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Topic: Re: Factoring
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Lou Talman

Posts: 876
Registered: 12/3/04
Re: Factoring
Posted: Apr 14, 1998 12:33 PM
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Lin McMullin offers us:

> Factoid (as our local TV station calls them): Of all possible quadratic
> polynomials whose coefficients are less than or equal to 20, only 7.6%

> of them factor.

The question is an interesting one, if only because there are several
ways to interpret it. Should we, for example, identify two polynomials
that differ only by a constant factor--for example,

x^2 + 2 x + 1


2 x^2 + 4 x + 2?

What about the quadratic polynomial 3 x^2? Does it count, or is it too
trivial to bother with?

Here are some facts: There are exactly 28000 quadratic polynomials of
the form a x^2 + b x + c with integer coefficients a, b, c for which 0 <
a <= 20, -20 <= b, c <= 20, a, b, and c are relatively prime, and b^2 +
c^2 > 0. Of these,

Sqrt[b^2 - 4 a c]

is integral in 2237 cases. That's 7.9893%.

If we drop the requirement that a, b, c be relatively prime, and allow
that 0 < Abs[a] <= 20, then there are 67200 such polynomials. Of these,
the discriminant has integer square root in 6312 cases. That's 9.328%.

--Lou Talman

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