It is known that with a very small number of exceptions, for a term of a Lehmer sequence {Un( âˆš R,Q)} to be prime its index must be prime. For example, F4 = U4(1,âˆ’1) = 3 is prime. Also, Un(1, 2) isâ€¦ (More)

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations,Un+2( âˆš R,Q)=RUn+1âˆ’QUn,U0 = 0,U1 = 1, and Vn+2( âˆš R,Q)=RVn+1âˆ’QVn,â€¦ (More)

A perfect number is any positive integer that is equal to the sum of its proper divisors. Several years ago, F. Luca showed that the Fibonacci and Lucas numbers contain no perfect numbers. In thisâ€¦ (More)

Twenty-five years ago, W. M. Snyder extended the notion of a repunit Rn to one in which for some positive integer b, Rn(b) has a b-adic expansion consisting of only ones. He then applied algebraicâ€¦ (More)

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 unawareâ€¦ (More)

A proof of the Lucasâ€“Lehmer test can be difficult to find, for most textbooks that state the result do not prove it. Over the past two decades, there have been some efforts to produce elementaryâ€¦ (More)