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A338800 Smallest number that is the sum of two distinct n-th powers of primes in two different ways.

%I A338800 #23 Jul 01 2024 15:05:00
%S A338800 16,410,6058655748,3262811042
%N A338800 Smallest number that is the sum of two distinct n-th powers of primes in two different ways.
%C A338800 The Lander, Parkin, and Selfridge conjecture implies that for n >= 5 a number can be the sum of two n-th powers of positive integers in at most one way, and in particular that a(n) does not exist for n >= 5. - _Robert Israel_, Nov 13 2020
%C A338800 a(5) > 10^31 if it exists. - _Michael S. Branicky_, Jul 01 2024
%H A338800 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture">Lander, Parkin, and Selfridge conjecture</a>.
%e A338800 16 = 3 + 13 = 5 + 11.
%e A338800 410 = 7^2 + 19^2 = 11^2 + 17^2.
%e A338800 6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3.
%e A338800 3262811042 = 7^4 + 239^4 = 157^4 + 227^4.
%p A338800 f:= proc(n) local S,P,p,pn,b;
%p A338800   S:= {}:
%p A338800   P:= {}:
%p A338800   p:= 1:
%p A338800   b:= infinity;
%p A338800   do
%p A338800    p:= nextprime(p);
%p A338800    pn:= p^n;
%p A338800    if pn > b then return b fi;
%p A338800    V:= select(`<`,map(`+`,P,pn),b);
%p A338800    newv:= V intersect S;
%p A338800    S:= S union V;
%p A338800    P:= P union {p^n};
%p A338800    if newv <> {} then
%p A338800      b:= min(newv);
%p A338800      S:= select(`<`,S,b);
%p A338800      P:= select(`<`,P, b);
%p A338800    fi;
%p A338800   od:
%p A338800 end proc:
%p A338800 map(f, [$1..4]); # _Robert Israel_, Nov 13 2020
%Y A338800 Cf. A016078, A046881, A338799.
%K A338800 nonn,hard,more
%O A338800 1,1
%A A338800 _Ilya Gutkovskiy_, Nov 10 2020