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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 .
A strong Fibonacci pseudoprime is a composite number n for which congruence holds for Q = −1 and all P. [17] It follows [17]: 460 that an odd composite integer n is a strong Fibonacci pseudoprime if and only if: n is a Carmichael number; 2(p + 1) | (n − 1) or 2(p + 1) | (n − p) for every prime p dividing n.
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 ...
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 property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird. [4] This leads to the definition of primitive weird numbers: weird numbers that are not a multiple of other weird numbers (sequence A002975 in the OEIS). Among the 1765 weird numbers less than one million, there are ...
The following facts, even the reciprocity laws, are straightforward deductions from the definition of the Jacobi symbol and the corresponding properties of the Legendre symbol. [2] 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.
Specifically, Kim and Pomerance showed the following: The probability that a random odd number n ≤ x is a Fermat pseudoprime to a random base < < is less than 2.77·10 −8 for x= 10 100, and is at most (log x) −197 <10-10,000 for x≥10 100,000.