How to generate factors given the b-value of

Angelina Proctor

Angelina Proctor

Answered question

2022-04-09

How to generate factors given the b-value of the quadratics is the hypotenuse of a Pythagorean Triple?
Given two quadratics with integer coefficients,
y=x2+bx+c and x2+bxc

Answer & Explanation

Juan Goodwin

Juan Goodwin

Beginner2022-04-10Added 10 answers

Step 1
Let
p±(x)=x2+bx±cZ[x],
with discriminant b24c.
As you observed, if p±
have integer roots, then for integers m,n we have
b2+4c=m2,
b24c=n2
so that m2+n2=2b2
This does not prove that b is the hypotenuse in a pythagorean triple. For example, the degenerate case
m=n=b
always solves the equation m2+n2=2b2,
but obviously not every positive integer is the hypotenuse in a pythagorean triple.

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