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A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2. 4


%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.

%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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Last modified February 22 17:14 EST 2020. Contains 332140 sequences. (Running on oeis4.)