Integer solutions of x^n + y^n = z^{n-1} - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:15:07Z http://mathoverflow.net/feeds/question/109125 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109125/integer-solutions-of-xn-yn-zn-1 Integer solutions of x^n + y^n = z^{n-1} joro 2012-10-08T09:13:49Z 2012-10-08T11:48:21Z <p>This is related to <a href="http://mathoverflow.net/questions/77022/diophantine-xpyqxyr" rel="nofollow">another question</a></p> <p>I am interested in the non-trivial integer solutions of $$x^n + y^n = z^{n-1}$$</p> <p>for $n \ge 4$. A solution is trivial if $xyz=0$ or $x = \pm y$. There are infinitely many rational solutions to $x^n + y^n = (x+y)^{n-1}$ parametrized in the linked question.</p> <p>For $n=5$ parametric solutions are $(-121 \cdot 2^{{\left(4 k + 3\right)}}, 363 \cdot 2^{{\left(4 k + 3\right)}},11^3 \cdot 2^{5k+4})$</p> <p>For $n > 5$ couldn't find any solution so far.</p> <blockquote> <p>Q1. Are there non-trivial solutions for $n > 5$?</p> <p>Q2. Are there $n$ for which non-trivial solutions don't exist?</p> <p>Q3. Is it possible for some $n > 5$ to find solutions without searching? Parametrizing all solutions (this might settle a case of Fermat-Catalan Conjecture)?</p> </blockquote> <p>Computationally the fastest way I found so far is pari's "t=thueinit(x^5+1,1);sol=thue(t,a^4);" though iterating over the divisors is another option. There are congruence conditions mod $\varphi^{-1}(n)$.</p> http://mathoverflow.net/questions/109125/integer-solutions-of-xn-yn-zn-1/109128#109128 Answer by Ilya Bogdanov for Integer solutions of x^n + y^n = z^{n-1} Ilya Bogdanov 2012-10-08T10:01:15Z 2012-10-08T11:48:21Z <p>Take any $a,b$ and set $c=a^n+b^n$. Then the triple $(ac^{n-2},bc^{n-2},c^{n-1})$ is a solution of your equation.</p> <p>Conversely, if $(x,y,z)$ is a solution and $d$ is their gcd, so $(x,y,z)=(ad,bd,cd)$, then you get $d(a^n+b^n)=c^{n-1}$. One of the solutions is presented above (with $d=c^{n-2}$). But there also exist smaller solutions --- they appear as soon as $a^n+b^n$ is not square-free.</p>