^{2} divides 2^{p − 1} − 1,^{[4]} therefore connecting these primes with Fermat's little theorem, which states that every odd prime *p* divides 2^{p − 1} − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.^{[5]}^{[6]}

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the *abc* conjecture.

As of April 2022^{[update]}, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).

The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation 2^{p -1} ≡ 1 (mod *p*^{2}). From the definition of the congruence relation on integers, it follows that this property is equivalent to the definition given at the beginning. Thus if a prime *p* satisfies this congruence, this prime divides the Fermat quotient . The following are two illustrative examples using the primes 11 and 1093:

- For
*p*= 11, we get which is 93 and leaves a remainder of 5 after division by 11, hence 11 is not a Wieferich prime. For*p*= 1093, we get or 485439490310...852893958515 (302 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.

Wieferich primes can be defined by other equivalent congruences. If *p* is a Wieferich prime, one can multiply both sides of the congruence 2^{p−1} ≡ 1 (mod *p*^{2}) by 2 to get 2^{p} ≡ 2 (mod *p*^{2}). Raising both sides of the congruence to the power *p* shows that a Wieferich prime also satisfies 2^{p2} ≡2^{p} ≡ 2 (mod *p*^{2}), and hence 2^{pk} ≡ 2 (mod *p*^{2}) for all *k* ≥ 1. The converse is also true: 2^{pk} ≡ 2 (mod *p*^{2}) for some *k* ≥ 1 implies that the multiplicative order of 2 modulo *p*^{2} divides gcd(*p*^{k} − 1, φ(*p*^{2})) = *p* − 1, that is, 2^{p−1} ≡ 1 (mod *p*^{2}) and thus *p* is a Wieferich prime. This also implies that Wieferich primes can be defined as primes *p* such that the multiplicative orders of 2 modulo *p* and modulo *p*^{2} coincide: ord_{p2} 2 = ord_{p} 2, (By the way, ord_{1093}2 = 364, and ord_{3511}2 = 1755).

H. S. Vandiver proved that 2^{p−1} ≡ 1 (mod *p*^{3}) if and only if .^{[7]}^{: 187 }

In 1902, Meyer proved a theorem about solutions of the congruence *a*^{p − 1} ≡ 1 (mod *p*^{r}).^{[8]}^{: 930 }^{[9]} Later in that decade Arthur Wieferich showed specifically that if the first case of Fermat's last theorem has solutions for an odd prime exponent, then that prime must satisfy that congruence for *a* = 2 and *r* = 2.^{[10]} In other words, if there exist solutions to *x*^{p} + *y*^{p} + *z*^{p} = 0 in integers *x*, *y*, *z* and *p* an odd prime with *p* ∤ *xyz*, then *p* satisfies 2^{p − 1} ≡ 1 (mod *p*^{2}). In 1913, Bachmann examined the residues of . He asked the question when this residue vanishes and tried to find expressions for answering this question.^{[11]}

The prime 1093 was found to be a Wieferich prime by W. Meissner for all primes *p* < 2000 and found this residue to be zero for *t* = 364 and *p* = 1093, thereby providing a counterexample to a conjecture by Grave about the impossibility of the Wieferich congruence.^{[12]} E. Haentzschel later ordered verification of the correctness of Meissner's congruence via only elementary calculations.^{[13]}^{: 664 } Inspired by an earlier work of Euler, he simplified Meissner's proof by showing that 1093^{2} | (2^{182} + 1) and remarked that (2^{182} + 1) is a factor of (2^{364} − 1).^{[14]} It was also shown that it is possible to prove that 1093 is a Wieferich prime without using complex numbers contrary to the method used by Meissner,^{[15]} although Meissner himself hinted at that he was aware of a proof without complex values.^{[12]}^{: 665 }

The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922^{[16]} and another proof of it being a Wieferich prime was published in 1965 by Guy.^{[17]} In 1960, Kravitz^{[18]} doubled a previous record set by Fröberg^{[19]} and in 1961 Riesel extended the search to 500000 with the aid of BESK.^{[20]} Around 1980, Lehmer was able to reach the search limit of 6×10^{9}.^{[21]} This limit was extended to over 2.5×10^{15} in 2006,^{[22]} finally reaching 3×10^{15}. It is now known that if any other Wieferich primes exist, they must be greater than 6.7×10^{15}.^{[23]}

In 2007–2016, a search for Wieferich primes was performed by the distributed computing project Wieferich@Home.^{[24]} In 2011–2017, another search was performed by the PrimeGrid project, although later the work done in this project was claimed wasted.^{[25]} While these projects reached search bounds above 1×10^{17}, neither of them reported any sustainable results.

