: the same but with four numbers.
Euler? demonstrates that for each D-tripla there is also another positive integer w so that xw, yw, zw are less 1 of being a square. So that xyzw is a D-quadrupla.
I. e. (1,3,8) and (1,3,8,120)
Knowing the D-tripla it is possible to calculate the forth member to have a D-quadrupla (or to extend from xyz to xyzw) with the formula:
(w+,w-) w=x²+y²+z²+2xyz±√(xy+1)(xz+1)(yz+1)
(note that w+ e w- are always integers)
in which w+,w- are the roots (two roots) of the quadratic equation:
x²+y²+z²+w²-2(xy+xz+xw+yz+yw+zw)-4xyzw-4=0
When xyz is a D-tripla the inferior root (w-) is 0, and the superior root (w+) is w: the four value of the D-quadrupla xyzw.
But what if xyz is not a D-tripla (not being squares xy+1... ) but nonetheless the product (xy+1)(xz+1)(yz+1) is an square? Is it possible? Is it also extensible to a quadrupla xyzw? Euler shows that none of this was possible. The product of non-squares xy+1,.... is not an square.
The demostration goes like this: if some of the elements (xy+1), (xz+1), (yz+1) are not squares (and then xyz is not a D-tripla) but their product is a square, it is posible to find a forth element w- so that the product (xy+1)(xw-+1)(yw-+1) is a square and xyw- is not D-tripla. Note that the value of w- is not 0, as it would be if xyz was to be a D-tripla.
w- is, instead, a positive integer and w-<z.
Follows that we have got the product of three non-squares (xy+1)(xw+1)(yw+1) resulting in an square but with inferior value, because w-<z.
But then we are again in the initial conditions and we can repeat the process to get again an inferior value, and this infinitely many times: infinite descent.
There are three steps to demonstrate this:
1) (xy+1)(xw-+1)(yw-+1) is a square and some of the factors are not square
2) w- is a positive integer
3)w- <z
Then xyw- is inferior than xyz but with the same properties. The infinite descent has begun.
Once understood this the demonstration is quite straightforward and it is possible to find in
http://www.jstor.org/stable/2691347#page_scan_tab_contents
How it goes
