Articles of elementary number theory

how to solve these questions with congruences

how can I solve these type of questions : using congruences find the last digit or last two digit of $27^{27^{26}}$ and find remainder when $53^{103}$ is divided by 7. I can solve 2nd question and a simpler case of first question with binomial theorem but in questions like 1st this does not seem to […]

The number of solutions to the equations $x^2-2y^2=\pm p$

Knowing that the two equations $x^2-2y^2=\pm1$ have infinitely many solutions in $\mathbb{Z}\times \mathbb{Z}$ (i.e., that there are infinitely many elements with norm $\pm 1$ in $\mathbb{Z}[\sqrt{2}]$), how does one prove that the two equations $x^2-2y^2=\pm p$ also have infinitely many solutions in $\mathbb{Z}\times \mathbb{Z}$ for every prime $p$ satisfying $p\equiv_8 \pm 1$?

Sums of two perfect squares

Show that if $q$ is a number that can be expressed as the sum of two perfect squares, then $2q$ and $5q$ can also be expressed as the sum of two perfect squares. EDIT: I’ve recently revisited this problem and I found an elementary answer which I posted as an answer below.

Let $p$ be an odd prime. Suppose that $a$ is an odd integer and also $a$ is a primitive root mod $p$. Show that $a$ is also a primitive root mod $2p$.

Let $p$ be an odd prime. Suppose that $a$ is an odd integer and also $a$ is a primitive root modulo $p$. Show that a is also a primitive root modulo $2p$. Any hints will be appreciated. Thanks very much.

Perhaps a Pell type equation

Find all pairs of positive integers $(a,b)$ that satisfy $13^a+3=b^2$. If $a$ is even then $3=(b-13^{a/2})(b+13^{a/2})$ which have no solutions. Now if the case $a=2k+1$ is odd then $b^2-13.(13^k)^{2}=3$ I cant proceed from here, please help. Any other methods for the latter case?

7 digit number consisting of 7s and 5s

Find all the 7 digit numbers that have only 5 and 7 as their digits and divisible by both 5 and 7. I have no clue how to use divisibility of 7 to solve this problem. DO i need to check all the 64 combinations?

Distributivity mod an integer

Let $a,b,c,m \in \mathbb{Z}$, is it always the case that $$a((b+c) \text{ mod } q) \text{ mod } q = (ab \text{ mod } q + ac \text{ mod } q) \text{ mod } q$$

Number of primes less than 2n

A series of questions. Explanations would be useful. I have done the first four parts. I am confused on how to go about the last two. Show that for any prime $p$ the largest power of $p$ that divides $n!$ is $$ \left\lfloor \frac{n}{p}\right\rfloor +\left\lfloor \frac{n}{p^2} \right\rfloor +\cdots +\left\lfloor\frac{n}{p^r}\right\rfloor$$ where $p^r\le n < p^{r+1}$ Use […]

Express $11!$ and $\binom{23}{11}$ as products of their prime factors

I’m a bit stuck on how to figure this question out without a calculator and what kind of working I’m supposed to show. Any help would be appreciated, thank you. $\ddot\smile$ Factorise $11!$ and $\binom{23}{11}$ into primes without a calculator in any way. Use this to calculate their $\gcd$ and $\rm{lcm}$, and confirm how these […]

Intuition – Divisibility results – If c divides some integers, then c divides any linear combination of them.

Not querying about proofs here. I don’t want to memorize, thence are there intuitions or illustrations for them? As a student, how else can I remember these results? Origin – Elementary Number Theory, Jones, p4, Exercise 1.3 and T 1.3