In 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously. The new project uses checksums to enable independent double-checking of each subinterval, thus minimizing the risk of missing an instance because of faulty hardware.^{[26]} As of March 2021,^{[update]} the leading edge of this search was over 3×10^{18}.^{[27]}

It has been conjectured (as for Wilson primes) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below *x* is approximately log(log(*x*)), which is a heuristic result that follows from the plausible assumption that for a prime *p*, the (*p* − 1)-th degree roots of unity modulo *p*^{2} are uniformly distributed in the multiplicative group of integers modulo *p*^{2}.^{[28]}

The following theorem connecting Wieferich primes and Fermat's Last Theorem was proven by Wieferich in 1909:^{[10]}

- Let
*p*be prime, and let*x*,*y*,*z*be integers such that*x*^{p}+*y*^{p}+*z*^{p}= 0. Furthermore, assume that*p*does not divide the product*xyz*. Then*p*is a Wieferich prime.

The above case (where *p* does not divide any of *x*, *y* or *z*) is commonly known as the first case of Fermat's Last Theorem (FLTI)^{[29]}^{[30]} and FLTI is said to fail for a prime *p*, if solutions to the Fermat equation exist for that *p*, otherwise FLTI holds for *p*.^{[31]}
In 1910, Mirimanoff expanded^{[32]} the theorem by showing that, if the preconditions of the theorem hold true for some prime *p*, then *p*^{2} must also divide 3^{p − 1} − 1. Granville and Monagan further proved that *p*^{2} must actually divide *m*^{p − 1} − 1 for every prime *m* ≤ 89.^{[33]} Suzuki extended the proof to all primes *m* ≤ 113.^{[34]}

Let *H _{p}* be a set of pairs of integers with 1 as their greatest common divisor,

A non-Wieferich prime is a prime *p* satisfying 2^{p − 1} ≢ 1 (mod *p*^{2}). J. H. Silverman showed in 1988 that if the *abc* conjecture holds, then there exist infinitely many non-Wieferich primes.^{[35]} More precisely he showed that the *abc* conjecture implies the existence of a constant only depending on *α* such that the number of non-Wieferich primes to base *α* with *p* less than or equal to a variable *X* is greater than log(*X*) as *X* goes to infinity.^{[36]}^{: 227 } Numerical evidence suggests that very few of the prime numbers in a given interval are Wieferich primes. The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by *W _{2}* and

It is known that the *n*th Mersenne number *M*_{n} = 2^{n} − 1 is prime only if *n* is prime. Fermat's little theorem implies that if *p* > 2 is prime, then *M*_{p−1} (= 2^{p − 1} − 1) is always divisible by *p*. Since Mersenne numbers of prime indices *M*_{p} and *M*_{q} are co-prime,

- A prime divisor
*p*of*M*_{q}, where*q*is prime, is a Wieferich prime if and only if*p*^{2}divides*M*_{q}.^{[40]}

- A prime divisor

Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free. If *q* is prime and the Mersenne number *M*_{q} is *not* square-free, that is, there exists a prime *p* for which *p*^{2} divides *M*_{q}, then *p* is a Wieferich prime. Therefore, if there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers with prime index that are not square-free. Rotkiewicz showed a related result: if there are infinitely many square-free Mersenne numbers, then there are infinitely many non-Wieferich primes.^{[41]}

Similarly, if *p* is prime and *p*^{2} divides some Fermat number *F*_{n} = 2^{2n} + 1, then *p* must be a Wieferich prime.^{[42]}

