How to prove $a^{1729}\:=_{1729}\:\:a$? I know it's a Carmichael number, but how can we show it directly without knowing it's a Carmichael number?
49938
Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
66861
Show that $11^{10} \equiv 1 (\mod 100)$
59788
Let T = {1000,1001...,9999}. How many numbers have at least one digit that is 0...
26233
$[K(\zeta_n):K]=\phi(n)$?
32381
How to calculate $276^{246} \mod 323$ without a calculator?
127198
What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
32929
Questions about proving $\lnot \lnot a = a$
37417
Why is $ 561 = 3*11*17 $ the smallest Carmichael number?
42780
Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
32
Is $n^{th}$ root of $2$ an irrational number?
81778
If $m|n$ and $a$ is a primitive root of $n$, show that $a$ is a primitive root of $m$ (understanding a tip)
1383
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
53918
whats the proof for $\lim_{x → 0} [(a_1^x + a_2^x + .....+ a_n^x)/n]^{1/x} = (a_1.a_2....a_n)^{1/n}$
147665
$a_1=3;\ a_{n+1}=3^{a_n}$; find $a_{2004}\bmod100$
80658
Prove that the number $19\cdot8^n+17$ is not prime, $n\in\mathbb{Z}^+$
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?