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How could I prove that $$7n^4 + (12s + 6)n^3 + (6s^2 + 6s + 1)n^2 - 2s^3 = 0$$ has no natural solutions?

The solutions lead to summation identities and solving this one would prove there are no cubic sequences. On the other hand, any proof that there are no cubic sequences would also prove this has no solutions.

I felt that since $s^3$ is the only negative term and it is dominated by $n^4$ there may be hope in solving this.

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up vote 2 down vote accepted

Define the two variable polynomial $$p(n,s)=7n^4+(12s+6)n^3+(6s^2+6s+1)n^2-2s^3.$$ Now make the change of variables $n=r,s=m-r$, and note this is an invertible change since we have $r=n,m=n+s$ as the way to get back from $(r,m)$ to the original variables $(n,s)$. Now $p$ has a simpler formula in the variables $r,m$, namely we have $$p(r,m-r)=r(r+1)[ r(r+1)+6m^2 ] - 2m^3.$$

Now if this were zero for some integers $r,m$ we would have $$2m^3 = r(r+1)[r(r+1)+6m^2],$$ and then rearranging and factoring we would have $$ 2m^2 [m-3r(r+1)]=[r(r+1)]^2.$$ Then since the right side here is positive we would have $m>3r(r+1)$, from which $$2m^2 >18 [r(r+1)]^2,$$ which is not possible because $2m^2$ is actually a factor of $[r(r+1)]^2$ in the previous equation.

Note: The change of variables here really amounts to using $m$ as the final cube in the longer sequence of cubes to the left of the proposed equation in the original search for examples like $3^3+4^3+5^3=6^3+7^3$ (which is false), so that the search is for consecutive sequences of cubes having total length odd, where the middle cube plus the ones before it sum to the same as the higher cubes.

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magnificent!!!! – sperners lemma Nov 17 '12 at 19:57
I had been looking via using the middle cube, since then one can set the middle cupe equal to a sum of differences of cubes above and below it, and saw the nice formula result. Took me a while to correlate this approach with the other one expressing the same search in a different way! – coffeemath Nov 17 '12 at 20:03
this proof is beautiful and i appreciate your notes on how you discovered it. – sperners lemma Nov 17 '12 at 20:04

Taking your item above as $f(n,s),$ with $n \geq 0$ I get $$ f(-1 - n, s + 2 n + 1 ) = f(n,s). $$ Just a thought.

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