For all integers $r,s>2$, there are infinitely many primes which can be written as the sum of an r-gonal number and an s-gonal number. In general, any linear combination of an r-gonal number and an s-gonal number with coprime positive integer coefficients produces infinitely many primes. However, unless $r=s=4$, such primes can not be described by finitely many congruences.
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Bhattacharya et al. (2024) studied this question.
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