
theorem
  for k be Integer holds
    (ex a,b be Integer st a|^2 - b|^2 = k) iff k mod 4 <> 2
  proof
    let k be Integer;
    L1: (ex a,b be Integer st a|^2 - b|^2 = k) implies k mod 4 <> 2
    proof
      assume ex a,b be Integer st a|^2 - b|^2 = k; then
      consider a,b be Integer such that
      A1: a|^2 - b|^2 = k;
      A2: (a - b)*(a + b) = k by A1,NEWTON01:1;
      parity (a - b) = parity (a + b) by NEWTON05:53; then
      (a - b is odd & a + b is odd) or (a - b is even & a + b is even); then
      k is odd or 2*2 divides k by A2,NEWTON02:2; then
      k mod 2*2 is odd or k mod 2*2 = 0 by INT_1:62;
      hence thesis;
    end;
    k mod 4 <> 2 implies (ex a,b be Integer st a|^2 - b|^2 = k)
    proof
      assume
      A1: k mod 4 <> 2;
      k mod (3+1) = 0 or ... or k mod (3+1) = 3 by NUMBER03:11; then
      per cases by A1;
      suppose k mod 4 = 0; then
        4 divides k by INT_1:62; then
        reconsider m = k/4 as Integer;
        (m+1)|^2 - (m - 1)|^2 = ((m + 1)+(m-1))*((m+1)-(m-1)) by NEWTON01:1
        .= 4*m;
        hence thesis;
      end;
      suppose k mod 4 = 1 or k mod 4 = 3; then
        k mod (2*2) = 2*0 + 1 or k mod (2*2) = 2*1 + 1; then
        k is odd; then
        reconsider m = (k-1)/2 as Integer;
        (m + 1)|^2 - m|^2 = (m+1+m)*(m+1 - m) by NEWTON01:1
        .= 2*m + 1;
        hence thesis;
      end;
    end;
    hence thesis by L1;
  end;
