Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester\u27s bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980. What motivated Sylvester\u27s sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888

Triangular repunit-there is but 1

summary:In this paper, we demonstrate that 1 is the only integer that is both triangular and a repunit

On the appearance of primes in linear recursive sequences

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, Un+2(R,Q)=RUn+1−QUn, U0=0, U1=1, and Vn+2(R,Q)=RVn+1−QVn, V0=2,  V1=R, where R and Q are relatively prime integers and n∈{0,1,…}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We add that the rank of apparition of an odd prime p in a specific Lehmer sequence is the index of the first term that contains p as a divisor. In this paper, we obtain results that pertain to the rank of apparition of primes of the form 2np±1. Upon doing so, we will also establish rank of apparition results under more explicit hypotheses for some notable special cases of the Lehmer sequences. Presently, there does not exist a closed formula that will produce the rank of apparition of an arbitrary prime in any of the aforementioned sequences

On the global asymptotic stability of xn+1=(a+bxn)/(A+xn−1)

We shall utilize a detailed study of the semicycles of the positive solutions to establish under appropriate hypotheses the global asymptomatic stability of the positive equilibrium