For a prime q and positive integers x, y, z, w , q divides yz, q divides xy and q doesn't divide yw implies that q^2 divides xz
834
If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
113248
Proving $r!$ divides the product of r succesive positive integers
156940
There does not exist a polynomial $p(x)$ with integer coefficients which gives a prime number $\forall x\in \mathbb{Z}$
42932
Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
36862
A question about proving that there is no greatest common divisor
10514
Proof that no polynomial with integer coefficients can only produce primes
113359
Proving that $x^2 + 4$ is not divisible by $3$
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
113994
Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$.
72144
Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$
62224
Need help with proof for Dedekind cuts on $\mathbb{Q}^+$
125604
Proof that $\mathbb{Z}$ has no zero divisors
131971
If $k>0$ is a positive integer and $p$ is any prime, when is $\mathbb Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in\mathbb Z_p\}$ a field.
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]
78455
Prove that a nonzero integer p is prime if and only if the ideal (p) is maximal in Z.
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$