In fact, there exists a natural number *n* and a prime *p* that *p*^{2} divides (where is the *n*-th cyclotomic polynomial) if and only if *p* is a Wieferich prime. For example, 1093^{2} divides , 3511^{2} divides . Mersenne and Fermat numbers are just special situations of . Thus, if 1093 and 3511 are only two Wieferich primes, then all are square-free except and (In fact, when there exists a prime *p* which *p*^{2} divides some , then it is a Wieferich prime); and clearly, if is a prime, then it cannot be Wieferich prime. (Any odd prime *p* divides only one and *n* divides *p* − 1, and if and only if the period length of 1/p in binary is *n*, then *p* divides . Besides, if and only if *p* is a Wieferich prime, then the period length of 1/p and 1/p^{2} are the same (in binary). Otherwise, this is *p* times than that.)

For the primes 1093 and 3511, it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number, because 364 and 1755 are neither prime nor powers of 2.^{[43]}

Scott and Styer showed that the equation *p*^{x} – 2^{y} = *d* has at most one solution in positive integers (*x*, *y*), unless when *p*^{4} | 2^{ordp 2} – 1 if *p* ≢ 65 (mod 192) or unconditionally when *p*^{2} | 2^{ordp 2} – 1, where ord_{p} 2 denotes the multiplicative order of 2 modulo *p*.^{[44]}^{: 215, 217–218 } They also showed that a solution to the equation ±*a*^{x1} ± 2^{y1} = ±*a*^{x2} ± 2^{y2} = *c* must be from a specific set of equations but that this does not hold, if *a* is a Wieferich prime greater than 1.25 x 10^{15}.^{[45]}^{: 258 }

Johnson observed^{[46]} that the two known Wieferich primes are one greater than numbers with periodic binary expansions (1092 = 010001000100_{2}=444_{16}; 3510 = 110110110110_{2}=6666_{8}). The Wieferich@Home project searched for Wieferich primes by testing numbers that are one greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of 3500 of the tested binary numbers generated by combination of bit strings with a bit length of up to 24 it has not found a new Wieferich prime.^{[47]}

It has been noted (sequence A239875 in the OEIS) that the known Wieferich primes are one greater than mutually friendly numbers (the shared abundancy index being 112/39).

It was observed that the two known Wieferich primes are the square factors of all non-square free base-2 Fermat pseudoprimes up to 25×10^{9}.^{[48]} Later computations showed that the only repeated factors of the pseudoprimes up to 10^{12} are 1093 and 3511.^{[49]} In addition, the following connection exists:

- Let
*n*be a base 2 pseudoprime and*p*be a prime divisor of*n*. If , then also .^{[31]}^{: 378 }Furthermore, if*p*is a Wieferich prime, then*p*^{2}is a Catalan pseudoprime.

For all primes *p* up to 100000, *L*(*p*^{n+1}) = *L*(*p*^{n}) only in two cases: *L*(1093^{2}) = *L*(1093) = 364 and *L*(3511^{2}) = *L*(3511) = 1755, where *L*(*m*) is the number of vertices in the cycle of 1 in the *doubling diagram* modulo *m*. Here the doubling diagram represents the directed graph with the non-negative integers less than *m* as vertices and with directed edges going from each vertex *x* to vertex 2*x* reduced modulo *m*.^{[50]}^{: 74 } It was shown, that for all odd prime numbers either *L*(*p*^{n+1}) = *p* · *L*(*p*^{n}) or *L*(*p*^{n+1}) = *L*(*p*^{n}).^{[50]}^{: 75 }

It was shown that and if and only if 2^{p − 1} ≢ 1 (mod *p*^{2}) where *p* is an odd prime and is the fundamental discriminant of the imaginary quadratic field . Furthermore, the following was shown: Let *p* be a Wieferich prime. If *p* ≡ 3 (mod 4), let be the fundamental discriminant of the imaginary quadratic field and if *p* ≡ 1 (mod 4), let be the fundamental discriminant of the imaginary quadratic field . Then and (*χ* and *λ* in this context denote Iwasawa invariants).^{[51]}^{: 27 }

