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SummaryIn the paper, the author presents an asymptotic formula for the arithmetic mean of the square roots of the first n positive integers using only tools from undergraduate calculus.

Let P(x,d,a) denote the number of primes pP(x;d,r) than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If P(x;d,n)>=P(x;d,r) for every x up to some large number, then one expects that N(x;d,n)>=N(x;d,r) for every x. Here N(x;d,a) denotes the number of integers n=N(x;4,1) for every x. In the process we express the so called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds true for a much larger class of constants. In a sequel to this paper the methods developed here will be used and somewhat refined to resolve a conjecture from P. Schmutz Schaller to the extent that the hexagonal lattice is `better' than the square lattice (see p. 201 of Bull. Amer. Math. Soc. 35 (1998), 193-214)., 26 pages

(2002). When Does a Sum of Positive Integers Equal Their Product? Mathematics Magazine: Vol. 75, No. 1, pp. 41-47.

The problem of representing integers as sums of terms of certain type is actual in number theory and its applications. We are interested in the average length of these expansions and the required number of auxiliary calculations. The paper deals with DBNS, chains, and the polyadic (factorial) expansions of positive integers.

The predictability of cycles in the series of Pythagorean triples led to an investigation that yielded numbers (x) that are associated with irrational square roots (√n). The cycles recur with geometric factors (cycle factors y) that are comprised of a positive integer x where y = x + √(x^2±1). On raising the cycle factors to the positive integer powers (ym), a series is generated where each consecutive member comes closer and closer to positive integers as the series progresses. A formula associates the square root (√n) with these series. Prime factorising the positive integers in the power series (xm) produces predictable patterns among the prime factors in the series. In general, power series that have each consecutive member in the series come closer to positive integers are limited to (x + √(x^2±r))m where x and r are positive integers and r < (x + 1)2 – x2 for the + r condition and r < x2 – (x – 1)2 for the – r condition.