<s>[INST] $1^2+17^2+19^2=3\cdot 7\cdot 31$ $1^2+17^2+19^2=3\cdot 7\cdot 31$.
$3$, $7$ and $31$ are the first three Mersenne primes
Let be $a$, $b$ and $c$ positive integers.
Let $M_n$ denote the n-th Mersenne prime.
Can be it proven that there are infinitely many positive a,b,c such that:
$a^2+b^2+c^2=P_n$, where $P_n$ is the product of the first $n$ Mersenne primes?
 [/INST] For $n \geq 3$, the product of the first $n$ Mersenne primes is congruent to $3 \times (-1)^{n-1},$ (mod $8$), since every Mersenne prime other than $3$ is congruent to $7$ (mod $8$).On the other hand, Legendre's three-square theorem asserts that the only positive integers which are not expressible as the sum of three integer squares are integers of the form $4^{a}(8b+7)$ with $a,b$ integers. No integer of the latter form is congruent to $\pm 3$ (mod $8$), so for $n \geq 3$, the product of the first $n$ Mersenne primes is expressible as the sum of three integer squares (but clearly, each such number is so expressible in only finitely many ways in this way).
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