On sums of primes and triangular numbers
Abstract
We study whether sufficiently large integers can be written in the form cp+Tx, where p is either zero or a prime congruent to r mod d, and Tx=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2ap-r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+Tx with x an integer and p a prime or zero; another conjecture asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a prime and x a positive integer.
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