Sum of Squares of ‘m’ Consecutive Woodall Numbers

On numbers which are the sum of two squares ∗

1. Arithmeticians are accustomed to investigating the nature of numbers in many ways where they show their source, either by addition or by multiplication. Of the aforementioned kind, the simplest is composition from units, by which all integers are understood to arise from units. Then numbers can also thus be considered as they are formed from the addition of two or more other integers, which ...

Fibonacci s-Cullen and s-Woodall Numbers

The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation Fn = ms m ± 1 has only f...

The Hankel Transform of the Sum of Consecutive Generalized Catalan Numbers

X iv :m at h/ 06 04 42 2v 1 [ m at h. C O ] 1 9 A pr 2 00 6 The Hankel Transform of the Sum of Consecutive Generalized Catalan Numbers Predrag Rajković, Marko D. Petković, University of Nǐs, Serbia and Montenegro Paul Barry School of Science, Waterford Institute of Technology, Ireland Abstract. We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of c...

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