# Volume 11, 2005, Number 2

**Volume 11** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The sum-of-divisors minimum and maximum functions**

*Original research paper. Pages 1—8*

József Sándor

**Note on some identities related to binomial coefficients**

*Original research paper. Pages 9—12*

Mladen V. Vassilev-Missana

**Fermat’s theorem on binary powers**

*Original research paper. Pages 13—22*

J. Leyendekkers and A. Shannon

Abstract

Modular rings are used to analyse integers of the form *N* = 2^{m} + 1. When *m* is odd, the integer structure prevents the formation of primes. When m is even, *N* ‘commonly’ has a right-end-digit of 5 and so is not a prime then. However, a sequence defined by *m* = 4 + 4*q*, *q* = 0, 1, 2, 3 can generate some primes as the right-end-digit is 7. Elements of this sequence satisfy the non-linear recurrence relation *G*_{m} = *G*_{m−1}^{2} − 2*G*_{m−1} + 2. Fermat numbers, where *m* = 2^{n} satisfy this recurrence relation. However, in this case, the integer structure reveals that primes are limited to *n* < 5.

**The birthday inequality**

*Original research paper. Pages 23—24*

K. Atanassov

**Volume 11** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4