reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th50:
  for p being odd Prime st n = (p-1)*(k*p+1) holds p divides CullenNumber n
  proof
    let p be odd Prime;
    assume
A1: n = (p-1)*(k*p+1);
    then
A2: 2|^n mod p = 1 by Th49;
    p-1 < p-0 by XREAL_1:15;
    then
A3: (p-1) mod p = p-1 by NAT_D:24;
A4: (k*p+1) mod p = 1 mod p by NAT_D:21
    .= 1 by INT_2:def 4,NAT_D:14;
A5: ((p-1)*(k*p+1)) mod p = (((p-1) mod p)*((k*p+1) mod p)) mod p by NAT_D:67
    .= (p-1) mod p by A4;
    (n*2|^n) mod p = ((n mod p)*(2|^n mod p)) mod p by NAT_D:67;
    then (CullenNumber n) mod p = (p-1+1) mod p by A1,A2,A3,A5,NAT_D:22
    .= 0 by NAT_D:25;
    hence thesis by INT_1:62;
  end;
