A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a …

Dec 3, 2025 · An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ...

Heath-Brown has shown in 1988 that every sufficiently large natural number is the sum of at most three powerful numbers. Probably the largest number which is not the sum of 3 powerful …

Welcome to the Prime Glossary: a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'powerful number.'

May 26, 1999 · Not every Natural Number is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently large Natural Number is the sum of at most three …

Theorem A powerful number is a positive integer such that each of its prime factors appears with multiplicity at least $2$. That is, each of its prime factors occurs at least squared.

For U2-1 to be powerful, (u-1)/2 and (u+l)/2 must be a powerful odd number and twice a powerful number, in either order. The two Pell equations produce examples in both orders.

Powerful Number Checker is a free online tool used to check if the given number is a powerful number.

Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as …

Dec 5, 2025 · A k-powerful (or k-full) number is a positive integer n such that for every prime number p dividing n, p^k also divides n. These are numbers of the form: 2-powerful...