Furthermore, the following result was obtained: Let *q* be an odd prime number, *k* and *p* are primes such that *p* = 2*k* + 1, *k* ≡ 3 (mod 4), *p* ≡ −1 (mod *q*), *p* ≢ −1 (mod *q*^{3}) and the order of *q* modulo *k* is . Assume that *q* divides *h*^{+}, the class number of the real cyclotomic field , the cyclotomic field obtained by adjoining the sum of a *p*-th root of unity and its reciprocal to the field of rational numbers. Then *q* is a Wieferich prime.^{[52]}^{: 55 } This also holds if the conditions *p* ≡ −1 (mod *q*) and *p* ≢ −1 (mod *q*^{3}) are replaced by *p* ≡ −3 (mod *q*) and *p* ≢ −3 (mod *q*^{3}) as well as when the condition *p* ≡ −1 (mod *q*) is replaced by *p* ≡ −5 (mod *q*) (in which case *q* is a Wall–Sun–Sun prime) and the incongruence condition replaced by *p* ≢ −5 (mod *q*^{3}).^{[53]}^{: 376 }

A prime *p* satisfying the congruence 2^{(p−1)/2} ≡ ±1 + *Ap* (mod *p*^{2}) with small |*A*| is commonly called a *near-Wieferich prime* (sequence A195988 in the OEIS).^{[28]}^{[54]} 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.^{[23]}^{[55]} The following table lists all near-Wieferich primes with |*A*| ≤ 10 in the interval [1×10^{9}, 3×10^{15}].^{[56]} This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.^{[22]}^{[57]} Bigger entries are by PrimeGrid.

p | 1 or −1 | A |
---|---|---|

3520624567 | +1 | −6 |

46262476201 | +1 | +5 |

47004625957 | −1 | +1 |

58481216789 | −1 | +5 |

76843523891 | −1 | +1 |

1180032105761 | +1 | −6 |

12456646902457 | +1 | +2 |

134257821895921 | +1 | +10 |

339258218134349 | −1 | +2 |

2276306935816523 | −1 | −3 |

82687771042557349 | -1 | -10 |

3156824277937156367 | +1 | +7 |

The sign +1 or -1 above can be easily predicted by Euler's criterion (and the second supplement to the law of quadratic reciprocity).

Dorais and Klyve^{[23]} used a different definition of a near-Wieferich prime, defining it as a prime *p* with small value of where is the Fermat quotient of 2 with respect to *p* modulo *p* (the modulo operation here gives the residue with the smallest absolute value). The following table lists all primes *p* ≤ 6.7 × 10^{15} with .

p |
||
---|---|---|

1093 | 0 | 0 |

3511 | 0 | 0 |

2276306935816523 | +6 | 0.264 |

3167939147662997 | −17 | 0.537 |

3723113065138349 | −36 | 0.967 |

5131427559624857 | −36 | 0.702 |

5294488110626977 | −31 | 0.586 |

6517506365514181 | +58 | 0.890 |

The two notions of nearness are related as follows. If , then by squaring, clearly . So if A had been chosen with small, then clearly is also (quite) small, and an even number. However, when is odd above, the related A from before the last squaring was not "small". For example, with , we have which reads extremely non-near, but after squaring this is which is a near-Wieferich by the second definition.

Main article: Fermat quotient |

A *Wieferich prime base a* is a prime *p* that satisfies

*a*^{p − 1}≡ 1 (mod*p*^{2}),^{[8]}with*a*less than*p*but greater than 1.

Such a prime cannot divide *a*, since then it would also divide 1.

It's a conjecture that for every natural number *a*, there are infinitely many Wieferich primes in base *a*.

Bolyai showed that if *p* and *q* are primes, *a* is a positive integer not divisible by *p* and *q* such that *a*^{p−1} ≡ 1 (mod *q*), *a*^{q−1} ≡ 1 (mod *p*), then *a*^{pq−1} ≡ 1 (mod *pq*). Setting *p* = *q* leads to *a*^{p2−1} ≡ 1 (mod *p*^{2}).^{[58]}^{: 284 } It was shown that *a*^{p2−1} ≡ 1 (mod *p*^{2}) if and only if *a*^{p−1} ≡ 1 (mod *p*^{2}).^{[58]}^{: 285–286 }

