Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: Does every integer $n>1$ have the form $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers? I conjecture that the answer is yes, i.e., each $n=2,3,\ldots$ can be written as the sum of two squares, a power of $3$ and a power of $5$. I have verified this for all $n=2,3,\ldots,6\times10^9$; for example, $$5 = 0^2 + 1^2 + 3^1 + 5^0\ \ \text{and}\ \ \ 25 = 1^2 + 4^2 + 3^1 + 5^1. $$ For the number of ways to write a positive integer $n$ as $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers and $a\leqslant b$, one may visit http://oeis.org/A303656.
Similarly, I guess that any integer $n>1$ can be written as the sum of two squares, a power of $2$ and a power of $3$. I also conjecture that any integer $n>5$ has the form $a^2+b^2+2^c+5\times2^d$ with $a,b,c,d$ nonnegative integers (cf. http://oeis.org/A303637). In contrast, R. C. Crocker [Colloq. Math. 112(2008), 235-267] proved that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of $2$. A more recent discussion, with references.
My above question looks quite challenging. Any ideas towards its solution?
