Recall that the triangular numbers are those integers $$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$ In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, which was conjectured by Fermat in 1638. On April 23, 2018, I noted that one of the three triangular numbers might be replaced by the sum of two powers of $5$.

QUESTION: Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?

I conjecture that the answer is yes. For example, $$4=T_1+T_1+5^0+5^0,\ \ 7 = T_0+T_1+5^0+5^1,\ \ 25 = T_0+T_5+5^1+5^1.$$ I have verified that each $n=2,3,\ldots,7\times10^9$ can be written as the sum of two triangular numbers and two powers of $5$. For the number of ways to write a positive integer $n$ as $T_a+T_b+5^c+5^d$ with $a,b,c,d\in\mathbb N$, $a\leqslant b$ and $c\leqslant d$, one may visit http://oeis.org/A303389. See also http://oeis.org/A303393 for the list of positive integers of the form $T_k+5^m\ (k,m\in\mathbb N)$.

I have some other similar conjectures, for example, I conjecture that every $n=2,3,\ldots$ can be written as the sum of two pentagonal numbers and two powers of $3$ (cf. http://oeis.org/A303401).

Gauss' triangular number theorem was proved via the theory of ternary quadratic forms. Are there any tools helpful to my question?


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