reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th50:
  not 341 divides 3|^341-3
  proof
A1: Euler 31 = 31-1 by EULER_1:20,NUMERAL2:24;
A2: 1 mod 31 = 1 by NAT_D:24;
A3: (3|^30)|^11 = 3|^(30*11) by NEWTON:9;
A4: 3|^330*3|^11 = 3|^(330+11) by NEWTON:8;
    (3|^Euler 31) mod 31 = 1 by EULER_2:18,NUMERAL2:24,PEPIN:41,INT_2:30;
    then (3|^30)|^11,1|^11 are_congruent_mod 31 by A1,A2,NAT_D:64,GR_CY_3:34;
    then 3|^330*3|^11,1*3|^11 are_congruent_mod 31 by A3,INT_4:11;
    then 3|^341,-18 are_congruent_mod 31 by A4,Lm1115,INT_1:15;
    then 3|^341-3,-18-3 are_congruent_mod 31;
    then not 31 divides 3|^341-3 or 31 divides -21 by Th4;
    then not 11*31 divides 3|^341-3 by Lm1116,Th3,INT_1:62;
    hence not 341 divides 3|^341-3;
  end;
