Réalis's equal sums of three squares parameterization

[Kieren]

Hello!

In Dickson's History (Vol. II, Ch. VIII, pg. 267), I ran across a
reference to a paper by Réalis, which "is said to give [a
parameterization of] all [integer] solutions" to the equation A^2 +
B^2 + C^2 = D^2 + E^2 + F^2.

When I located the original paper (in Nouv. Ann. Math, courtesy of
Google Book Search), I found that Réalis's solution appears — at best
— to be limited to solutions where all of A, B, C, D, E, and F are of
the same parity (easy to show, by pairwise addition or subtraction of
the expressions).

Regardless, I'm looking for a proof or disproof of the following
Proposition, and wondered if anyone knew of an existing one, or could
suggest an approach that I should take.

Every integer solution to the [2.3.3] equation
A^2 + B^2 + C^2 = D^2 + E^2 + F^2
where A, B, C, D, E, and F have the same parity, is given by the
parameterization
A = r^2 + s^2 + t^2 - u^2 - v^2
B = (r-u)^2 + (s-u)^2 + (t-u)^2 - (v-u)^2 - u^2
C = (r-v)^2 + (s-v)^2 + (t-v)^2 - (v-u)^2 - v^2
D = r^2 + (r-u)^2 + (r-v)^2 - (r-s)^2 - (r-t)^2
E = s^2 + (s-u)^2 + (s-v)^2 - (s-r)^2 - (s-t)^2
F = t^2 + (t-u)^2 + (t-v)^2 - (t-r)^2 - (t-s)^2
where r, s, t, u, and v are integers.

Many thanks!
Kieren MacMillan.

[isr...@math.ubc.ca]

It's not true. For example, try A=3, B=5, C=5, D=1, E=3, F=7.
This fits the parametrization only with v = (+/-) sqrt(1/12),
u = v, t = -5 v, s = -3 v, r = -2 v (according to Maple).

Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Kieren]

Hello Robert,

> It's not true. For example, try A=3, B=5, C=5, D=1, E=3, F=7.
> This fits the parametrization only with v = (+/-) sqrt(1/12),
> u = v, t = -5 v, s = -3 v, r = -2 v (according to Maple).

Thanks for the quick reply/disproof.

I note, though, that

(r,s,t,u,v) = (1,0,-2,-1,-1)

results in

(3)^2 + (5)^2 + (5)^2 = (-1)^2 + (-3)^2 + (-7)^2.

I wonder if one the following restricted Proposition is true?

Every integer solution to the [2.3.3] equation
A^2 + B^2 + C^2 = D^2 + E^2 + F^2
where A, B, C, D, E, and F have the same parity, is given by the
parameterization

A = +- ( r^2 + s^2 + t^2 - u^2 - v^2 )
B = +- ( (r-u)^2 + (s-u)^2 + (t-u)^2 - (v-u)^2 - u^2 )
C = +- ( (r-v)^2 + (s-v)^2 + (t-v)^2 - (v-u)^2 - v^2 )
D = +- ( r^2 + (r-u)^2 + (r-v)^2 - (r-s)^2 - (r-t)^2 )
E = +- ( s^2 + (s-u)^2 + (s-v)^2 - (s-r)^2 - (s-t)^2 )
F = +- ( t^2 + (t-u)^2 + (t-v)^2 - (t-r)^2 - (t-s)^2 )

where r, s, t, u, and v are integers.

Best wishes,
Kieren.

[Kieren]

Hello all,

A better statement of the restricted Proposition I am looking to prove/
disprove is the following:

If A, B, C, D, E, and F are integers of the same parity, and
A^2 + B^2 + C^2 = D^2 + E^2 + F^2,
then there exist integers r, s, t, u, and v such that
A^2 = ( r^2 + s^2 + t^2 - u^2 - v^2 )^2
B^2 = ( (r-u)^2 + (s-u)^2 + (t-u)^2 - (u-v)^2 - u^2 )^2
C^2 = ( (r-v)^2 + (s-v)^2 + (t-v)^2 - (u-v)^2 - v^2 )^2
D^2 = ( r^2 + (r-u)^2 + (r-v)^2 - (r-s)^2 - (r-t)^2 )^2
E^2 = ( s^2 + (s-u)^2 + (s-v)^2 - (s-r)^2 - (s-t)^2 )^2
F^2 = ( t^2 + (t-u)^2 + (t-v)^2 - (t-r)^2 - (t-s)^2 )^2.

Using maxima, I have not yet been able to find a counterexample.

Many thanks,
Kieren.

[Kieren]

Hello again,

Just an update, for those who are interested...

There *does* appear to be a subset of possibilities that the Réalis
parameterization cannot produce, even under the restrictions of my
previous post. Specifically, I had maxima generate a list of ALL sums
of three non-zero squares of equal parity, i.e.,
(2a)^2 + (2b)^2 + (2c)^2
and
(2a-1)^2 + (2b-1)^2 + (2c-1)^2
for a,b,c = 1...N, and generated the set of Réalis parameterizations
for r,s,t,u,v = -M...M (removing those that require terms equal to
zero)

For example, using N=20 and M=10, I got
Direct = {3, 11, 12, 19, 24, 27, 35, 36, 43, 44, 48, 51, 56, 59,
67, 68, 72, 75,...}
Réalis = {3, 11, 12, 19, 27, 35, 43, 44, 48, 51, 59, 67, 75,
76,...}

Réalis's method (at least for M=10) does not produce the set {24, 36,
56, 68, ...}. I'll be doing more number-crunching (in maxima and on
paper) to try to figure out whether this is just a result of the
parameter(s) being too small, and if not what the pattern of "holes"
is precisely.

Best wishes,
Kieren.