Math: Prime location!

[Monte314]

Can you find (positive) primes p, q, and r so that the following polynomial has only integer roots?


px^2 – qx + r = 0

[roninpro]

Hi Monte, it has been a while!

If the polynomial does have integer roots, we can write px^2 - qx + r = (px + a)(x + b) for some integers a and b. In particular, we must have a + b = q and ab = r (by Viete's formulas) and p divides a (since the roots must be integers). Because r is prime, one of a or b must be 1, and the other must be r. Combining this with p divides a, we have that a = ±p = r and b = ±1. Then, a + b = q is the same as p + 1 = q or p + 1 = -q. This shows that p and q are consecutive primes, which must be ±2 or ±3. So a quick check of the polynomials with those coefficients and satisfying the above conditions shows that none of them give only integer roots. (Though, we do come close with polynomials like 2x^2 + 3x -2.)

[Glathannus]

Damn.
I had a simple c# app (with nested FOR loops) crunching away at this for most of the night, but then a Windows update triggered an automatic reboot - so I'll have to start all over. Even then, I still might not be able to get an answer anytime soon, if X isn't a whole number.

[Monte314]

roninpro has solved this in the same way I did. Excellent work, as usual.

Glathannus has taken a reasonable tack in doing a scan for solutions; this is something I do myself once I have reason to believe that solutions probably exist.

[Reivan]

How about 1x^2-2x+1.

[Glathannus]

^
I don't think 1 counts as a prime number.

[roninpro]

It doesn't. If 1 were a prime number, then the Fundamental Theorem of Arithmetic would be false.

[Monte314]

roninpro has noted the central issue. We want prime factorization in the Integers to be unique.

Integers fall into three broad categories with respect to their factorization properties: Units, Composite numbers, and Prime numbers:

An integer L is a unit if it has a multiplicative inverse that is an integer. In the Integers, the only units are +1 and -1.

An integer L is composite if it can be written as a product of integers L = MN where neither M nor N is a unit.

An integer L is prime if L whenever MN is a multiple of L, then either:
1.) M is a multiple of L
or
2.) N is a multiple of L

(In other words, factorization of MN cannot "split" L across M and N. All of L has to be either in M or in N).

Sometimes people require primes to be positive, but this is unnecessary if you define "unique factorization" properly.

An integer L is irreducible if whenever L=MN, then exactly one of M or N is a unit.

This last notion, that "the only factors of a prime are 1 and itself", is what most people think prime means. They are able to get away with this because an integer is prime if and only if it is irreducible. This is NOT necessarily true in other mathematical structures, where, in fact, numbers may have multiple different prime factorizations!