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Wieferich pairs. A Wieferich pair is a pair of primes p and q that satisfy. pq − 1 ≡ 1 (mod q2) and qp − 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 only 7 known Wieferich pairs.
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n. The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then ...
Ruth–Aaron pair. In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal: 714 = 2 × 3 × 7 × 17, 715 = 5 × 11 × 13, and. 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29. There are different variations in the definition, depending on how many times to ...
Twin prime. A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime ...
Amicable numbers. Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s (a)= b and s (b)= a, where s (n)=σ (n)- n is equal to the sum of positive divisors of n except n itself (see also divisor function). The smallest pair of amicable ...
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Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number , with n at least 2, is of the form + + (see Proth number), where k is a positive integer. By itself, this makes it easy to prove the primality of the ...