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Or to put it algebraically, writing the sequence of prime numbers as (p 1, p 2, p 3, ...) = (2, 3, 5, ...), p n is a strong prime if p n > p n − 1 + p n + 1 / 2 . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime. The first few ...
A prime number q is a strong prime if q + 1 and q − 1 both have some large (around 500 digits) prime factors. For a safe prime q = 2p + 1, the number q − 1 naturally has a large prime factor, namely p, and so a safe prime q meets part of the criteria for being a strong prime.
A powerful number is a positive integer m such that for every prime number p dividing m, p 2 also divides m.Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a 2 b 3, where a and b are positive integers.
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.For example: 7 = 7 1, 9 = 3 2 and 64 = 2 6 are prime powers, while 6 = 2 × 3, 12 = 2 2 × 3 and 36 = 6 2 = 2 2 × 3 2 are not.
However, the number of Pythagorean primes up to is frequently somewhat smaller than the number of non-Pythagorean primes; this phenomenon is known as Chebyshev's bias. [1] For example, the only values of n {\displaystyle n} up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes less than or equal to n are 26861 and 26862.
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