On Sums of Primes and Triangular

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 n 6= 216 can be written in the form p + Tx with x ∈ Z 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.