Known solutions of *a*^{p−1} ≡ 1 (mod *p*^{2}) for small values of *a* are:^{[59]} (checked up to 5 × 10^{13})

*a*primes *p*such that*a*^{p − 1}= 1 (mod*p*^{2})OEIS sequence 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes) A000040 2 1093, 3511, ... A001220 3 11, 1006003, ... A014127 4 1093, 3511, ... 5 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... A123692 6 66161, 534851, 3152573, ... A212583 7 5, 491531, ... A123693 8 3, 1093, 3511, ... 9 2, 11, 1006003, ... 10 3, 487, 56598313, ... A045616 11 71, ... 12 2693, 123653, ... A111027 13 2, 863, 1747591, ... A128667 14 29, 353, 7596952219, ... A234810 15 29131, 119327070011, ... A242741 16 1093, 3511, ... 17 2, 3, 46021, 48947, 478225523351, ... A128668 18 5, 7, 37, 331, 33923, 1284043, ... A244260 19 3, 7, 13, 43, 137, 63061489, ... A090968 20 281, 46457, 9377747, 122959073, ... A242982 21 2, ... 22 13, 673, 1595813, 492366587, 9809862296159, ... A298951 23 13, 2481757, 13703077, 15546404183, 2549536629329, ... A128669 24 5, 25633, ... 25 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... 26 3, 5, 71, 486999673, 6695256707, ... A306255 27 11, 1006003, ... 28 3, 19, 23, ... 29 2, ... 30 7, 160541, 94727075783, ... A306256 31 7, 79, 6451, 2806861, ... A331424 32 5, 1093, 3511, ... 33 2, 233, 47441, 9639595369, ... 34 46145917691, ... 35 3, 1613, 3571, ... 36 66161, 534851, 3152573, ... 37 2, 3, 77867, 76407520781, ... A331426 38 17, 127, ... 39 8039, ... 40 11, 17, 307, 66431, 7036306088681, ... 41 2, 29, 1025273, 138200401, ... A331427 42 23, 719867822369, ... 43 5, 103, 13368932516573, ... 44 3, 229, 5851, ... 45 2, 1283, 131759, 157635607, ... 46 3, 829, ... 47 ... 48 7, 257, ... 49 2, 5, 491531, ... 50 7, ...

For more information, see^{[60]}^{[61]}^{[62]} and.^{[63]} (Note that the solutions to *a* = *b ^{k}* is the union of the prime divisors of

The smallest solutions of *n*^{p−1} ≡ 1 (mod *p*^{2}) are

- 2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... (The next term > 4.9×10
^{13}) (sequence A039951 in the OEIS)

There are no known solutions of *n*^{p−1} ≡ 1 (mod *p*^{2}) for *n* = 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, 1002, 1023, 1130, 1136, 1138, ....

It is a conjecture that there are infinitely many solutions of *a*^{p−1} ≡ 1 (mod *p*^{2}) for every natural number *a*.

The bases *b* < *p*^{2} which *p* is a Wieferich prime are (for *b* > *p*^{2}, the solutions are just shifted by *k*·*p*^{2} for *k* > 0), and there are *p* − 1 solutions < *p*^{2} of *p* and the set of the solutions congruent to *p* are {1, 2, 3, ..., *p* − 1}) (sequence A143548 in the OEIS)

*p*values of *b*<*p*^{2}2 1 3 1, 8 5 1, 7, 18, 24 7 1, 18, 19, 30, 31, 48 11 1, 3, 9, 27, 40, 81, 94, 112, 118, 120 13 1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168 17 1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288 19 1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360 23 1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528 29 1, 14, 41, 60, 63, 137, 190, 196, 221, 236, 267, 270, 374, 416, 425, 467, 571, 574, 605, 620, 645, 651, 704, 778, 781, 800, 827, 840

The least base *b* > 1 which prime(*n*) is a Wieferich prime are

- 5, 8, 7, 18, 3, 19, 38, 28, 28, 14, 115, 18, 51, 19, 53, 338, 53, 264, 143, 11, 306, 31, 99, 184, 53, 181, 43, 164, 96, 68, 38, 58, 19, 328, 313, 78, 226, 65, 253, 259, 532, 78, 176, 276, 143, 174, 165, 69, 330, 44, 33, 332, 94, 263, 48, 79, 171, 747, 731, 20, ... (sequence A039678 in the OEIS)

We can also consider the formula , (because of the generalized Fermat little theorem, is true for all prime *p* and all natural number *a* such that both *a* and *a* + 1 are not divisible by *p*). It's a conjecture that for every natural number *a*, there are infinitely many primes such that .

