Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides. 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.
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.
, the only known Wieferich primes are 1093 and 3511.
Equivalent definitions
The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation. 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:Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence by 2 to get. Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies, and hence for all. The converse is also true: for some implies that the multiplicative order of 2 modulo p2 divides gcd, φ, that is, 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 p2 coincide:,.
H. S. Vandiver proved that if and only if.
History and search status
In 1902, W. F. Meyer proved a theorem about solutions of the congruence ap − 1 ≡ 1. 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 xp + yp + zp = 0 in integers x, y, z and p an odd prime with p ∤ xyz, then p satisfies 2p − 1 ≡ 1. In 1913, Bachmann examined the residues of. 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 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. E. Haentzschel later ordered verification of the correctness of Meissner's congruence via only elementary calculations. Inspired by an earlier work of Euler, he simplified Meissner's proof by showing that 10932 | and remarked that is a factor of. 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, although Meissner himself hinted at that he was aware of a proof without complex values.
The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922 and another proof of it being a Wieferich prime was published in 1965 by Guy. 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. This limit was extended to over 2.5 in 2006, finally reaching 3. It is now known that if any other Wieferich primes exist, they must be greater than 6.7.
In 2007–2016, a search for Wieferich primes was performed by the distributed computing project Wieferich@Home. In 2011–2017, another search was performed by the PrimeGrid project, although later the work done in this project was claimed wasted. While these projects reached search bounds above 1, neither of them reported any sustainable results.
It has been conjectured that infinitely many Wieferich primes exist, and that the number of Wieferich primes below x is approximately log, which is a heuristic result that follows from the plausible assumption that for a prime p, the 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:The above case is commonly known as the first case of Fermat's last theorem 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.
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. Granville and Monagan further proved that p2 must actually divide for every prime m ≤ 89. Suzuki extended the proof to all primes m ≤ 113.
Let Hp be a set of pairs of integers with 1 as their greatest common divisor, p being prime to x, y and x + y, p−1 ≡ 1, being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers.
From uniqueness of factorization of ideals in Q it follows that if the first case of Fermat's last theorem has solutions x, y, z then p divides x+y+z and, and are elements of Hp.
Granville and Monagan showed that ∈ Hp if and only if p is a Wieferich prime.
Connection with the ''abc'' conjecture and non-Wieferich primes
A non-Wieferich prime is a prime p satisfying. J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes. 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 as X goes to infinity. 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 W2 and W2c respectively, are complementary sets, so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are proper subsets of the set of prime numbers. It was later shown that the existence of infinitely many non-Wieferich primes already follows from a weaker version of the abc conjecture, called the ABC- conjecture. Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers as well as if there exists a real number ξ such that the set is of density one, where the index of composition λ of an integer n is defined as and, meaning gives the product of all prime factors of n.Connection with Mersenne and Fermat primes
It is known that the nth Mersenne number is prime only if n is prime. Fermat's little theorem implies that if is prime, then Mp−1 is always divisible by p. Since Mersenne numbers of prime indices Mp and Mq are co-prime,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 Mq is not square-free, that is, there exists a prime p for which p2 divides Mq, 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.
Similarly, if p is prime and p2 divides some Fermat number Fn, then p must be a Wieferich prime.
In fact, there exists a natural number n and a prime p that p2 divides if and only if p is a Wieferich prime. For example, 10932 divides, 35112 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 ; and clearly, if is a prime, then it cannot be Wieferich prime.
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.
