Quote:
Originally Posted by philmoore
I second Pace's suggestion, but I am still trying to understand the process of how both of you come up with these beautiful parametrized solutions. I see this sort of thing occasionally on the NMBRTHRY listserve, but I have no idea how people come up with them! Care to share any secrets?

Wouldn't they want submitters to be subscribers  or at least regular readers? I don't know how to submit it without at least reading  the web doesn't seem to offer that information to nonsubscribers.

I started with x=b, y=z=1, and hoped I could extend it. A quadratic extension would give me six "free" variables and only four constraints, so I started looking for solutions to the multiplication for
x = x1*s^2 + x2*s + b
y = y1*s^2 + y2*s + 1
z = z1*s^2 + z2*s + 1
The trickiest part was finding integer solutions for the s^4 term,
x1*y1+x1*z1+y1*z1=0
x1 =  y1 * z1 / (y1 + z1)
I did a quick search on small values of y1 and z1 looking for integer values of x1. I ran into a dead end with y1 = z1 = 2, so I tried the smallest "off diagonal" solution, y1=3 z1=6. I think Pace used this same case. I imagine other families of solutions can be made from the other off diagonal solutions.
I set x1=2d, y1=3d, z1=6d and then looked for similar tricks to force the s^3, s^2, and s terms of the product to zero. I used the symbolic manipulation package in Mathcad (a Maple variant) at each step to express the product as a polynomial in s, then looked for a way to force one more coefficient to be zero  substitute that into the definition and repeat.
Once I had a parameterized solution to the multiplication, I made "s" a multiple of (2b+1) to easily enforce the gcd condition.
William