Known solutions for small *a* are: (checked up to 4 × 10^{11}) ^{[64]}

primes such that 1 1093, 3511, ... 2 23, 3842760169, 41975417117, ... 3 5, 250829, ... 4 3, 67, ... 5 3457, 893122907, ... 6 72673, 1108905403, 2375385997, ... 7 13, 819381943, ... 8 67, 139, 499, 26325777341, ... 9 67, 887, 9257, 83449, 111539, 31832131, ... 10 ... 11 107, 4637, 239357, ... 12 5, 11, 51563, 363901, 224189011, ... 13 3, ... 14 11, 5749, 17733170113, 140328785783, ... 15 292381, ... 16 4157, ... 17 751, 46070159, ... 18 7, 142671309349, ... 19 17, 269, ... 20 29, 162703, ... 21 5, 2711, 104651, 112922981, 331325567, 13315963127, ... 22 3, 7, 13, 94447, 1198427, 23536243, ... 23 43, 179, 1637, 69073, ... 24 7, 353, 402153391, ... 25 43, 5399, 21107, 35879, ... 26 7, 131, 653, 5237, 97003, ... 27 2437, 1704732131, ... 28 5, 617, 677, 2273, 16243697, ... 29 73, 101, 6217, ... 30 7, 11, 23, 3301, 48589, 549667, ... 31 3, 41, 416797, ... 32 95989, 2276682269, ... 33 139, 1341678275933, ... 34 83, 139, ... 35 ... 36 107, 137, 613, 2423, 74304856177, ... 37 5, ... 38 167, 2039, ... 39 659, 9413, ... 40 3, 23, 21029249, ... 41 31, 71, 1934399021, 474528373843, ... 42 4639, 1672609, ... 43 31, 4962186419, ... 44 36677, 17786501, ... 45 241, 26120375473, ... 46 5, 13877, ... 47 13, 311, 797, 906165497, ... 48 ... 49 3, 13, 2141, 281833, 1703287, 4805298913, ... 50 2953, 22409, 99241, 5427425917, ...

Main article: Wieferich pair |

A Wieferich pair is a pair of primes *p* and *q* that satisfy

*p*^{q − 1}≡ 1 (mod*q*^{2}) and*q*^{p − 1}≡ 1 (mod*p*^{2})

so that a Wieferich prime *p* ≡ 1 (mod 4) will form such a pair (*p*, 2): the only known instance in this case is *p* = 1093. There are only 7 known Wieferich pairs.^{[65]}

- (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence OEIS: A282293 in OEIS)

Start with a(1) any natural number (>1), a(*n*) = the smallest prime *p* such that (a(*n* − 1))^{p − 1} = 1 (mod *p*^{2}) but *p*^{2} does not divide a(*n* − 1) − 1 or a(*n* − 1) + 1. (If *p*^{2} divides a(*n* − 1) − 1 or a(*n* − 1) + 1, then the solution is a trivial solution) It is a conjecture that every natural number *k* = a(1) > 1 makes this sequence become periodic, for example, let a(1) = 2:

- 2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}.

Let a(1) = 83:

- 83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}.

Let a(1) = 59 (a longer sequence):

- 59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ..., it also gets 5.

However, there are many values of a(1) with unknown status, for example, let a(1) = 3:

- 3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).

Let a(1) = 14:

- 14, 29, ? (There are no known Wieferich prime in base 29 except 2, but 2
^{2}= 4 divides 29 − 1 = 28)

Let a(1) = 39 (a longer sequence):

- 39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)

It is unknown that values for a(1) > 1 exist such that the resulting sequence does not eventually become periodic.

