Any integer solutions to $3n^{2}+3n+1=m^{3}$?
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
||||
|
|
|
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 $$n=0 \text{ gives us }m=1$$ $$n=-1 \text{ gives us }m=1$$ |
||||
|
|
