reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem Th24:
  5 satisfies_Sierpinski_problem_121_for 1
  proof
    set n = 5;
    2|^32 + 1 >= 1 + 1 by XREAL_1:6,NAT_1:14;
    hence 1 * 2|^(2|^n) + 1 is composite by Lm5,NAT_6:44;
    let m be positive Nat;
    assume m < n;
    then per cases by Th3;
    suppose m = 0;
      hence thesis;
    end;
    suppose m = 1;
      hence thesis by Lm2,XPRIMES1:5;
    end;
    suppose m = 2;
      hence thesis by Lm2,Lm4,XPRIMES1:17;
    end;
    suppose m = 3;
      hence thesis by Lm3,Lm6,XPRIMES1:257;
    end;
    suppose m = 4;
      hence thesis by Lm4,Lm7,PEPIN:62;
    end;
  end;
