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

16, 410, 6058655748, 3262811042

Offset: 1

Author

Ilya Gutkovskiy, Nov 10 2020

Keywords

• nonn
• hard
• more

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

Examples

			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.
		

Links

• Wikipedia, Lander, Parkin, and Selfridge conjecture.

Crossrefs

Cf. A016078, A046881, A338799.

Programs

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