Articles of elementary number theory

Sequence Of Primes

Hello I have a basic number theory question. I want to find a list of primes of the form a, a + d, a + 2d, … , a + 5d So a sequence of at least 6 or greater if I want to select a = 101 then what would I choose as my […]

Inductive proof of the identity $\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$

This question already has an answer here: Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$ 3 answers

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the equation$$\text{N}(a) = 1$$where now we take $a \in \mathbb{Q}(i)$. What is a method for producing solutions to this equation using the arithmetic […]

How to find inverse Modulo?

Find the inverse modulo, Modulo inverse of $5991 \pmod{2014}$ ? I am aware of the Euclid algorithm, but I am not sure how to apply it here?

How to find all binary numbers in base $10$ s.t. that its divisible by its own numerical value in base $2$?

Consider $N=1010$. Its numerical value in base-$2$ is $2^3+2=10$ and $10 \mid 1010$. How to find all positive integers like $N$? Its easy to see that $n$ is in the form $10^{n_1}+10^{n_2}+…+10^{n_k}$, where $n_1>n_2>…>n_k \ge 0$. We need $$ 2^{n_1}+2^{n_2}+…+2^{n_k} \mid 10^{n_1}+10^{n_2}+…+10^{n_k} $$ I’ll be very grateful if you could share your thoughts on this […]

Square of four digit number $a$

A natural number $a$ had four digits and $a^{2}$ ends with same four digits as that of $a$. Find the value of $(10080-a)$ Please provide some hints to get start with this problem.

Does Bezout's lemma work both ways.

I know that if $a$ and $b$ have a highest common factor $h$ then you can write $h=ax+by$ for some $x,y \in \mathbb{Z}$ but how about if you can write $h=ax+by$ for some $x,y \in \mathbb{Z}$ then can you write $h=\gcd(a,b)$? I.e. Is $h=\gcd(a,b) \iff h=ax+by,~ x,y \in \mathbb{Z}$

For prime $p>2: 1^23^25^2\cdot\cdot\cdot(p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \pmod p$

Possible Duplicate: Why is the square of all odds less than an odd prime $p$ congruent to $(-1)^{(p+1)/(2)}\pmod p$? If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$ I’d love your help with proving the following claim: For prime $p>2$: $$1^23^25^2\cdot\cdot\cdot(p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \pmod p.$$ I […]

Proof that the euler totient function is multiplicative, correctness?

I’ve tried proving that $\varphi(mn) = \varphi(m)\varphi(n)$ (if $gcd(mn)=1$). The proof I try to setup doesn’t look like the proof I find in textbooks, where am I going wrong? Proof: We try to show that the number of relative primes to $mn$ in $\{1, 2, \ldots , mn-1\}$ is equal to the number of relative […]

Prove that every number between two factors of primes is composite.

I am looking for some help with this problem: Let $p_1,p_2,\dots,p_{n+1}$ be the first $n+1$ primes in order. Prove that every number between $p_1p_2p_3\dots p_n+1$ and $p_1p_2p_3\dots p_n + p_{n+1}-1$ is composite (inclusive of the second term). How does this show that there are gaps of arbitrary length in the sequence of primes? I know […]