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: : 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 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 ''p''2 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 ''p''2 coincide: , (By the way, ord10932 = 364, and ord35112 = 1755). H. S. Vandiver proved that if and only if .History and search status
In 1902, 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 ''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. The prime 1093 was found to be a Wieferich prime by 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 byProperties
Connection with Fermat's Last Theorem
The following theorem connecting Wieferich primes andConnection 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(''X'') 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. 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''-(''k'', ''ε'') ''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''(''n'') 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 ''n''th Mersenne number is prime only if ''n'' is prime. Fermat's little theorem implies that if is prime, then ''M''''p''−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''. 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. Similarly, if ''p'' is prime and ''p''2 divides someConnection with other equations
Scott and Styer showed that the equation ''p''x − 2y = ''d'' has at most one solution in positive integers (''x'', ''y''), unless when ''p''4 , 2ord''p'' 2 − 1 if ''p'' ≢ 65 (mod 192) or unconditionally when ''p''2 , 2ord''p'' 2 − 1, where ord''p'' 2 denotes the multiplicative order of 2 modulo ''p''. They also showed that a solution to the equation ±''a''''x''1 ± 2''y''1 = ±''a''''x''2 ± 2''y''2 = ''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 × 1015.Binary periodicity of ''p'' − 1
Johnson observed that the two known Wieferich primes are one greater than numbers with periodic binary expansions (1092 = 0100010001002=44416; 3510 = 1101101101102=66668). 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 (the shared abundancy index being 112/39).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: :Let ''n'' be a base 2 pseudoprime and ''p'' be a prime divisor of ''n''. If , then also . Furthermore, if ''p'' is a Wieferich prime, then ''p''2 is a Catalan pseudoprime.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 2''x'' 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 (''χ'' and ''λ'' in this context denote Iwasawa invariants). 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 (in which case ''q'' is aGeneralizations
Near-Wieferich primes
A prime ''p'' satisfying the congruence 2(''p''−1)/2 (mod ''p''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 , 3 This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch. Bigger entries are by PrimeGrid. 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 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'' (theBase-''a'' Wieferich primes
A ''Wieferich prime base a'' is a prime ''p'' that satisfies : , 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 , , then . Setting ''p'' = ''q'' leads to . It was shown that if and only if . Known solutions of for small values of ''a'' are: (checked up to 5 × 1013) : For more information, see and. (Note that the solutions to ''a'' = ''bk'' is the union of the prime divisors of ''k'' which does not divide ''b'' and the solutions to ''a'' = ''b'') The smallest solutions of 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 > 4.9×1013) There are no known solutions of 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 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 solutions < ''p''2 of ''p'' and the set of the solutions congruent to ''p'' are : 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, ... 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 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 × 1011) :Wieferich pairs
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 . There are only 7 known Wieferich pairs. : (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence in OEIS)Wieferich sequence
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: . : Let a(1) = 83: :83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: . 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 22 = 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)Wieferich numbers
A Wieferich number is an odd natural number ''n'' satisfying the congruence 2(''n'') ≡ 1 (mod ''n''2), where denotes the Euler's totient function (according to Euler's theorem, 2(''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, ... 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''(''n'') ≡ 1 (mod ''n''2). 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: :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, ... 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 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, ...Wieferich prime with order ''n''
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.Lucas–Wieferich primes
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).Wieferich places
Let ''K'' be a global field, i.e. a number field or a function field in one variable over aSee also
*References
Further reading
* * * * * *External links
*