Order 3 Bimagic Series Impossibility

Theorem. There are no order 3 bimagic series.
Proof.
S1 = 15
S2 = 95

x2 ≡ 0 (mod 4) when x is even.
x2 ≡ 1 (mod 4) when x is odd.

S2 ≡ 3 (mod 4),
thus all 3 odd entries must be odd.

x2 ≡ 1 (mod 8) when x is odd.

S2 ≡ 7 (mod 8),
but the squares of 3 odd entries add to 3 (mod 8).