Sums of squares of primes

[all at abouthugo.de]

"Pfoertner, Hugo" <Hugo.Pfoertner at muc.mtu.de> schrieb am 22.10.2003,
16:14:01:
> SeqFans,
> 
> yesterday I've found the thread in sci.math:
> "Sum of unique prime squares?"
> http://mathforum.org/discuss/sci.math/t/547283
> 
> I wrote a little program with the idea to find a new sequence
> "Minumum number of terms needed to represent n as a sum of
> the squares of primes, admitting also negative terms".
> 
> From my preliminary result the sequence should start (from n=0):
> 4,3,4,4,1,2,?,?,4,1,4,4,3,2,4,3,2,?,?,4,3,...
[...] 
> Question:
> Can we replace all ? with "5"? Can 6,7,17,18,... be written as sum or
> difference
> of 5 distinct squares of primes? Is 5 terms the maximum?
> 
> I'd be happy if someone could check (and extend) my results.
>

Edwin Clark found that all missing entries could be represented by
5 terms. If this is the maximum and if no better representations can
be found for some of the terms needs to be proved. I have now submitted:
%S A088910 4 3 4 4 1 2 5 5 4 1 4 4 3 2 4 3 2 5 5 4 3 2 4 3 2 1 4 4 3 2 3
5 4 3 2 4
3 4 3 3 2 5 5 4 3 2 4 3 2 1 4 4 3 2 3 5 4 3 2 4 5 4 3 3 4 3 5 4 3 4 3 3
2 3 2 4 3 4 3 4 4 3 4 3 5 4 4 3 4 5 5 4 3 4 4 3 2 3 4 4 3 4 5 5 4 3 4 4
3 4 4 3 2 5 5 4 3 2 4 3 2 1 4 4 3 2 3 5 4 3 2 4 5 4 3 3 4 3 5 4 3 4 3 3
2 3 2 4 3 4 3 4 4 3 4 3 3 4 4 3 2 3 5 4 3 2 3 3 2 1 2 4 3 2 3 4 4 3 2 3
%N A088910 Conjectured minimal required number k of terms in a
representation n=sum_(i=1..k)e_i*(p_i)^2 by distinct primes p_i,
where e_i is 1 or -1.
%C A088910 It is conjectured that all sequence terms are <=5.
The sequence terms with a(n)=5 were provided by Edwin Clark
(eclark(at)math.usf.edu).
%D A088910 Robert E. Dressler, Louis Pigno, Robert Young, Sums of
squares of primes.
Nordisk Mat. Tidskr. 24 (1976), no. 1, 39-40.
%H A088910 Hugo Pfoertner, <a
href="http://www.randomwalk.de/sequences/a088910.txt">Conjectured
minimal representations of n by squares of distinct primes.</a> Table
for n<=400.
%e A088910 The following are representations with the minimal number of
terms:
a(0)=4: 0=7^2-11^2-17^2+19^2, a(1)=3: 1=7^2+11^2-13^2, a(4)=1: 4=2^2,
a(5)=2: 5=3^2-2^2, a(6)=5: 6=-(2^2)+3^2+7^2+11^2-13^2.
%Y A088910 Cf. A088934 maximum required prime in representation,
A048261, A088908, A088909.
%O A088910 0
%K A088910 ,nonn,

Many thanks to Edwin.

Hugo Pfoertner