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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.
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 ...
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.
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 = 2 p + 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.
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. The integer 360 is not an Achilles number because it is ...
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
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