0803 . 3737 Conjectures on Sums of Primes and Triangular Numbers

We raise the following new conjecture: Each natural number n 6= 216 can be written in the form p + Tx, where p is 0 or a prime, and Tx = x(x + 1)/2 is a triangular number. Some extensions and variants of this conjecture are also discussed. 1. On integers of the form p + Tx Since 1+ 2+ · · ·+ n = n(n+1)/2, those integers Tx = x(x+1)/2 with x ∈ Z are called triangular numbers. Here is a list of triangular numbers not exceeding 200: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190. For n, x ∈ N = {0, 1, 2, . . .}, clearly Tn 6 x ⇐⇒ 2n+ 1 6 √ 8x+ 1. Thus, for any x > 0, there are exactly ⌊( √ 8x+ 1 − 1)/2⌋ + 1 triangular numbers not exceeding x. Here is an important observation of Fermat. Fermat’s Assertion. Each n ∈ N can be written as a sum of three triangular numbers. An equivalent version of this assertion states that for each n ∈ N the number 8n + 3 is a sum of three squares (of odd integers). This is a consequence of the following profound theorem (see, e.g., [G, pp. 38–49] or [N, pp. 17-23]) due to Gauss and Legendre: A natural number can be written as a sum of three squares of integers if and only if it is not of the form 4(8l + 7) with k, l ∈ N. Prime numbers play key roles in number theory. By the prime number theorem, for x > 2 the number π(x) of primes not exceeding x is approximately x/ log x (in fact, limx→∞ π(x)/(x/ logx) = 1). Here is a famous result due to I. M. Vinogradov [V]. 2000 Mathematics Subject Classification. Primary 11P99; Secondary 05A05, 11A41. 1