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| first_terms = 1093, 3511 | largest_known_term = 3511 | OEIS = A001220 }} In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 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. 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: : 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''p''2 ≡2''p'' ≡ 2 (mod ''p''2), and hence 2''p''k ≡ 2 (mod ''p''2) for all ''k'' ≥ 1. The converse is also true: 2''p''k ≡ 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: , (By the way, ord10932 = 364, and ord35112 = 1755). H. S. Vandiver proved that if and only if . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wieferich prime」の詳細全文を読む スポンサード リンク
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