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I'm working steadily through this mathematical puzzle but I'm completely stuck on 13 Down - I can't get anything to fit in. Could someone give me the answer to it? Thanks a lot.

I was saving it for this afternoon so can't help yet sorry - where is the best place to start? 4d and/or 19d ??

Yes - start with the 2 number answers which initially have 3 possibilities. Just the sort of puzzle for a miserable wet Bank Holiday!

8272

Thanks KT!

You could start with 4dn/19dn, but that won't help you enter anything in the grid immediately. 1 dn has only 2 possible entries, all the other options being 4 digits. One of those is quickly eliminated by considering 6 across.
Another useful clue is 20ac, which has only one answer that yields a 4-digit product with a 3-digit entry, so you can enter three digits in the grid without waiting for any crossing confirmation.

Finished! Quite a few of the low primes are 5 or 7. Very enjoyable test of logic. I'm glad I don't send in a solution as I think the answers would need a lot of checking.

I think you're all brilliant. I usually try the number puzzles (have only completed two) but I am at a loss to understand the instructions. Back to the knitting....

Yes indeed, understanding the preamble is the hard bit. Once you get a few answers, there are very few alternatives for the rest and all the solutions quickly fall into place

rosalind, using 1 down as an example, the sum of the two primes must be 124, 126 or 128, being (21 x 6) plus or minus 0 or 2. As their product has 4 digits but the grid entry only has 3 digits, they must both be 2-digit primes, differing by less than 9. Only 2 of the 15 combinations yield a pair of prime numbers - the correct choice is determined by checking cells. By tackling each clue in this fashion the possible choices can be reduced by cross-checking to give the unique solution. Simples !