query: if p>=5 is a prime number, then p^2 +2 is a composite.
positive_passages.docid: 148713
positive_passages.text: If $p\geq 5$ is a prime number, show that $p^2+2$ is composite.
negative_passages.docid: 65511
negative_passages.text: Proving the following false: $f(x)=2x+1$ produces a prime number if and only if x is prime
negative_passages.docid: 85042
negative_passages.text: Write $1/1 + 1/2 + ...1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$ is prime [Wolstenholme's theorem]
negative_passages.docid: 44019
negative_passages.text: Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
negative_passages.docid: 59036
negative_passages.text: If $a^{n}-1$ is prime then $a=2$ and $n$ is prime?
negative_passages.docid: 10425
negative_passages.text: Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$.
negative_passages.docid: 96027
negative_passages.text: Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
negative_passages.docid: 35553
negative_passages.text: If $n$ is composite, so is $2^n-1$
negative_passages.docid: 54547
negative_passages.text: Prove this number must be a prime number or 1.
negative_passages.docid: 91285
negative_passages.text: Prove $a+b+c+d $ is composite
negative_passages.docid: 56328
negative_passages.text: Find p is the prime number which $\frac{p+1}{2}$ and $\frac{p^2+1}{2}$ both are square number.
negative_passages.docid: 80399
negative_passages.text: Is $29$ the only prime of the form $p^p+2$?
negative_passages.docid: 3500
negative_passages.text: Compositeness of $n^4+4^n$
negative_passages.docid: 27170
negative_passages.text: $a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
negative_passages.docid: 86594
negative_passages.text: If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
negative_passages.docid: 121890
negative_passages.text: Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
negative_passages.docid: 29251
negative_passages.text: Show that the square of any prime number is the Factof of some integer