Articles of elementary number theory

A “fast” and “manual” approach for sorting a set of fractions

What could be the fastest manual approach for sorting (ascending) these fractions: $$\frac{117}{229},\frac{143}{291},\frac{123}{241},\frac{109}{219}$$ I would also be very grateful if somebody explain a general manual approach for sorting fractions which doesn’t really follow any pattern.

Let n>30. Then prove there exists a natural number $ 1< m\leq n$ such that $(n,m)=1$ and $m$ is not prime.

Let $n>30$. Then prove there exists a natural number $1<m\leq n$ such that $(n,m)=1$ and $m$ is not prime. $(m,n)$ denotes the greatest common divisor of $m$ and $n$. Thanks

Show that $a^{13} \equiv a \pmod{3 \cdot 7 \cdot 13}$.

Show that $a^{13} \equiv a \pmod{3 \cdot 7 \cdot 13}$. I want to know if my attempt is correct. First $a^{13} \equiv (a^3)^4 \cdot a \equiv a^4 \cdot a \equiv a^3 \cdot a^2 \equiv a \cdot a^2 \equiv a^3 \equiv a \pmod 3$. Second, $(a^7)^2 \cdot a^{-1} \equiv a^2 \cdot a^{-1} \equiv a \pmod 7$. […]

Find integer numbers $x,y>1$ such that $x|3^y+1$ and $y|3^x+1$

Find all integer numbers $x,y>1$ such that $x|3^y+1$ and $y|3^x+1$. I tried to use $\text{ord}$ but still no clue.

A formula for the least common multiple less than n?

Can anyone please tell me a exact formula for $\operatorname{lcm}(1,2,3,…n)$ and if its not possible then asymptotic expansion will do.

Sums of Primitive Roots and Quadratic Residues when $p \equiv 3\pmod 4$

Define $$R_{p}=\{ r \mid r: \text{primitive root of p}, 1 \le r \le p \}$$ and also $$Q_{p}=\{ a \mid a: \text{quadratic residue of p}, 1 \le a \le p \}$$ $$Q_p^c=\{a \mid a: \text{quadratic nonresidue of p}, 1 \le a \le p \}$$ When $p$ is a prime number and $p=4k+3$ for some $k […]

double check my steps to find multiplicative inverse?

assume we want to find the multiplicative inverse for $117$ in $Z337$. i know that in order to find the the multiplicative inverse we use Euclidean and Extended Euclidean algorithm. Eculidian: $337 = 2*117 + 103$ $117 = 1*103 + 14$ $103 = 7*14 + 5$ $14 = 2*5 + 4$ $5 = 1*4 + […]

If $p$ is a prime integer, prove that $p$ is a divisor of $\binom p i$ for $0 < i < p$

I was thinking of using the definition for combinations and use the fact that $p$ appears in the expansion of $\binom pi$ and hence $p$ is a divisor. I don’t know whether I am on the right track!

How many distinct factors of $n$ are less than $x$?

For some (squarefree) integer $n$ and some integer $x$, I would like to find an expression that gives, for all $n$ and $x$, a good upper bound on the function $$f(n, x) = \sum_{d|n, d < x} 1$$ which counts the number of distinct factors of $n$ that are less than $x$. For example, taking […]

Prove that there are no positive integers $a, b$ and $n >1$ such that $a^n – b^n$ divides $ a^n + b^n$.

This question already has an answer here: Why does $a^n – b^n$ never divide $a^n + b^n$? 3 answers