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I recently encountered this following proposition:

For every polynomial, there is some positive integer for which it is composite.

What is the most elementary proof of this?

- Prime Numbers And Perfect Squares
- Prove congruence using fermat's thm
- prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$
- Prove $∑^{(p−1)/2}_{k=1} \left \lfloor{\frac{2ak}{p}}\right \rfloor \equiv ∑^{(p−1)/2}_{k=1} \left \lfloor{\frac{ak}{p}}\right \rfloor ($mod $2)$
- A Variation on the Coin Problem
- Sum involving the product of binomial coefficients

- Limit inferior of the quotient of two consecutive primes
- Estimation of sums with number theory functions
- Expressing a positive integer as a sum of positive integers
- Proof of no prime-representing polynomial in 2 variables
- integer partitions
- What exactly *is* the Riemann zeta function?
- Find a line which is tangent to the curve $y=x^4-4x^3$ at 2 points
- Mere coincidence? (prime factors)
- How many values does the expression $1 \pm 2 \pm 3 \pm \cdots \pm n$ take?
- natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$

Suppose $P$ is a polynomial, then it is periodic mod $m$: I mean $P(a) = P(a+m) \pmod m$.

Suppose it takes on different prime values like $p$ and $q$.

Then $P(x) \equiv 0 \pmod p$ for some $x$, and $P(y) \equiv 0 \pmod q$ for some $y$. By periodicity we can find a $z$ such that $pq|P(z)$ using periodicity.

Not quite true, look at the polynomial $17$. And the usual theorem specifies that the polynomial has integer coefficients.

Let $P(x)$ be a non-constant polynomial with integer coefficients. Without loss of generality we can assume that its lead coefficient is positive.

It is not hard to show that there is a positive integer $N$ such that for all $n\ge N$, we have $P(n)\gt 1$, and such that $P(x)$ is **increasing** for $x\ge N$. (For large enough $x$, the derivative $P'(x)$ is positive.)

Let $P(N)=q$. Then $P(N+q)$ is divisible by $q$. But since $P(x)$ is increasing in $[N,\infty)$, we have $P(N+q)\gt q$. Thus $P(N+q)$ is divisible by $q$ and greater than $q$, so must be composite.

**Remark:** One can remove the “size” part of the argument. For any $b$, the polynomial equation $P(x)=b$ has at most $d$ solutions, where $d$ is the degree of $P(x)$. So for almost all integers $n$, $P(n)$ is not equal to $0$, $1$, or $-1$.

Let $N$ be a positive integer such that $P(n)$ is different from $0$, $1$, or $-1$ for all $n\ge N$. Let $P(N)=q$. Consider the numbers $P(N+kq)$, where $k$ ranges over the non-negative integers. All the $P(N+kq)$ are divisible by $q$. But since the equations $P(n)=\pm q$ have only finitely many solutions, there is a $k$ (indeed there are infinitely many $k$) such that $P(N+kq)$ is not equal to $\pm q$, but divisible by $q$. Such a $P(N+kq)$ cannot be prime.

I prefer using considerations of size.

- Density and expectation of the range of a sample of uniform random variables
- Calculate $1\times 3\times 5\times \cdots \times 2013$ last three digits.
- Are sequences with Cesaro mean a closed subset of $\ell_\infty$?
- Torsion elements do not form a submodule.
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- Prove that the product of four consecutive positive integers plus one is a perfect square
- Prove that $\sqrta+\sqrtb+\sqrtc=0$
- How does $e^{i x}$ produce rotation around the imaginary unit circle?
- Positive semidefinite matrix proof
- Convergence Proof: $\lim_{x\rightarrow\infty} \sqrt{4x+x^2}- \sqrt{x^2+x}$
- What's the thing with $\sqrt{-1} = i$
- Characters of the symmetric group corresponding to partitions into two parts
- What's the best 3D angular co-ordinate system for working with smartphone apps
- Solution to a linear recurrence