Articles of elementary number theory

Using Euler's Totient Function, how do I find all values n such that $\phi(n)=12$?

How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works?

Rational solutions to $a+b+c=abc=6$

The following appeared in the problems section of the March 2015 issue of the American Mathematical Monthly. Show that there are infinitely many rational triples $(a, b, c)$ such that $a + b + c = abc = 6$. For example, here are two solutions $(1,2,3)$ and $(25/21,54/35,49/15)$. The deadline for submitting solutions was July […]

If $\operatorname{ord}_ma=10$, find $\operatorname{ord}_ma^6$

If $\operatorname{ord}_ma=10$, find $\operatorname{ord}_ma^6$. Justify you’re answer. I’m not sure what I should say for my answer to be justified. However, I expect $\operatorname{ord}_ma^6=5$, because $5\cdot 6=30$ and $10\mid 30$. Like I said, no idea if this is how to go about solving this problem.

Prove that distinct Fermat Numbers are relatively prime

This question already has an answer here: Fermat numbers and GCD 3 answers

$ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$

How can we prove that $ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$?

Is that true that all the prime numbers are of the form $6m \pm 1$?

Is that true that all the prime numbers are of the form $6m \pm 1$ ? If so, can you please provide an example? Thanks in advance.

Proof of Wilson's Theorem using concept of group.

I am studying group theory so I do it by using the concept of group. What I am trying to prove is if p is prime then $(p-1)!\equiv-1\mod p$ Note that $\mathbb{Z_p}$ forms a multiplicative group. Hence $\forall a \in \{1,2,\dots,p-1\},\exists a^{-1}\in \{1,2,\dots,p-1\}$ This means that $aa^{-1}\equiv 1\mod p$ If $a=a^{-1}, 1 \equiv aa^{-1}= a^2 […]

Very elementary number theory and combinatorics books.

I know the basics of logic, sets, relations and the like, so studying intros to abstract algebra and real analysis is not that hard. That said, I have a deficiency when it comes to elementary number theory and combinatorics topics. Things like divisibility, modular arithmetic, congruences, prime numbers and such like regularly pop up as […]

How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?

For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$: Case one: $7\mid a^2$ and $7 \mid b^2$ Case two (which (I think) is not possible): $7$ does not divide $a^2$ and $7$ does not divide $b^2$, but their sum. I’ve shown that case $1$ implies $7\mid a$ and $7\mid […]

How to show $\binom{2p}{p} \equiv 2\pmod p$?

how to prove $\forall p$ prime : $\binom{2p}{p} \equiv 2 \pmod p$ we have: $\binom{2p}{p} = \frac{2p (2p-1)(2p-3)…1}{p!p!}$ but how to continue?