query: Prove that there are prime numbers such as $p_{i}$ such that $A = p_{1}^{a_{1}}*p_{2}^{a_{2}} * ... * p_{k}^{a_{k}} , a_{i}> 0 $
positive_passages.docid: 144592
positive_passages.text: how to prove the existence and uniqueness of the prime factorization of any natural number.
negative_passages.docid: 45793
negative_passages.text: How do you prove a number is prime?
negative_passages.docid: 111037
negative_passages.text: Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
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: 39843
negative_passages.text: Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
negative_passages.docid: 58182
negative_passages.text: How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 79940
negative_passages.text: $\binom{p}{i}$ divisible by $p$, with $p$ prime
negative_passages.docid: 68221
negative_passages.text: Question regardles primes and the fundamental theorem of arithmetic
negative_passages.docid: 111490
negative_passages.text: Proof that every number has at least one prime factor
negative_passages.docid: 94917
negative_passages.text: Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
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: 111967
negative_passages.text: Let $p$ be a prime number. Prove that there exists $x∈Z$ for which $p|x^2−x+ 3$ if and only if there exists $y∈Z$ for which $p|y^2−y+ 25$.
negative_passages.docid: 130846
negative_passages.text: Prove or disprove: If $a \in \mathbb{Z}$, $a > 0$, and $p$ is a prime such that $p^4 \mid a^3$, then $p^2 \mid a$.
negative_passages.docid: 50412
negative_passages.text: Prime elements in $\mathbb{Z}/n\mathbb{Z}$
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.