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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… (More)

- John H. Jaroma
- 2004

It is known that with a very small number of exceptions, for a term of a Lehmer sequence {Un( √ R,Q)} to be prime its index must be prime. For example, F4 = U4(1,−1) = 3 is prime. Also, Un(1, 2) is prime for n = 6, 8, 9, 10, 15, 25, 25, 65, while Vn(1, 2) is prime for n = 9, 12, and 20. This criterion extends to the companion Lehmer sequences {Vn( √ R,Q)},… (More)

- JOHN H. JAROMA
- 2005

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,… (More)

- John H. Jaroma, Kamaliya N. Reddy
- The American Mathematical Monthly
- 2007

- Jennifer T. Betcher, John H. Jaroma
- The American Mathematical Monthly
- 2003

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 algebraic number theory in order to determine the pairs of integers under which Rn(b) has a prime divisor congruent to 1 modulo n. In this paper, we show how Snyder’s… (More)

A proof of the Lucas–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 elementary versions of this famous result. However, the two that we acknowledge in this note did so by using either algebraic numbers or group theory. It also appears that in… (More)

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