OFFSET
1,1
COMMENTS
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
a(5) > 10^31 if it exists. - Michael S. Branicky, Jul 01 2024
LINKS
EXAMPLE
16 = 3 + 13 = 5 + 11.
410 = 7^2 + 19^2 = 11^2 + 17^2.
6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3.
3262811042 = 7^4 + 239^4 = 157^4 + 227^4.
MAPLE
f:= proc(n) local S, P, p, pn, b;
S:= {}:
P:= {}:
p:= 1:
b:= infinity;
do
p:= nextprime(p);
pn:= p^n;
if pn > b then return b fi;
V:= select(`<`, map(`+`, P, pn), b);
newv:= V intersect S;
S:= S union V;
P:= P union {p^n};
if newv <> {} then
b:= min(newv);
S:= select(`<`, S, b);
P:= select(`<`, P, b);
fi;
od:
end proc:
map(f, [$1..4]); # Robert Israel, Nov 13 2020
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Ilya Gutkovskiy, Nov 10 2020
STATUS
approved