For the present investigation Poetasis gave me the necessary impetus in his answer on my related question here. I analyzed a large dataset of triples $w,x,y$ which fulfill the following diophantine system of equations:
$$x^2-w^2= \square_1\qquad y^2-x^2= \square_2\qquad y^2-w^2= \square_3$$
For this I included also very large triples such as $(w,x,y)=(11689849600,62507610880,64825529600)$ or $(w,x,y)=(17179869168,48806446500,49940015580)$.
I observed the following behavior to be very reliable/robust:
- The number of prime factors of the form $4k+1$ that $w$ contains range between $0$ and $8$
- The number of prime factors of the form $4k+1$ that $x$ contains range between $1$ and $8$
- The number of prime factors of the form $4k+1$ that $y$ contains range between $2$ and $8$
Hence $x$ must contain at least $1$, $y$ must contain at least $2$ of such prime factors and in every case the number of such prime factors (contained in $w,x,y$) cannot exceed $8$. Whether this observed upper limit of 8 really applies in general, I am not sure - even though I have really examined many such triples. To make it a little more vivid, here is my observation summarized in a table (I have included all triples containing integers up to a size of 12 million and then a few extra large triples up to almost $2^{36}$):
\begin{array}{|l|l|l|l|} \hline & \text{occurrences in }w & \text{occurrences in }x & \text{occurrences in }y \\ \hline 0 & 33051 & 0 & 0 \\ \hline 1 & 91340 & 12396 & 0 \\ \hline 2 & 92632 & 82493 & 40292 \\ \hline 3 & 52476 & 116551 & 107706 \\ \hline 4 & 19985 & 63612 & 98890 \\ \hline 5 & 4259 & 16554 & 38642 \\ \hline 6 & 576 & 2500 & 7810 \\ \hline 7 & 56 & 257 & 952 \\ \hline 8 & 5 & 17 & 88 \\ \hline \end{array}
My Question: Can we explain this?
And as a follow-up question, it would be intersting to know: Assuming we consider corresponding quadruples $(w,x,y,z)$ instead of these triples $(w,x,y)$, would $z$ have to contain $3$ such primes (of the form $4k+1$) as a factor or would the rules be much stricter for all integers $w,x,y,z$?
Update: According the hint of Gerry Myerson we can conclude that $x$ must contain at least one prime factor of the form $4k+1$, since $x$ is a sum of two squares. A proof for this is, for example, given in "Proofs from the book (Fifth edition)" by Aigner and Ziegler (p. 21, chapter "Representing numbers as sums of two squares"), where he shows that every prime of the form $4k+1$ is a sum of two squares. Another reference is Fermat's 4n+1 Theorem (Wolfram MathWorld). Consequently, as mentioned by Gerry Myerson, $y$ must be divisible by at least two $4k+1$ primes, since $y$ is a sum of squares in two different ways. Continuing this idea for a quadruple $(w,x,y,z)$, $z$ must contain three distinct prime factors of the form $4k+1$, since $z$ is a sum of squares in three different ways.