ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 22 Apr 2021 23:08:18 +0200Matrix solutions to $A^k + B^k = C^k$https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.Thu, 22 Apr 2021 20:02:36 +0200https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/Comment by vdelecroix for <p>Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.</p>
https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56778#post-id-56778Repeating John question: what is the link with the Sage software?Thu, 22 Apr 2021 23:08:18 +0200https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56778#post-id-56778Comment by vdelecroix for <p>Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.</p>
https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56777#post-id-56777`A = diag(1,0,0)`, `B = diag(0,1,0)` and `C = diag(1,1,0)` works for all k.Thu, 22 Apr 2021 23:07:30 +0200https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56777#post-id-56777Comment by John Palmieri for <p>Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.</p>
https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56774#post-id-56774There are basically trivial examples, for example with n=2: $A = e_{11}$, $B = e_{22}$, $C = I$ (where $e_{11}$ has a 1 in the (1,1) spot, zeroes elsewhere). What does this have to do with Sage? And how do you characterize "trivial" so as to exclude such examples?Thu, 22 Apr 2021 21:06:15 +0200https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56774#post-id-56774Comment by rewi for <p>Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.</p>
https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56776#post-id-56776actually n is strictly greater than 2 and none of A,B,C are non zero and non nilpotent matrixThu, 22 Apr 2021 21:10:08 +0200https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56776#post-id-56776Comment by John Palmieri for <p>Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.</p>
https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56775#post-id-56775An even more trivial example: all of the matrices could be zero, or $A=0$ and $B=C$.Thu, 22 Apr 2021 21:07:39 +0200https://ask.sagemath.org/question/56773/matrix-solutions-to-ak-bk-ck/?comment=56775#post-id-56775