On powerful numbers

The Power of Powerful Numbers

In this note we discuss recent progress concerning powerful numbers, raise new questions and show that solutions to existing open questions concerning powerful numbers would yield advancement of solutions to deep, long-standing problems such as Fermat's Last Theorem. This is primarily a survey article containing no new, unpublished results. A powerful number is a positive integer n which is div...

On a Conjecture of Erdos on 3-powerful Numbers

A conjecture of P. Erdos says that the diophantine equation x+y = z has infinitely many solutions with (x,y) — 1 and such that if a prime p divides xyz, then p divides xyz. In this paper, we give a proof of this conjecture. Let k ^ 2 be an integer. A /c-powerful number is a positive integer x with the property that if the prime p divides x, then p divides x (2-powerful numbers were called initi...

A conjecture of Erdös on 3-powerful numbers

Erdös conjectured that the Diophantine equation x + y = z has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers n for which p | n implies p3 | n. This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers x, y and z none of which is a perfect cube. This is now demonst...

On generalized fuzzy numbers

This paper first improves Chen and Hsieh’s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the λ-cutting sets or generalized λ-cutting sets, so that the generalized fuzzy numbers are integrat...

The Few, the Strong: Rat Cortex Features Small Numbers of Powerful Connections