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
KEYWORD
nonn
AUTHOR
Anatoly E. Voevudko, Aug 31 2015
STATUS
approved