Connection with other equations
Scott and Styer showed that the equation px – 2y = d has at most one solution in positive integers, unless when p4 | 2ordp 2 – 1 if p ≢ 65 or unconditionally when p2 | 2ordp 2 – 1, where ordp 2 denotes the multiplicative order of 2 modulo p. They also showed that a solution to the equation ±ax1 ± 2y1 = ±ax2 ± 2y2 = 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 1015.Binary periodicity of ''p'' − 1
Johnson observed that the two known Wieferich primes are one greater than numbers with periodic binary expansions. 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.Abundancy of ''p'' − 1
It has been noted that the known Wieferich primes are one greater than mutually friendly numbers.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. Later computations showed that the only repeated factors of the pseudoprimes up to 1012 are 1093 and 3511. In addition, the following connection exists:Connection with directed graphs
For all primes up to, only in two cases: and, where is the number of vertices in the cycle of 1 in the doubling diagram modulo. 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 2x reduced modulo m. It was shown, that for all odd prime numbers either or.Properties related to number fields
It was shown that and if and only if 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, let be the fundamental discriminant of the imaginary quadratic field and if, let be the fundamental discriminant of the imaginary quadratic field. Then and .Furthermore, the following result was obtained: Let q be an odd prime number, k and p are primes such that 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. This also holds if the conditions and are replaced by and as well as when the condition is replaced by and the incongruence condition replaced by.
Generalizations
Near-Wieferich primes
A prime p satisfying the congruence 2/2 with small |A| is commonly called a near-Wieferich prime. 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 . 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 |
The sign +1 or -1 above can be easily predicted by Euler's criterion.
Dorais and Klyve 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 following table lists all primes p ≤ 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 had been chosen with small, then clearly is also small, and an even number. However, when is odd above, the related 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.
Base-''a'' Wieferich primes
A Wieferich prime base a is a prime p that satisfiesSuch 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,, then. Setting p = q leads to. It was shown that if and only if.
Known solutions of for small values of a are:
For more information, see and.
The smallest solutions of are
There are no known solutions of for n = 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 347, 355, 435, 454, 542, 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 for every natural number a.
The bases b < p2 which p is a Wieferich prime are, and there are solutions < p2 of p and the set of the solutions congruent to p are
The least base b > 1 which prime is a Wieferich prime are
We can also consider the formula,. It's a conjecture that for every natural number a, there are infinitely many primes such that.
Known solutions for small a are:
Wieferich pairs
A Wieferich pair is a pair of primes p and q that satisfyso that a Wieferich prime p ≡ 1 will form such a pair : the only known instance in this case is. There are only 7 known Wieferich pairs.
Wieferich sequence
Start with a any natural number, a = the smallest prime p such that p − 1 = 1 but p2 does not divide a − 1 or a + 1. − 1 or a It is a conjecture that every natural number k = a > 1 makes this sequence become periodic, for example, let a = 2:Let a = 83:
Let a = 59 :
However, there are many values of a with unknown status, for example, let a = 3:
Let a = 14:
Let a = 39 :
It is unknown that values for a > 1 exist such that the resulting sequence does not eventually become periodic.
When a=k, a will be : 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, ?,...
Wieferich numbers
A Wieferich number is an odd natural number n satisfying the congruence 2 ≡ 1, where denotes the Euler's totient function ≡ 1. If Wieferich number n is prime, then it is a Wieferich prime. The first few Wieferich numbers are: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.
More generally, a natural number n is a Wieferich number to base a, if a ≡ 1.
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:
As above, if Wieferich number q is prime, then it is a Wieferich prime.
Weak Wieferich prime
A weak Wieferich prime to base a is a prime p satisfies the conditionEvery 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
Wieferich prime with order ''n''
For integer n ≥2, a Wieferich prime to base a with order n is a prime p satisfies the conditionClearly, 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 equivent 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.
Lucas–Wieferich primes
Let P and Q be integers. The Lucas sequence of the first kind associated with the pair is defined byfor all. A Lucas–Wieferich prime associated with is a prime p such that Up−ε ≡ 0, where ε equals the Legendre symbol. All Wieferich primes are Lucas–Wieferich primes associated with the pair.
Fibonacci–Wieferich primes
Let Q = −1. For every natural number P, the Lucas–Wieferich primes associated with are called P-Fibonacci–Wieferich primes or P-Wall–Sun–Sun primes. If P = 1, they are called Fibonacci–Wieferich primes. If P = 2, they are called Pell–Wieferich primes.For example, 241 is a Lucas–Wieferich prime associated with, 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, 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.