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That's a better solution, alwayspuzzled (hadn't heard of that down entry before - but it exists).

Sorry, do not agree, alwayspuzzled. Putting the symbol you suggest at the end of 13 down throws the whole sum out. Also do not agree that 28 across is a maths term as it does not follow the pattern of the other three unclueds.

In spite of ALL suggestions on this forum so far, I remain totally in the dark.

I'm still clueless about the unclued lights despite the extra clues here. Why 'full' rather than full in the Preamble? What is to be enumerated and how? Etc.

This gets my vote for the most unfair Spectator puzzle yet. I never expected to say it, but bring back Doc and Brewer's to give those of us who have solved all the clued lights a sporting chance of finishing.

Hi again all, I thought my hint would have been sufficient but clearly not.
Every number can be expressed in terms of multiplication of prime numbers (hence the puzzle title) and here 2525 (the puzzle number) = 101 x 5 x 5 which appears here in the middle rows as 101 X 5 squared.

Sorry, ginge but don't agree with that either. Now you have two 5s and an unclued which does not fit the other three. You are right in the main, in my opinion, but your logic for 'squared' seems not quite right.

Hi todorovia, I only have one "5" in my grid with the entries as per alwayspuzzled's details.

Not 'squared' but 'squares' - which is a description of the number.

As this is an answers site, todorovia - could you give some more detail for your alternative solution (two options is bad enough, but three!)?

I opted for the first 'solution' - using the same two numbers - because the TV programme started life at the down entry and that by treating the numbers with the maths term (separately) you get the sum in a simple side by side arrangement, as well as by multiplication - using just one of them.
But I think I prefer the pure maths option and I can't see why it doesn't work - can you please explain (maths isn't my strong suit... by a long way).

I should have added (as per alwayspuzzled's) amended ...