[seqfan] can all of these be square? a^2+b^2, a^2+c^2, b^2+c^2, a^2+b^2+c^2

[David Rabahy]

I have been searching for a set of three positive integers, say, a, b, & c,
such that the following sums are all square at the same time;

a^2+b^2
a^2+c^2
b^2+c^2
a^2+b^2+c^2

Here's an example that almost works;

a = 104, a^2 = 10816
b = 153, b^2 = 23409
c = 672, c^2 = 451584

a^2+b^2 = 34225 (square of 185)
a^2+c^2 = 462400 (square of 680)
b^2+c^2 = 474993 (not square)
a^2+b^2+c^2 = 485809 (square of 697)

Using the following Python program;

start = 1
end   = 40000
print('[',start,end,')')

for a in range(1,end+1):
 aa = a*a
 for b in range(a,end+1):
  bb = b*b
  ab = aa+bb
  t = ab**0.5
  if t == int(t):
   for c in range(max(b,start),end+1):
    cc = c*c
    ac = aa+cc
    t = ac**0.5
    if t == int(t):
     abc = ab+cc
     t = abc**0.5
     if t == int(t):
      bc = bb+cc
      t = bc**0.5
      if t == int(t):
       print(a,b,c,ab,ac,bc,abc)

I have searched through 40000 (and increasing) to no avail yet.  I can
continue to search but perhaps I am missing some crucial insight.  Is there
some reason all 4 sums cannot be square at the same time?