Wieferich prime

Wieferich prime

Named after	Arthur Wieferich
Publication year	1909
Author of publication	Wieferich, A.
No. of known terms	2
Subsequence of	Crandall numbers Wieferich numbers Lucas-Wieferich primes 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 p² divides 2^p − 1 − 1, 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.

Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511 (sequence A001220 in the OEIS).

Explanation of the Wieferich property

The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation, namely the congruence 2^{p − 1} ≡ 1 (mod p²). 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 ${\tfrac {2^{p-1}-1}{p}}$. The following are two illustrative examples using the primes 11 and 1093:

For 11 we get ${\tfrac {2^{10}-1}{11}}$ which is 93 and leaves a remainder of 5 after division by 11, hence 11 is not a Wieferich prime. For 1093 we get ${\tfrac {2^{1092}-1}{1093}}$ or 530585362....3096656895 (320 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.

History and search status

In 1902 W. F. Meyer proved a theorem about solutions of the congruence a^{p − 1} ≡ 1 (mod p^r). 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. 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 p²). In 1913, Bachmann examined the residues of ${\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$. He asked the question when this residue vanishes and tried to find expressions for answering this question.

The prime 1093 was found to be a Wieferich prime by Waldemar Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of ${\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p$ 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 Grawe about the impossibility of the Wieferich congruence. E. Haentzschel later ordered verification of the correctness of Meissners congruence via only elementary calculations. Inspired by an earlier work of Euler, he simplified Meissners proof by showing that 1093² | (2¹⁸² + 1) and remarked that (2¹⁸² + 1) is a factor of (2³⁶⁴ − 1). The prime 3511 was found to be a Wieferich prime by N. G. W. H. Beeger in 1922. In 1960, Kravitz doubled a previous record set by Fröberg and in 1961 Riesel extended the search to 500000 with the aid of BESK. Around 1980, Lehmer was able to reach the search limit of 6×10⁹. This limit was extended to over 2.5×10¹⁵ in 2006, finally reaching 3×10¹⁵. It is now known, that if any other Wieferich primes exist, they must be greater than 6.7×10¹⁵. The search for new Wieferich primes is currently performed by the distributed computing project Wieferich@Home. In December 2011, another search was started by the PrimeGrid project. As of May 2012, PrimeGrid has extended the search limit to 17×10¹⁵ and continues.

Chris Caldwell conjectured that only a finite number of Wieferich primes exist. It has also 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 p² are uniformly distributed in the multiplicative group of integers modulo p².

Properties

Connection with Fermat's last theorem

The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:

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) and we say that FLTI fails for a prime p, if solutions to the Fermat equation exist for that p, otherwise FLTI holds for p. In 1910, Mirimanoff expanded the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p² must also divide 3^p − 1 − 1. Granville and Monagan further proved that p² must actually divide m^p − 1 − 1 for every prime m ≤ 89. Suzuki extended the proof to all primes m ≤ 113.

Connection with the abc conjecture

A non-Wieferich prime is a prime p satisfying 2^p − 1 ≢ 1 (mod p²). J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes. Numerical evidence suggests that very few of the prime numbers in a given interval are Wieferich primes. A proof of the abc conjecture would not automatically prove that there are only finitely many Wieferich primes, since the set of Wieferich primes and the set of non-Wieferich primes could possibly both be infinite and the finiteness or infiniteness of the set of Wieferich primes would have to be proven separately. The existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers.

Connection with Mersenne and Fermat primes

It is known that the nth Mersenne number M_n = 2ⁿ − 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 p² divides M_q.

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 a Mersenne number M_q is not square-free, that is, there exists a prime p for which p² 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 that are not square-free. Rotkiewicz showed that the converse is also true, that is, if there are infinitely many square-free Mersenne numbers, then there are infinitely many non-Wieferich primes.

Similarly, if p is prime and p² divides some Fermat number F_n = 2²ⁿ + 1, then p must be a Wieferich prime.

For the primes 1093 and 3511 it was shown that neither of them is a factor of any Mersenne or Fermat number.

Connection with other equations

Scott and Styer showed that if p is a Wieferich prime greater than 10¹⁵, then the Diophantine equation p^x−2^y=p^u−2^v has solutions in p, x, y, u, v with x≠u. They were later able to extend this result to other equations.

Binary periodicity of p−1

Johnson observed that the two known Wieferich primes are one greater than numbers with periodic binary expansions (1092 = 010001000100₂; 3510 = 110110110110₂). The Wieferich@Home project searches 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.

Equivalent congruences

Wieferich primes can be defined by other congruences equivalent to the one usually used. If p is a Wieferich prime, one can multiply both sides of the congruence 2^p-1 ≡ 1 (mod p²) with 2 to get 2^p ≡ 2 (mod p²). Thus a Wieferich prime satisfies 2^p2 ≡ 2 (mod p²), since it must satisfy 2^kp ≡ 2 (mod p²) for all integers k ≥ 1.

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 p²). It was shown that a^p2-1 ≡ 1 (mod p²) if and only if a^p-1 ≡ 1 (mod p²).

Connection with pseudoprimes

It was observed that the two known Wieferich primes are the square factors of all non-square free base-2 Fermat pseudoprimes up to 25×10⁹. Later computations showed that the only repeated factors of the pseudoprimes up to 10¹² are 1093 and 3511. Furthermore if p is a Wieferich prime, then p² is a Catalan pseudoprime.

