If $p~ \mid~ m^p + n^p$, prove $p~ \mid~ \frac{m^p + n^p}{m+n}$.
11693
If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
34692
Prove if $ p$ is a prime of the form $3n+1$, then $p$ is also of the form $6m+1$
129331
Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
114812
Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+...+a_0$ be a complex polynomial satisfying $|p(z)| \leq 1$ for $|z|\leq 1$, then prove that $p(z)=z^n$
145884
Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
65534
Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$
48997
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
90916
Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
49217
How do I show that $a^p + b^p > (a + b)^p$ for $a,b>0$ and $0<p<1$?
87522
If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$
100558
Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
57639
Show there exists an $\alpha\in F_{p^n}$ such that $\alpha +\alpha^p+\alpha^{p^2}+\cdots+\alpha^{p^{n-1}}\neq 0$
32166
Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$
80826
To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$