Order 5 Tetramagic Series Impossibility

Theorem. There are no order 5 tetramagic series.
Proof.
S1 = 65
S2 = 1105    ≡ 7 (mod 9)
S3 = 21125   ≡ 2 (mod 9)
S4 = 430729  ≡ 7 (mod 9)

From the Modulo 9/3 Tetramagic Series Lemma,
this tetramagic series must have
  7 entries of 2 (mod 3).

An order 5 tetramagic series must have
at least 7 entries.