Order 15 Tetramagic Series Impossibility

Theorem. There are no order 15 tetramagic series.
Proof.
S1 = 1695
S2 = 254815
S3 = 43095375
S4 = 7774354687

S4 ≡ 15 (mod 16) so all 15 entries are odd.

Let 8Aj+1 be the square of an odd entry, j=1..15.
Let Tn = sum(j=1..15) (Aj)n, n=1,2.

  S2 = 8T1 + 15
or
  (S2 - 15)/8 = T1 = 31850
thus
  T1 is even.

  S4 = 64T2 + 16T1 + 15
or
  (S4 - 15)/16 = 4T2 + T1 = 485897167
thus
  T1 is odd.

T1 can't be both odd and even.