Order 23 Pentamagic Series Impossibility

Theorem. There are no order 23 pentamagic series.
Proof.
S1 = 6095
S2 = 2151535
S3 = 854427575
S4 = 361935090463
S5 = 159703621325975

S4 ≡ 15 (mod 16), thus there are
15 odd entries and 8 even entries.

Let 2Aj+1 be an odd entry, j=1..15
Let 2Bj be an even entry, j=1..8
Let Tn = sum (Aj)n, j = 1..15,  n=1..5
Let Un = sum (Bj)n, j = 1..8, n=1..5

S1 = 2T1 + 15 + 2U1
S2 = 4T2 + 4T1 + 15 + 4U2
S3 = 8T3 + 12T2 + 6T1 + 15 + 8U3
S4 = 16T4 + 32T3 + 24T2 + 8T1 + 15 + 16U4
S5 = 32T5 + 80T4 + 80T3 + 40T2 + 10T1 + 15 + 32U5

(S3 - 15)/2 =
  4T3 + 6T2 + 3T1 + 4U3 = 427213780
thus
  T1 .. T5 are even.

(S1 - 15)/2 =
  T1 + U1 = 3040
thus
  U1 .. U5 are even.

(S3 - 15)/4 =
  2T3 + 3T2 + 3T1/2 + 2U3 = 213606890
thus
  T1/2 is even.

(S4 - 15)/16 =
  T4 + 2T3 + 3T2/2 + T1/2 + U4 = 22620943153
thus
  T2/2 is odd.

Let T2/2 = 2z+1 so that T2 = 4z+2.

((S3 - 15)/4 - 6)/2 =
  T3 + 6z + 3T1/4 + U3 = 106803442
thus
  T1/4 is even

((S5 - 15)/4 - 20)/2 =
  4T5 + 10T4 + 10T3 + 20z + 5T1/4 + 4U5 = 19962952665735
thus
  T1/4 is odd

T1/4 can't be both even and odd.