Show that $1+2+...+(n-1)\equiv 0 \mod n$
57746
Prove that, for any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.
30630
Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
98017
If $a_{n+1}/a_n\to\ell$ show $a_n^{1/n}\to\ell$
72600
Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
51858
Showing $\gcd(2^m-1,2^n+1)=1$
127816
Number theory proof mod p
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
14648
Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer
26478
Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
40968
show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
37871
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
77076
Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
48333
Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
45037
Show $(2^m-1,2^n+1)=1$ if $m$ is odd
93427
Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
127901
If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.