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%I #39 Jun 18 2021 12:40:43
%S 16,32,64,128,243,256,512,1024,2048,2744,4096,6561,8192,16384,32768,
%T 65536,131072,177147,185193,262144,474552,524288,614656,810000,941192,
%U 1048576,1124864,1419857,1500625,2097152,3241792,4194304
%N Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.
%C Beal's conjecture states that A, B, and C have a common prime factor.
%C Theorem. If A, B are odd and x, y are even, Beal's conjecture has no counterexample. Proof: Let D be odd, D > 1 and let w be even, w > 2. Then D^w == 9 (mod 24) while D == 0 (mod 3); otherwise, D^w == 1 (mod 24) (trivial). Any even C^z == {0; 8; 16} (mod 24): if C == 0 (mod 3), C^z == 0 (mod 24); if C == 1 (mod 3), C^z == 16 (mod 24); if C == 2 (mod 3), C^z == 8 (mod 24), while z is odd, and C^z == 16 (mod 24), while z is even (trivial). But C^z == (x'+y') (mod 24) where A^x = x' (mod 24), B^y = y' (mod 24); since (x'+y') = {2; 10; 18}, C^z == {2; 10; 18} (mod 24), which cannot be a counterexample to Beal's conjecture. - _Sergey Pavlov_, May 08 2021
%H Anatoly E. Voevudko and Charles R Greathouse IV, <a href="/A261782/b261782.txt">Table of n, a(n) for n = 1..1229</a> (first 196 terms from Voevudko)
%H American Mathematical Society, <a href="http://www.ams.org/profession/prizes-awards/ams-supported/beal-prize">Beal Prize</a>
%H Anatoly E. Voevudko, <a href="/A245713/a245713.txt">Description of all powers in b245713</a>
%H Anatoly E. Voevudko, <a href="/A261782/a261782.txt">Description of all powers in b261782</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Beal%27s_conjecture">Beal's conjecture</a>
%e 2^3 + 2^3 = 2^4.
%o (PARI) is(n)=if(ispower(n)<3, return(0)); for(x=3,logint((n+1)\2,2), for(A=2,sqrtnint(n,x), if(ispower(n-A^x)>2, return(1)))); 0 \\ _Charles R Greathouse IV_, Sep 03 2015
%o (PARI) list(lim)=my(v=List(),u=v,t); for(z=3,logint(lim\=1,2), for(C=2,sqrtnint(lim,z), listput(v,C^z))); v=Set(v); for(i=1,#v, for(j=i,#v, t=v[i]+v[j]; if(t>lim, break); if(setsearch(v,t), listput(u,t)))); Set(u) \\ _Charles R Greathouse IV_, Sep 03 2015
%Y Subsequence of A076467.
%Y Cf. A245713.
%K nonn
%O 1,1
%A _Anatoly E. Voevudko_, Aug 31 2015
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