Any integer solutions to $3n^{2}+3n+1=m^{3}$?
Are there any integer solutions to $3n^{2}+3n+1=m^{3}$?
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number-theory
1 Answers
16
We have that $3n^2 + 3n + 1 = m^3$ Adding $n^3$ to both sides, we get that $n^3 + 3n^2 + 3n+1 = m^3 + n^3$ $(n+1)^3 = m^3 + n^3$ Hence, no solution exists except for trivial solutions. (This is of-course the Fermat's theorem where the exponent is $3$. The proof for the exponent $3$ is actually not that hard.)
And the trivial solutions are
$n=0 \text{ gives us }m=1$
$n=-1 \text{ gives us }m=1$