How can I find out if two parabolas will share an integer in their sequence?

Question

I have two parabolas:

$x = 4y^2 + 48y + 24$

$x = 4y^2 + 4y$

When I graph $4y^2+4y$ it hits the number $x=24$ at $y = -3$ and $y = 2$

When I graph

$4y^2 + 48y + 24$ it hits $x = 24$ at $y = -12$ and $y = 0$

I'd like to be able to tell no matter what the two parabola equations are if they will share a common integer. And where the first common integer hits. But these porabolas may not necessarily intersect but still share a common x value like 24.

It might be very similar to these: How to find the meeting number of two sequences? How can I find a common factor of two polynomials?

https://en.wikipedia.org/wiki/Polynomial_remainder_theorem

Or this: https://en.wikipedia.org/wiki/Diophantine_equation

But I can't seem to plug it in.

Do I set them equal like this and solve for y? $4y^2+48y+24 = 4y^2+4y$

Do I set them equal but with a different variable and solve for each? $4y^2+48y+24 = 4x^2+4x$

Answer 1

You are trying to solve $4y_1^2+48y_1+24 = 4y_2^2+4y_2$ for integer solutions. This can be rewritten as: $4(y_1+6)^2-(2y_2+1)^2=119$ or $(2y_1+2y_2+13)(2y_1-2y_2+11)=119$

All possible solutions now must be from one of the allowable factorisations: $119 = 1\times 119 = 7\times 17$, where ordering and $\pm1$ could be used to generate more cases. From these eight cases, we get all the solutions $(y_1, y_2) \in \{ (24, -30), (-36, 29), (24, 29), (-36, -30), (0, -3), (-12, 2), (0, 2), (-12, -3)\}$

Answer 2

A little help:

In the case of integer inputs, coefficient gcd helps in the above case; they both produce only multiples of 4. We can then ask if $y^2+12y+6$ and $y^2+y$ can both be 6. If not, we can give up on the originals ever being 24. You can check for divisibility by its divisors first, starting with prime divisors, continuing with prime powers; if it fails for any of them, or has contradicting conditions on them, then it fails.

Parity arguments and modular arithmetic come in handy here. Via parity arguments, we can show that if a polynomial is odd or even depends on the parity, of the cardinality, of the set of its odd coefficients; if even in number, parity falls to the constant term. Otherwise, parity is opposite the constant term on odd inputs, and the same parity as the constant only at even inputs. This immediately says for example, that $3x^2+5x+7$ and either one above will never both be the same number with integer inputs and outputs; one is always odd, and one is always even.

Polynomial remainder theorem can be used for algebraic a values; it still involves testing the first certain number of values though.

Chinese remainder theorem allows you to find out that if the polynomial is only divisible by 5 at inputs that have remainder 4 on division by 5, and divisible by 7 at inputs with remainder on division by 7 of 6, that divisibility by 35 or its multiples only comes up at inputs that are remainder 34 on division by 35.