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Murky -

To be fair, assuming you have 1a, 9a, 19a, and 7d, the choices for 5d are very small and none will allow the bar between 13a and 14a to be part of the partition, given those primes that you have already used to that point. So that assumption is ruled out fairly early on in the solving process.

Actually, given the clue to 13a, you can determine 5d immediately from 1a, 9a and 19a and it becomes obvious that bar cannot be part of the partition.

I'm sure I must be missing a more efficient way of tackling this, because I can't really see a rational way of paring down the possibilities towards the end, other than trial and error.

I've got 3 possibilities for 2 down:
9739737
9991737
9311337

Could someone tell me which, if any, of these is correct please?

Demeter, the correct answer is 9,991,737. It cannot be 9,311,337 or 9,,739,737 because neither the 3rd nor the 5th number can be 3; this is because the palindrome for 11 down is 3_3 and you cannot have more than one 3 in the 3rd and 5th rows due to the rule for 10a and 15a.

Please could somebody confirm the sum of the digits for 3d as I'm not sure I'm correct?

Thanks crypticben.

Sum of digits for 3dn is 33.

I got 33 as the sum of the 7 digits in the middle row (starting with 12ac), please could someone confirm if this is right?

Correct CrypticBen

I've got 33 for the sum of the digits of the middle row and 37 for the sum of the digits of 3dn but I'm slightly confused as to where I have gone wrong. I'd be very grateful if somebody would be willing to engage in email correspondence as I can't say any more without giving away quite a lot of information which I know to be correct, possibly ruining the puzzle for others.

My email is mcdonnell . bc @ gmail . com
(please ignore the spaces in the email address, I had to insert them as it would not let me post any links due to me being a new member)