Connection with directed graphs

For all primes up to 100000 L(pⁿ⁺¹) = L(pⁿ) only for two cases: L(1093²) = L(1093) = 364 and L(3511²) = L(3511) = 1755, where m is the modulus of the doubling diagram and L(m) gives the number of vertices in the cycle of 1. The term doubling diagram refers to the directed graph with 0 and the natural numbers less than m as vertices with arrows pointing from each vertex x to vertex 2x reduced modulo m. It was shown, that for all odd prime numbers either L(pⁿ⁺¹) = p × L(pⁿ) or L(pⁿ⁺¹) = L(pⁿ).

Alternative definition

Let H_p be a set of pairs of integers with 1 as their greatest common divisor, p being prime to x, y and x + y, (x + y)^p-1 ≡ 1 (mod p²), (x + ξy) being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q(ξ) is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers (such an extension is known as a number field or in this particular case, where ξ is a root of unity, a cyclotomic number field).

From uniqueness of factorization of ideals in Q(ξ) it follows that if the first case of Fermat's last theorem has solutions x, y, z then p divides x+y+z and (x, y), (y, z) and (z, x) are elements of H_p.

Granville and Monagan showed that (1, 1) ∈ H_p if and only if p is a Wieferich prime.

Generalizations

Near-Wieferich primes

A prime p satisfying the congruence 2^(p−1)/2 ≡ ±1 + Ap (mod p²) with small |A| is commonly called a near-Wieferich prime (sequence A195988 in the OEIS). 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| ≤ 10 in the interval [1×10⁹, 3×10¹⁵]. This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.

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

Dorais and Klyve used a different definition of a near-Wieferich prime, defining it as a prime p with small value of $\left|{\tfrac {\omega (p)}{p}}\right|$ where $\omega (p)={\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$ 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 × 10¹⁵ with $\left|{\tfrac {\omega (p)}{p}}\right|\leq 10^{-14}$.

p	$\omega (p)$	$\left|{\tfrac {\omega (p)}{p}}\right|\times 10^{14}$
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

Base-a Wieferich primes

Main article: Fermat quotient

A Wieferich prime base a is a prime p that satisfies

a^p − 1 ≡ 1 (mod p²).

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

Wieferich pairs

Main article: Wieferich pair

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

p^{q − 1} ≡ 1 (mod q²) and q^{p − 1} ≡ 1 (mod p²)

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 6 known Wieferich pairs.

Wieferich numbers

A Wieferich number is an odd integer w ≥ 3 satisfying the congruence 2^φ(w) ≡ 1 (mod w²), where φ(·) denotes the Euler function. If Wieferich number w is prime, then it is a Wieferich prime. The first few Wieferich numbers are:

1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, … (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.

Another definition specifies a Wieferich number as positive odd integer q such that q and ${\tfrac {2^{m}-1}{q}}$ are not coprime, where m is the multiplicative order of 2 modulo q. The first of these numbers are:

21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, … (sequence A182297 in the OEIS)

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

Lucas-Wieferich primes

A Lucas-Wieferich prime associated with the pair (P, Q) is a prime p such that U_p-ε ≡ 0 (mod p²), where U_p-ε denotes the Lucas sequence of the first kind and P and Q are integers. All Wieferich primes are Lucas-Wieferich primes associated to the pair (3, 2).

See also

• Wall–Sun–Sun prime – a hypothetical type of prime number defined in the study of Fermat's last theorem (FLT)
• Wolstenholme prime – another type of prime number which in the broadest sense also resulted from the study of FLT
• Table of congruences - lists other congruences satisfied by prime numbers

Further reading

• Haussner, R. (1926), "Über die Kongruenzen 2^p-1-1 ≡ 0 (mod p²) für die Primzahlen p=1093 und 3511", Archiv for Mathematik og Naturvidenskab (in German), 39 (5): 7, JFM 52.0141.06, DNB 363953469
• Haussner, R. (1927), "Über numerische Lösungen der Kongruenz u^p-1-1 ≡ 0 (mod p²)", Journal für die reine und angewandte Mathematik (Crelle's Journal). (in German), 1927 (156): 223–226, doi:10.1515/crll.1927.156.223
• Ribenboim, P. (1983), "1093", The Mathematical Intelligencer, 5 (2): 28–34, doi:10.1007/BF03023623
• Silverman, J. H. (1988), "Wieferich's criterion and the abc-conjecture", Journal of Number Theory, 30 (2): 226–237, doi:10.1016/0022-314X(88)90019-4
• Ribenboim, P. (1979), Thirteen lectures on Fermat's Last Theorem, Springer-Verlag, pp. 139, 151, ISBN 0-387-90432-8
• Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), Springer Verlag, p. 14, ISBN 0-387-20860-7
• Garza, G.; Young, J. (2004), "Wieferich Primes and Period Lengths for the Expansions of Fractions", Math. Mag., 77 (4): 314–319, doi:10.2307/3219294
• Crandall, R. E.; Pomerance, C. (2005), Prime numbers: a computational perspective (PDF), Springer Science+Business Media, Inc., pp. 31–32, ISBN 0-387-25282-7
• Ribenboim, P. (1996), The new book of prime number records, Springer-Verlag New York, Inc., pp. 333–346, ISBN 0-387-94457-5
• Broughan, K. (2006), "Relaxations of the ABC Conjecture using integer k'th roots" (PDF), New Zealand J. Math., 35 (2): 121–136

External links

• Weisstein, Eric W. "Wieferich prime". MathWorld.
• Fermat-/Euler-quotients (a^p−1 − 1)/p^k with arbitrary k
• A note on the two known Wieferich primes
• PrimeGrid's Wieferich Prime Search project page