Order 19 Pentamagic Series Impossibility

Theorem. There are no order 19 pentamagic series.
Proof.
S1 = 3439
S2 = 828799
S3 = 224707699
S4 = 64985300791
S5 = 19576759444579

S4 ≡ 7 (mod 16)
so there must be 7 odd entries and 12 even entries.

Let 2Aj+1 be an odd entry, j=1..7
Let 2Bj be an even entry, j=1..12
Let Tn = sum (Aj)n, j = 1..7,  n=1..5
Let Un = sum (Bj)n, j = 1..12, n=1..5

S1 = 2T1 + 7 + 2U1
S2 = 4T2 + 4T1 + 7 + 4U2
S3 = 8T3 + 12T2 + 6T1 + 7 + 8U3
S4 = 16T4 + 32T3 + 24T2 + 8T1 + 7 + 16U4
S5 = 32T5 + 80T4 + 80T3 + 40T2 + 10T1 + 7 + 32U5

(S3 - 7)/2 =
  4T3 + 6T2 + 3T1 + 4U3 = 112353846
thus
  T1 is even and T2..T5 are even.

(S1 - 7)/2 =
  T1 + U1 = 1716
thus
  U1 is even and U2..U5 are even.

(S3 - 7)/4 =
  2T3 + 3T2 + 3T1/2 + 2U3 = 56176923
thus
  T1/2 is odd

(S4 - 7)/16 =
  T4 + 2T3 + 3T2/2 + T1/2 + U4 = 4061581299
thus
  T2/2 is even

Let T1/2 = 2z+1

((S3 - 7)/4 - 3)/2 =
  T3 + 3T2/2 + 3z + U3 = 28088460
thus z is even

((S5 - 7)/4 - 5)/2 =
  4T5 + 10T4 + 10T3 + 5T2 + 5z + 4U5 = 2447094930569
thus z is odd

z can't be both even and odd.