Lenny Jones

A curious connection between fermat numbers and finite groups

1. INTRODUCTION. In the seventeenth century, Fermat defined the sequence of numbers F n = 2 2 n + 1 for n ≥ 0, now known as Fermat numbers. If F n happens to be prime, F n is called a Fermat prime. Fermat showed that F n is prime for each n ≤ 4, and he conjectured that F n is prime for all n (see Brown [1] or Burton [2, p. 271]). Almost one hundred years passed before Euler demonstrated in 1732 that F 5 is in fact composite. Ironically, it is now known that F n is composite for many values of n and, as of the date this article was written, no new Fermat primes had been discovered. In this paper we solve a problem in finite groups whose solution relies heavily on techniques from elementary number theory. While it is not unusual for this phenomenon to occur, the main result is surprisingly a direct consequence of the fact that F 5 is composite.

When Does Appending the Same Digit Repeatedly on the Right of a Positive Integer Generate a Sequence of Composite Integers

Abstract Let k be a positive integer, and suppose that k = a1a2 … at, where ai is the ith digit of k (reading from left to right). Let d ∈ {0,1,…,9}. For n ≥ 1, define In this article, we examine when sn is composite for all n.

On the Iteration of a Function Related to Euler's φ-Function

Abstract A unit x in a commutative ring R with identity is called exceptional if 1 − x is also a unit in R. For any integer n ≥ 2, define φe (n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Eulers φ-function, we develop analogous results on iterations of the function φe , when restricted to a particular subset of the positive integers.

On primitive covering numbers

In 2007, Zhi-Wei Sun defined a covering number to be a positive integer L such that there exists a covering system of the integers where the moduli are distinct divisors of L greater than 1. A covering number L is called primitive if no proper divisor of L is a covering number. Sun constructed an infinite set ℒ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given L ∈ℒ, we derive a formula that gives the exact number of coverings that have L as the least common multiple of the set M of moduli, under certain restrictions on M. Additionally, we disprove Sun’s conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.

Fibonacci Variations of a Conjecture of Polignac

Abstract. In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form , for some integer and some prime p. In 1950, Erdős constructed infinitely many counterexamples to Polignacs conjecture. In this article, we show that there exist infinitely many positive integers that cannot be written in either of the forms or , where is a Fibonacci number, and p is a prime.

Characterizing Finite Groups Using the Sum of the Orders of the Elements

We give characterizations of various infinite sets of finite groups under the assumption that and the subgroups of satisfy certain properties involving the sum of the orders of the elements of and . Additionally, we investigate the possible values for the sum of the orders of the elements of .

Sierpiński Numbers in Imaginary Quadratic Fields

Abstract. In 1960, Sierpiński proved that there exist infinitely many odd positive rational integers k such that is composite in for all . Any such integer k is now known as a Sierpiński number, and the smallest value of k produced by Sierpińskis proof is . In 1962, John Selfridge showed that is also a Sierpiński number, and he conjectured that this value of k is the smallest Sierpiński number. This conjecture, however, is still unresolved today. In this article, we investigate the analogous problem in the ring of integers of each imaginary field having class number one. More precisely, for each , with , that has unique factorization, we determine all , with minimal odd norm larger than 1, such that is composite in for all . We call these numbers Selfridge numbers in honor of John Selfridge.

A Variation on the Money-Changing Problem

Abstract The classical money-changing problem is to determine what amounts of money can be made with a given set of denominations. We present a variation on this problem and ask the following question: For what denominations of money a1, a2, …, at is there exactly one way, using the fewest number of coins possible, to make change for every amount that can be made? We provide a solution to this problem when we have at most three denominations.

POLYNOMIAL VARIATIONS ON A THEME OF SIERPIŃSKI

In 1960, Sierpinski proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdos and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

An infinite family of ninth degree dihedral polynomials

For any integer