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

[Allan Wechsler]

This is a famous problem with many names. For a brief discussion see
http://mathworld.wolfram.com/PerfectCuboid.html. No solutions are known;
searches have been done up into the 10^10 range.

On Mon, Feb 1, 2016 at 10:14 AM, David Rabahy <DavidRabahy at comcast.net>
wrote:

> 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?
>
>
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