Sums of squares of primes

[Edwin Clark]

On Wed, 22 Oct 2003, David Wilson wrote:

> This thread got me thinking about expressing nonnegative integers as sums of distinct
> squared primes.  My inclination was to think that any sufficiently large integer would be
> so expressible, this is probably provable from the prime number theorem.  Empirically,
> I checked up to 1000000, and i found 2438 numbers which were not sums of distinct
> squared primes, the largest being 17163.  This agrees with A048261.

A reference for this fact is 
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MR0419352 (54 #7373) 
Dressler, Robert E.; Pigno, Louis; Young, Robert 
Sums of squares of primes. 
Nordisk Mat. Tidskr. 24 (1976), no. 1, 39--40.
Abstract:
It is known [R. Sprague, Math. Z. 51 (1948), 289--290; MR 10, 283] that
the largest integer not representable as a sum of distinct squares is
128. In this paper, the authors announce that the largest integer not
representable as a sum of distinct squares of primes is 17163. The authors
use a lemma due to H.-E. Richert [Nordisk Mat. Tidskr. 31 (1949),
120--122; MR 11, 646] with the inequality $p_{i+1}^2\leq 2p_i{}^2 (i\geq
5)$ ($p_i$ denoting the $i$th prime), and computational verification for
the numbers $n$ in $17163<n\leq 17163+503^2$.
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It is easy to check that these 2438 numbers which are not the sum of
distinct primes squared are all of the form 

(*)  sum_i e_i*q_i   

where e_i is 1 or -1 and the q_i's are distinct primes.

It follows then that every positive integer can be written in the
form (*). Hugo's question of whether or not 5 terms suffices in (*)
remains. 


--Edwin