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SummaryWe study some sequences of multi-polygonal numbers, specifically the square triangular, oblong triangular, pentagonal triangular, and pentagonal square numbers. To do so, we state and prove theorems that relate the factors of a given triangular number to a pair of larger triangular numbers with a triangular sum, and relate a triangular number that is three times another triangular number (or three times a square) to a larger pair of triangular numbers with the same property.

Summary. We show that the sum of any triangular number and a term in each of two triangular number subsequences is also a perfect square.

It is well known that the sum of two consecutive terms from the sequence of triangular numbers is a perfect square. We show that the sum of two consecutive terms from a subsequence of that sequence...

conditions (1) and (2). While it is beyond what students could reasonably be expected to solve during the exam, clearly the elegant question is: What is the maximum order of a subset S c U in which no two elements commute? We shall show that this order is 32,390, so that any subset of 32,391 matrices in U must include a commuting pair. The method we use can be applied to numerous variations of the problem posed. For our purposes, the requirement of perfect square entries creates an extra level of difficulty. Consequently, let us first solve a variation that seems more natural. We shall then return to the original question of finding the largest totally noncommuting set S when the entries must be perfect squares. Specifically, we ask what is the order of a maximum noncommuting set among all 2 X 2 matrices (a b) with nonnegative integer entries bounded by 0 < a, b, c, d < n. In this case our universal set U has (n + 1)4 matrices in it.

(1991). Tetrahedral Numbers as Sums of Square Numbers. Mathematics Magazine: Vol. 64, No. 2, pp. 104-108.

Since all the numbers listed in (1) and (2) are perfect squares, it follows, by formula (3), that the triangle with sides a, b, and c is a solution of the PSTP (once "integers" has been replaced by "rationals" in the statement of the problem). Conversely, assume now that the PSTP has a solution. Let a, b, and c be the sides of a triangle that solves this problem. We may assume that gcd(a, b, c) = 1. Indeed, otherwise, let d = gcd(a, b, c). Since a, b, and c are all perfect squares, so is d. Then the triangle with sides a/d, b/d, and c/d still solves the PSTP, and gcd(a/d, b/d, c/d) = 1.

Computer experiments suggest conjectures which may be confirmed probabilistically and then proved by exact analytic means.

(1973). Perfect Square Patterns in the Pascal Triangle. Mathematics Magazine: Vol. 46, No. 4, pp. 203-208.

(1964). Consecutive Integers Whose Sum of Squares is a Perfect Square. Mathematics Magazine: Vol. 37, No. 1, pp. 19-32.