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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 ...
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 n is a strong pseudoprime to at most one quarter of all bases below n; [3] [4] thus, there are no "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases. Thus given a random base, the probability that a number is a strong pseudoprime to that base is less than 1/4, forming the basis of the widely used ...
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 ...
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.
The Jacobi symbol is defined only when the upper argument ("numerator") is an integer and the lower argument ("denominator") is a positive odd integer. 1. If n is (an odd) prime, then the Jacobi symbol ( a / n ) is equal to (and written the same as) the corresponding Legendre symbol. 2.