Sums of squares of primes

[Pfoertner, Hugo]

SeqFans,

yesterday I've found the thread in sci.math:
"Sum of unique prime squares?"
http://mathforum.org/discuss/sci.math/t/547283

In an answer Robert Israel wrote:
<<
Actually it seems that all positive integers at least up to 1000 can be 
written as differences of sums of squares of distinct primes.  I wouldn't
be surprised if this was true for all positive integers, but I don't 
immediately see a way to prove it.
>>

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,...


I checked the first 100 primes and four terms in the sum with the following
result (sorry, 101 lines):

  n  Terms needed
         Primes squared plus sign

   0 4   7 -11  19 -17
   1 3   7  11 -13
   2 4   5   7  17 -19
   3 4   2  -7 -11  13
   4 1   2
   5 2   3  -2
   6 ?
   7 ?
   8 4   3  -7 -11  13
   9 1   3
  10 4   3   7  11 -13
  11 4  -2  -3  -5   7
  12 3  -2  -3   5
  13 2   3   2
  14 4  -3  -5 -11  13
  15 3  -3  -5   7
  16 2   5  -3
  17 ?
  18 ?
  19 4   2  -3  -5   7
  20 3   2  -3   5
  21 2   5  -2
  22 4  -5 -11 -19  23
  23 3  -5 -11  13
  24 2   7  -5
  25 1   5
  26 4   5   7  11 -13
  27 4   2  -5 -11  13
  28 3   2  -5   7
  29 2   5   2
  30 3  -2   3   5
  31 ?
  32 4   3  -5 -11  13
  33 3   3  -5   7
  34 2   5   3
  35 4  -2  -3 -11  13
  36 3  -2  -3   7
  37 4   2   3  -5   7
  38 3   2   3   5
  39 3  -3 -11  13
  40 2   7  -3
  41 ?
  42 ?
  43 4   2  -3 -11  13
  44 3   2  -3   7
  45 2   7  -2
  46 4  -5  -7 -13  17
  47 3  -5  -7  11
  48 2  13 -11
  49 1   7
  50 4   7  11  13 -17
  51 4   2  -5  -7  11
  52 3   2 -11  13
  53 2   7   2
  54 3  -2   3   7
  55 ?
  56 4   3  -5  -7  11
  57 3   3 -11  13
  58 2   7   3
  59 4  -2  -3  -7  11
  60 ?
  61 4  -2  -3   5   7
  62 3   2   3   7
  63 3  -3  -7  11
  64 4  -3   5 -11  13
  65 3  -3   5   7
  66 ?
  67 4   2  -3  -7  11
  68 3  -2  -7  11
  69 4   2  -3   5   7
  70 3  -2   5   7
  71 3  -7 -13  17
  72 2  11  -7
  73 3   5 -11  13
  74 2   7   5
  75 4   2  -7  17 -13
  76 3   2  -7  11
  77 4  -2   3  -7  11
  78 3   2   5   7
  79 4  -2   3   5   7
  80 4   3  -7  17 -13
  81 3   3  -7  11
  82 4   3   5 -11  13
  83 3   3   5   7
  84 ?
  85 4   2   3  -7  11
  86 4  -3  -5  -7  13
  87 3  -3  -5  11
  88 4  -3   5  -7  11
  89 ?
  90 ?
  91 4   2  -3  11  -5
  92 3  -2  -5  11
  93 4  -2   5  -7  11
  94 4  -5  -7 -11  17
  95 3  -5  -7  13
  96 2  11  -5
  97 3   5  -7  11
  98 4   5  19  71 -73
  99 4   2  -5  13  -7
 100 3   2  -5  11

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.

Thanks

Hugo Pfoertner