For any integer n greater than 1, $4^n+n^4$ is never a prime number.
3500
Compositeness of $n^4+4^n$
51928
Use an induction argument to prove that for any natural number $n$, the interval $(n,n+1)$ does not contain any natural number.
63702
There is a prime between $n$ and $n^2$, without Bertrand
113359
Proving that $x^2 + 4$ is not divisible by $3$
119630
For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
53525
Infiniteness of non-twin primes.
35837
How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
96844
Prove that 10101...10101 is NOT a prime.
46946
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
50596
$p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
79208
Is the number $-1$ prime?
69830
How often is $1+\prod_{k=1}^n p_k$ not prime?
71935
Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.
80658
Prove that the number $19\cdot8^n+17$ is not prime, $n\in\mathbb{Z}^+$
123544
Show that $n = 3^{100} + 2$ is not a prime number.
70523
Proving that $x^4 - 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$.
73612
Is there a Non-prime Number that is Divisible only by Numbers Greater than its Square Root?