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

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

Offset: 1

Author

Anatoly E. Voevudko, Aug 31 2015

Keywords

• nonn

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

Examples

			2^3 + 2^3 = 2^4 = 16, so 16 is in the sequence.
		

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

Crossrefs

Subsequence of A076467.

Cf. A245713.

Programs

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