Except for $\ p=2\ $ primes split into two disjoint classes, $\ p\equiv1\mod4\ $ and $\ p\equiv3\mod4.\ $ Squares respect this partition, odd prime $\ p=m^2+n^2\ \Leftrightarrow\ p\equiv1\mod4.\ $ On the other hand, *triangles* $\ \binom k2\ $ are oblivious to the $\mod4\ $ classification, as well as to the other classification $\mod6\ $ (each prime different from $2$ and $3$ is congruent to $1$ or $-1$).


**Question**   What is a simple characterization of primes of the form $\ p=\binom m2+\binom n2\ ?\ $ or, in a sense, there are no simple characterizations (?).



Equation $\ p=\binom m2+\binom n2$
$p={m\choose 2}+{n\choose 2}$ is equivalent to $8p+2=(2m-1)^2+(2n-1)^2$. On the other hand, if $8p+2$ is a sum of two squares, these squares must be both odd (this is seen modulo 4). So, applying the criterion for representability as a sums of two squares, we get that *$p$ is a sum of two triangular numbers if and only if the prime factorization of $4p+1$ does not contain prime divisors $\equiv 3\pmod 4$ in odd power.* I do not think that there is a more explicit characterization.


