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%I #18 Feb 12 2021 03:48:45
%S 776151559,3518958160000
%N 3-powerful numbers that can be written as the sum of two coprime 3-powerful numbers.
%C Any counterexample to the Beal conjecture, i.e., the statement that the Diophantine equation A^x + B^y = C^z has no solution for pairwise coprime A, B, C and x, y, z > 2, has to be a term from this sequence.
%H A. Nitaj, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.108.4355">On a conjecture of Erdős on 3-powerful numbers</a>, Bull. London Math. Soc. 27 (1995), no. 4, 317-318.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Beal_conjecture">Beal conjecture</a>
%e 3518958160000 = 1392672604221 + 2126285555779 = 3^4 * 29^3 * 89^3 + 7^3 * 11^3 * 167^3 = 2^7 * 5^4 * 353^3.
%o (PARI) is_a036966(n) = my(e=factor(n)[, 2]~); if(#e==0 || vecmin(e) < 3, return(0)); 1
%o is(n) = if(!is_a036966(n), return(0)); my(x=1, y=n-1); while(x < y, if(gcd(x, y)==1 && n==x+y && is_a036966(x) && is_a036966(y), return(1)); x++; y--); 0
%Y Cf. A036966.
%K nonn,hard,bref,more
%O 1,1
%A _Felix Fröhlich_, Jan 07 2018
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