Timeline for For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Nov 10, 2015 at 2:55 | comment | added | Tito Piezas III | By the way, if you want, can you add the list of $n\leq100$ you found that are solvable? It will be a useful piece of information to your answer. | |
| Nov 9, 2015 at 13:26 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Made some simplifications.
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| Nov 9, 2015 at 13:19 | vote | accept | Tito Piezas III | ||
| Nov 9, 2015 at 13:18 | comment | added | Tito Piezas III | Thanks! With $n=13$, I get, $$-a+13b+c = -d+13e+f$$ where $$(-p + 7 q - 6 u)^k+(6 p - u)^k+( 6 q + 7 u)^k = \\(-p + 7 q + 6 u,)^k+(6 p + u)^k+( 6 q - 7 u)^k$$ and already true for $k=1,2$. It is also for $k = 6$ if $$37 p^3 + 75 p^2 q + 99 p q^2 + 221 q^3 = -(37 p + 221 q) u^2$$ From your transformations, I get initial points $p/q = -565/173$ and $p/q = -1049/193$. No wonder I was not able to find this $n$ since I only limited the search to $p < \pm500$ and $q<200$. | |
| Nov 9, 2015 at 9:23 | history | answered | Allan MacLeod | CC BY-SA 3.0 |