I have a function involving polynomials and the centre of the Binomial Triangle and I'd like to prove that the function produces a positive integer infinitely many times. I don't have any interest in what values the integers take so much, merely that they exist.
The function I have is:
$$f(n) = \dfrac{15n^5+23n^4-20n^3-56n^2-48n-16}{n^3(n+1)(n+2)^4}\begin{pmatrix}2n \\ n \end{pmatrix} + 4\dfrac{3n^2+6n+4}{n^3 (n+2)^3}$$
The only method I have thought of is that I could try to prove that $\sin\left(\pi f(n)\right)$ has an infinte number of roots. But that feels like kicking the can down the road as I don't know how to do that either!
Thank you for any and all help. Ben
Using partial fractions I get:
$$f(n) = \frac{\left( 4n^3+28n^2+84n+76 \right) \begin{pmatrix} 2n \\ n \end{pmatrix} - 2n-4}{(n+2)^4} - \frac{\begin{pmatrix} 2n \\ n \end{pmatrix} - 2}{n^3} - \frac{4\begin{pmatrix} 2n \\ n \end{pmatrix}}{n+1} $$
From here I can find conditions for each fraction separately.
$\frac{\begin{pmatrix} 2n \\ n \end{pmatrix} - 2}{n^3} \in \mathbb{N} \implies n \text{ is prime > 3 (I found this on A000984)}$
$\frac{4\begin{pmatrix} 2n \\ n \end{pmatrix}}{n+1} \in \mathbb{N} \implies n \in \mathbb{N} \text{ (Always true)}$
$$f(n) = \frac{\left( 4n^3+28n^2+84n+76 \right) \begin{pmatrix} 2n \\ n \end{pmatrix} - 2n-4}{(n+2)^4} - \color{Blue}{\frac{\begin{pmatrix} 2n \\ n \end{pmatrix} - 2}{n^3} - \frac{4\begin{pmatrix} 2n \\ n \end{pmatrix}}{n+1}} $$
This leaves the final condition:
$\frac{\left( 4n^3+28n^2+84n+76 \right) \begin{pmatrix} 2n \\ n \end{pmatrix} - 2n-4}{(n+2)^4} \in \mathbb{N} \\ \implies \left( 4n^3+28n^2+84n+76 \right) \begin{pmatrix} 2n \\ n \end{pmatrix} \equiv 2n+4\pmod{n^4+8n^3+24n^2+32n+16}$
This seems to hold true when $n+2$ is a prime.
So if this function outputs an infinite number of integers, the twin-prime conjecture should be true. If $n$ is the lower of a pair of twin primes, then $f(n)$ is an integer.
Here is a proof that if both $n$ and $n+2$ are prime ($n>3$), then the output is integer: clearly it suffices to show that if $n+2$ is prime then $$ \left(4n^3+28n^2+84n+76\right)\binom{2n}{n} \equiv 2(n+2) \bmod{(n+2)^4}.$$ Let $p=n+2$ be prime, then this is equivalent to showing that: $$\tag1 2\left(p^3+p^2+5p-3\right)\binom{2p-4}{p-2} \equiv p \bmod{p^4}.$$
Proof
Since $n>3$, $p>3$ and by Wolstenhome's theorem we have: \begin{align} \binom{2p-1}{p-1}&\equiv 1 \bmod p^3\\ p\binom{2p-1}{p-1}&\equiv p \bmod p^4\\ p\frac{2p-1}{p-1}\frac{2p-2}{p-2}\frac{2p-3}{p-3}\binom{2p-4}{p-4}&\equiv p \bmod p^4. \end{align}
But $$ \binom{2p-4}{p-2}=\binom{2p-4}{p-4}\frac{p-1}{p-3}\frac{p}{p-2}.$$ Then \begin{align} p\frac{2p-1}{p-1}\frac{2p-2}{p-2}\frac{2p-3}{p-3}\frac{p-3}{p-1}\frac{p-2}{p}\binom{2p-4}{p-2}&\equiv p \bmod p^4 \end{align} That is \begin{align} \frac{(2p-1)(2p-2)(2p-3)}{(p-1)^2}\binom{2p-4}{p-2}&\equiv p \bmod p^4 \end{align}
But by doing the power expansion of $\frac{1}{(p-1)^2}$ , we see that $$ \frac{1}{(p-1)^2} \equiv 1+2p+3p^2+4p^3 \bmod p^4$$ and we are done since it is easy to see by expansion that $$(2p-1)(2p-2)(2p-3)(1+2p+3p^2+4p^3) \equiv 2(-3+5p+p^2+p^3) \bmod p^4.$$
I does seem that for the OP function to bring an integer, the argument must be the smallest prime of a twin prime pair. But proving this seems very difficult. What about trying to disprove it by computer search of a counterexemple ?