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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
 16, 32, 64, 128, 243, 256, 512, 1024, 2048, 2744, 4096, 6561, 8192, 16384, 32768, 65536, 131072, 177147, 185193, 262144, 474552, 524288, 614656, 810000, 941192, 1048576, 1124864, 1419857, 1500625, 2097152, 3241792, 4194304 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Beal's conjecture states that A, B, and C have a common prime factor. 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 LINKS Anatoly E. Voevudko and Charles R Greathouse IV, Table of n, a(n) for n = 1..1229 (first 196 terms from Voevudko) American Mathematical Society, Beal Prize Anatoly E. Voevudko, Description of all powers in b245713 Anatoly E. Voevudko, Description of all powers in b261782 Wikipedia, Beal's conjecture EXAMPLE 2^3 + 2^3 = 2^4 = 16, so 16 is in the sequence. PROG (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 (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 CROSSREFS Subsequence of A076467. Cf. A245713. Sequence in context: A317475 A335161 A239751 * A256818 A048170 A340624 Adjacent sequences: A261779 A261780 A261781 * A261783 A261784 A261785 KEYWORD nonn AUTHOR Anatoly E. Voevudko, Aug 31 2015 STATUS approved

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Last modified September 10 19:55 EDT 2024. Contains 375794 sequences. (Running on oeis4.)