A213721

[Robert Israel]

A213721 is " Smallest prime that is the sum of four nonzero squares in exactly n ways."

Here "ways" doesn't count permutations of the squares as different (perhaps there should be a comment to that effect).  There seems to be no reason to think such a prime should always exist, so I'm going to propose adding "or -1 if no such prime exists" to the Name.
In particular, my brute-force computations seem to show that for n = 88 there is no such prime.  

My question is, is this provable?  This might be done using a lower bound on the number of ways to write a prime as the sum of four nonzero squares.  I am aware of Jacobi's four-squares theorem, according to which prime p can be expressed as a sum of four squares of integers in 8*(p+1) ways (including both positive and negative integers as separate "ways", and also counting permutations as different).  In order for a sufficiently large p to have only 88 ways in the sense of  A213721, nearly all of those 8*(p+1) would have to correspond to cases where one or two of the squares are 0.  I wouldn't be surprised if
this could be ruled out by good bounds on the number of ways of writing p as the sum of two or three squares.  Are suitable bounds "well-known"?

Cheers,

Robert