reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  5|^4+(5+1)|^4 is composite &
  not ex n being Nat st n < 5 & n|^4+(n+1)|^4 is composite
  proof
    set x = 5|^4+(5+1)|^4;
    thus 2 <= x by Lm6,Lm7;
    x = 17 * 113 by Lm6,Lm7;
    hence x is non prime by Th13,PEPIN:60;
    given n being Nat such that
A1: n < 5 and
A2: n|^4+(n+1)|^4 is composite;
    n <= 5-1 by A1,INT_1:52;
    then n = 0 or ... or n = 4;
    hence thesis by A2,Th56,Th57,Th58,Th59;
  end;
