And belated thanks, Crosswhit, for your confirmation re the ending of 25A.
I found the puzzle mildly interesting in the early stages, it was nice that there was exposure to Fermat, Mersenne & Pascal. But really that was as far as my praise stretches. After the early phase of identifying properties for locations, the puzzle turned into fairly typical fare for numerics, therein lies a major criticism of the genre, the possible list of values for one entry excluding some of the possible list of values for another entry...eventually the whole business being confirmed by a lack of degrees of freedom about 5 clues later. Which is just a bore.
Another criticism of the genre is that the puzzles are rarely self-checking, at least not to the level of self-checking provided by definition/wordplay - can anyone please confirm that I did indeed finish the puzzle correctly, and that there weren't possible values in tables which I somehow missed. Which has happened before.
Eventually I could solve the NW corner after identifying only 2 possible values for 26 : 522 & 527, but only 522 is the sum of 2 squares. Which led in turn to filling all the remaining entries quite quickly, the tetrahedral number being 26235 by my reckoning. The other entry I wouldn't mind confirmation of is 21D, which I have as 12403.