Talk:Powerful number

This article includes too many statements without derivation, "clearly"s, and "since a, b" which assumes the reader knows the property being used.--Walt 19:26, 22 June 2006 (UTC)

Untagging since I believe the edits I made today address this complaint. —David Eppstein 00:31, 12 September 2006 (UTC)

Confusion about the opening definition

“ A powerful number is a positive integer m 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 number m of the form m = a²b³, where a and b are positive integers. ”

How does the second sentence follow from the first? Surely the definition in the first sentence is satisfied by any square number? Colonies Chris (talk) 13:38, 25 February 2008 (UTC)

Any square number also satisfies the second definition, with b = 1. In general, one can take b to be the product of the prime numbers that have an odd exponent in the prime factorization of m, and a = √m/b³. —David Eppstein (talk) 15:55, 25 February 2008 (UTC)

But the first sentence completely rules out odd-exponent primes. It says, if p divides m, so does p². That means that all the prime factors of m have to have even exponents. (Of course some of the exponents might be divisible by three as well, but nothing in that sentence requires it or even hints at it). Now I'm even more baffled. Colonies Chris (talk) 12:14, 27 February 2008 (UTC)

p² divides p³. Why would you think it doesn't? —David Eppstein (talk) 15:21, 27 February 2008 (UTC)

Sorry, I had got it into my head that the exponent of each prime factor had to be even, rather than simply >= 2. Now it makes sense. Could the explanation of the equivalence be done less 'technically' by considering each prime factor's exponent modulo 6? (0 & 3 are cubes, 2 and 4 are squares, 5 is a square times a cube, 1 is a cube times a square provided the exponent is > 1, which it always is). Colonies Chris (talk) 12:19, 28 February 2008 (UTC)

I also do not understand the equivalent equation. Why must it be a square times a cube? 36 is on the list. 36 = 2²3². It meets the requirement that every prime number diving it (2 and 3), then the square also divides it (4 and 9). But it is not a square times a cube. 12.13.126.253 (talk) 18:18, 26 August 2008 (UTC)Tim Forbis —Preceding unsigned comment added by 12.13.126.253 (talk) 18:08, 26 August 2008 (UTC)

36 = 6² × 1³. —David Eppstein (talk) 18:11, 26 August 2008 (UTC)

I would not consider 1 a necessary prime factor. I think that Chris has it right: The powers of each prime factor need to be >= 2, simply stated. Making it a "requirement" that it is a square times a cube becomes overly confusing especially when you factor in the fact that an identity is used. The only benefit I see, which I believe was the original purpose of the extra definition, is to break each powerful number into a square number and a square-free number. 12.13.126.253 (talk) 18:33, 26 August 2008 (UTC)Tim Forbis

Why do you think being prime has anything to do with the representation as a product of a square and a cube? It doesn't say that the cubes and squares are prime numbers, or even products of primes — just that they are positive integers. 1 is a positive integer. —David Eppstein (talk) 18:56, 26 August 2008 (UTC)

Just a quick question, I'm probably wrong but thought I'd ask anyway. Seeing as 1 is not a prime number (is it?), then surely 1 has no prime numbers dividing it, so there is no p, so no p²? in which case, 1 should not be on the list.88.108.217.171 (talk) 18:14, 6 July 2008 (UTC)

It is a vacuous truth that every prime dividing 1 divides it with an exponent of two or greater. Put another way, a number is not powerful if there is some prime that divides it only once. But for 1, there is no such prime, so it is powerful. —David Eppstein (talk) 18:33, 6 July 2008 (UTC)

Meaning?

"A powerful number is a positive integer m such that for every prime number p dividing m,"

Any prime number will divide m. But not integrally. Examinator~enwiki (talk) 22:34, 8 June 2015 (UTC)

In this context it should be obvious that "dividing" means that it divides it evenly. I think anyone who understands your objection will also understand what is intended here. —David Eppstein (talk) 22:44, 8 June 2015 (UTC)