The Sum of Three Consecutive Squares

Three Consecutive Almost Squares

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.

The Sum of Two Squares

An Incremental DC Algorithm for the Minimum Sum-of-Squares Clustering

Here, an algorithm is presented for solving the minimum sum-of-squares clustering problems using their difference of convex representations. The proposed algorithm is based on an incremental approach and applies the well known DC algorithm at each iteration. The proposed algorithm is tested and compared with other clustering algorithms using large real world data sets.

The Sum of Digits Function of Squares

We consider the set of squares n2, n < 2k, and split up the sum of binary digits s(n2) into two parts s[<k](n 2) + s[≥k](n 2), where s[<k](n 2) = s(n2 mod 2k) collects the first k digits and s[≥k](n 2) = s(bn2/2kc) collects the remaining digits. We present very precise results on the distribution on s[<k](n 2) and s[≥k](n 2). For example, we provide asymptotic formulas for the numbers #{n < 2k ...

A 'sum of Squares' Theorem for Visibility Complexes a 'sum of Squares' Theorem for Visibility Complexes

We present a new and simpler method to implement in constant amortized time the ip operation of the so-called 'Greedy Flip Algorithm', an optimal algorithm to compute the visibility graph/complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses only the incidence structure of the visibility complex and only the primitive predicate which states tha...