When a(*n* − 1)=*k*, a(*n*) will be (start with *k* = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For *k* = 21, 29, 47, 50, even the next value is unknown)

A **Wieferich number** is an odd natural number *n* satisfying the congruence 2^{φ(n)} ≡ 1 (mod *n*^{2}), where *φ* denotes the Euler's totient function (according to Euler's theorem, 2^{φ(n)} ≡ 1 (mod *n*) for every odd natural number *n*). If Wieferich number *n* is prime, then it is a Wieferich prime. The first few Wieferich numbers are:

- 1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, ... (sequence A077816 in the OEIS)

It can be shown that if there are only finitely many Wieferich primes, then there are only finitely many Wieferich numbers. In particular, if the only Wieferich primes are 1093 and 3511, then there exist exactly 104 Wieferich numbers, which matches the number of Wieferich numbers currently known.^{[2]}

More generally, a natural number *n* is a **Wieferich number to base** *a*, if *a*^{φ(n)} ≡ 1 (mod *n*^{2}).^{[66]}^{: 31 }

Another definition specifies a **Wieferich number** as odd natural number *n* such that *n* and are not coprime, where *m* is the multiplicative order of 2 modulo *n*. The first of these numbers are:^{[67]}

- 21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, ... (sequence A182297 in the OEIS)

As above, if Wieferich number *q* is prime, then it is a Wieferich prime.

A weak Wieferich prime to base *a* is a prime *p* satisfies the condition

*a*^{p}≡*a*(mod*p*^{2})

Every Wieferich prime to base *a* is also a weak Wieferich prime to base *a*. If the base *a* is squarefree, then a prime *p* is a weak Wieferich prime to base *a* if and only if *p* is a Wieferich prime to base *a*.

Smallest weak Wieferich prime to base *n* are (start with *n* = 0)

- 2, 2, 1093, 11, 2, 2, 66161, 5, 2, 2, 3, 71, 2, 2, 29, 29131, 2, 2, 3, 3, 2, 2, 13, 13, 2, 2, 3, 3, 2, 2, 7, 7, 2, 2, 46145917691, 3, 2, 2, 17, 8039, 2, 2, 23, 5, 2, 2, 3, ...

For integer *n* ≥2, a Wieferich prime to base *a* with order *n* is a prime *p* satisfies the condition

*a*^{p−1}≡ 1 (mod*p*^{n})

Clearly, a Wieferich prime to base *a* with order *n* is also a Wieferich prime to base *a* with order *m* for all 2 ≤ *m* ≤ *n*, and Wieferich prime to base *a* with order 2 is equivalent to Wieferich prime to base *a*, so we can only consider the *n* ≥ 3 case. However, there are no known Wieferich prime to base 2 with order 3. The first base with known Wieferich prime with order 3 is 9, where 2 is a Wieferich prime to base 9 with order 3. Besides, both 5 and 113 are Wieferich prime to base 68 with order 3.

Let *P* and *Q* be integers. The Lucas sequence of the first kind associated with the pair (*P*, *Q*) is defined by

for all . A **Lucas–Wieferich prime** associated with (*P*, *Q*) is a prime *p* such that *U*_{p−ε}(*P*, *Q*) ≡ 0 (mod *p*^{2}), where *ε* equals the Legendre symbol . All Wieferich primes are Lucas–Wieferich primes associated with the pair (3, 2).^{[3]}^{: 2088 }

Let *Q* = −1. For every natural number *P*, the Lucas–Wieferich primes associated with (*P*, −1) are called ** P-Fibonacci–Wieferich primes** or

For example, 241 is a Lucas–Wieferich prime associated with (3, −1), so it is a 3-Fibonacci–Wieferich prime or 3-Wall–Sun–Sun prime. In fact, 3 is a *P*-Fibonacci–Wieferich prime if and only if *P* congruent to 0, 4, or 5 (mod 9),^{[citation needed]} which is analogous to the statement for traditional Wieferich primes that 3 is a base-*n* Wieferich prime if and only if *n* congruent to 1 or 8 (mod 9).

Let *K* be a global field, i.e. a number field or a function field in one variable over a finite field and let *E* be an elliptic curve. If *v* is a non-archimedean place of norm *q _{v}* of