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 Th37:
  3|^m divides 2|^(3|^m)+1
  proof
    defpred P[Nat] means 3|^$1 divides 2|^(3|^$1)+1;
A1: P[0]
    proof
      3|^0 = 1 by NEWTON:4;
      hence thesis by INT_2:12;
    end;
A2: for m st P[m] holds P[m+1]
    proof
      let m;
      set a = 3|^m;
      given k being Nat such that
A3:   2|^a+1 = a*k;
      set z = a*k;
      per cases;
      suppose
A4:     m = 0;
        2|^(3|^1)+1 = 2*2*2+1 by POLYEQ_5:2
        .= 3*3;
        then 3|^1 divides 2|^(3|^1)+1;
        hence thesis by A4;
      end;
      suppose 0 < m;
        then reconsider m1 = m-1 as Nat;
        set t = 3|^m1*3|^m*k*k*k - 3*(3|^m1*k*k) + k;
A5:     3|^(m+1) = 3*a by NEWTON:6;
A6:     z|^3 = z*z*z by POLYEQ_5:2
        .= a*a*a*k*k*k
        .= 3|^(m+(m1+1))*3|^m*k*k*k by NEWTON:8
        .= 3|^(m+1+m1)*3|^m*k*k*k
        .= 3|^(m+1)*3|^m1*3|^m*k*k*k by NEWTON:8
        .= 3|^(m+1)*(3|^m1*3|^m*k*k*k);
A7:     z|^2 = z*z by POLYEQ_5:1
        .= a*a*k*k
        .= 3|^(m+(m1+1))*k*k by NEWTON:8
        .= 3|^(m+1+m1)*k*k
        .= 3|^(m+1)*3|^m1*k*k by NEWTON:8
        .= 3|^(m+1)*(3|^m1*k*k);
        2|^(3|^(m+1)) = (z-1)|^3 by A3,A5,NEWTON:9
        .= z|^3 - 3*z|^2*1 + 3*1|^2*z - 1|^3 by POLYEQ_5:5
        .= 3|^(m+1)*t-1 by A5,A6,A7;
        then 3|^(m+1) divides 2|^(3|^(m+1))+1;
        hence thesis;
      end;
    end;
    for m holds P[m] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
