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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. Powerful numbers are also known as squareful, square-full, or 2-full.
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 ...
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.
For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. ... A prime number q is a strong prime if q + 1 and q − 1 both have ...
These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and () is the third cyclotomic polynomial, defined as + +.
For example, there are only 13 numbers less than 25·10 9 that are strong pseudoprimes to bases 2, 3, and 5 simultaneously. They are listed in Table 7 of. [2] The smallest such number is 25326001. This means that, if n is less than 25326001 and n is a strong probable prime to bases 2, 3, and 5, then n is prime.
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As an example, 108 is a powerful number. Its prime factorization is 2 2 · 3 3, and thus its prime factors are 2 and 3.Both 2 2 = 4 and 3 2 = 9 are divisors of 108. However, 108 cannot be represented as m k, where m and k are positive integers greater than 1, so 108 is an Achilles number.