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A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
An odd prime number p is defined to be regular if it does not divide the class number of the p th cyclotomic field Q (ζp), where ζp is a primitive p th root of unity. The prime number 2 is often considered regular as well. The class number of the cyclotomic field is the number of ideals of the ring of integers Z (ζp) up to equivalence.
Formula for primes. In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what such a "formula" can and cannot be.
In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (p1, p2, p3, ...) = (2, 3, 5, ...), pn is a strong prime if pn > pn ...
Thus, all Mersenne numbers M4k +1 are congruent to 11 modulo 20 and end in 11, 31, 51, 71 or 91, while Mersenne numbers M4k −1 ≡ 7 (mod 20) and end in 07, 27, 47, 67, or 87. For the perfect numbers, define Pn = 2n−1Mn be the value which is perfect if Mn is prime. When n = 4k +1 and k > 0, 24k ≡ 16 (mod 20), so Pn ≡ 16×11 ≡ 16 (mod ...
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In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers: [note 1] Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b. For example, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019, and since this ...