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If n is an odd composite integer that satisfies the above congruence, then n is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime) to base a. As long as a is not a multiple of n (usually 2 ≤ a < n ), then if a and n are not coprime, n is definitely composite, as 1 < gcd ( a , n ) < n is a factor of n .
In mathematics, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and / ()(where mod refers to the modulo operation).. The motivation for this definition is the fact that all prime numbers p satisfy the above equation which can be deduced from Fermat's little theorem.
A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime ...
Even and odd numbers: An integer is even if it is a multiple of 2, and is odd otherwise. Prime number: A positive integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Composite number: A positive integer that can be factored into a product of smaller positive ...
So if it is unknown whether a number n is prime or composite, we can pick a random number a, calculate the Jacobi symbol ( a / n ) and compare it with Euler's formula; if they differ modulo n, then n is composite; if they have the same residue modulo n for many different values of a, then n is "probably prime".
More extensive calculations show that, with this method of choosing D, P, and Q, there are only five odd, composite numbers less than 10 15 for which congruence is true. [ 8 ] If Q ≠ ± 1 {\displaystyle Q\neq \pm 1} (and GCD( n , Q ) = 1), then an Euler–Jacobi probable prime test to the base Q can also be implemented at minor computational ...
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.
If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these.