Mathematics

Given a positive integer n, we let sfp(n) denote the squarefree part of n. We determine all positive integers n for which max{sfp(n),sfp(n+1),sfp(n+2)}<=150 by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many n for which max{sfp(n),sfp(n+1),sfp(n+2)}<n^(1/3).

We consider matrix-vector equations of the form Ax = f(x) that are motivated by nonlinear oscillating systems such as the Tacoma Narrows Bridge. We identify a particular set, called the Fučı´k Spectrum, which is relevant to questions of solvability, and we develop theorems to describe the spectrum and show how it relates to the solvability of the matrix equation.