Articles of elementary number theory

Proving infintely many primes of the form 6k-1

I have seen the past threads but I think I have another proof, though am not entirely convinced. Suppose there are only finitely many primes $p_1, …, p_n$ of the form $6k-1$ and then consider the number $N = (p_1.p_2. … .p_n)^2 – 1$ If all of the primes dividing N are of the form […]

Not a perfect square of the form for any integer x.

Now a days, I become good fan of this site, as this site making me to learn more math..hahaha. Okay! Can we prove that $x^3 + 7$ cannot be perfect square for any positive/negative or odd/even integer of $x$. I checked with number up to x = 1,…1000. I realized that, not a perfect. But, […]

Polynomial equations in $n$ and $\phi(n)$

If $n$ is a positive integer, let $\phi(n)$ the Euler function. ( if $n=p_1^{\alpha_1}\dots p_k^{\alpha_k}$ with $p_i$ distinct primes, we have $\phi(n)=p_1^{\alpha_1-1} \dots p_k^{\alpha_k-1}(p_1-1)\dots(p_k-1)$ ) Let $P$ a polynomial in $\mathbb{Z}[X,Y]$. We suppose there exists an infinite number of positive integer $n$ such that $P(n,\phi(n))=0$ Is $P$ reducible in factors of degree one ? Thanks […]

Fermat: Prove $a^4-b^4=c^2$ impossible

Prove by infinite descent that there do not exist integers $a,b,c$ pairwise coprime such that $a^4-b^4=c^2$.

For what $n$ is it true that $(1+\sum_{k=0}^{\infty}x^{2^k})^n+(\sum_{k=0}^{\infty}x^{2^k})^n\equiv1\mod2$

Let $A:=\sum_{k=0}^{\infty}x^{2^k}$. For what $n$ is it true that $(A+1)^n+A^n\equiv1\mod2$ (here we are basically working in $\mathbb{F}_2$.) The answer is all powers of 2, and it’s fairly simple to see why they work, but the hard part is proving all non-powers of 2 don’t work. In the solution, it says “If $n$ is not a […]

Problem from Olympiad from book Arthur Engel

Each of the numbers $a_1 ,a_2,\dots,a_n$ is $1$ or $−1$, and we have $$S=a_1a_2a_3a_4+a_2a_3a_4a_5 +\dots+ a_na_1a_2a_3=0$$ Prove that $4 \mid n$. If we replace any $a_i$ by $−a_i$ , then $S$ does not change $\mod\, 4$ since four cyclically adjacent terms change their sign. Indeed, if two of these terms are positive and two negative, […]

Necessary and sufficient conditions for the sum of two numbers to divide their product

How to find necessary and sufficient conditions for the sum of two numbers to divide their product. Thanks in advance.

Counting the number of integers $i$ such that $\sigma(i)$ is even.

Define $\sigma(i)$ to be the sum of all the divisors of $i$. For example, $σ(24) = 1+2+3+4+6+8+12+24 = 60$. Given an integer $n$, how can we count the number of integers $i$, less than or equal to $n$, such that $\sigma(i)$ is even?

Prove that the congruence $x^2 \equiv a \mod m$ has a solution if and only if for each prime $p$ dividing $m,$ one of the following conditions holds

Let $m$ be odd and let $a \in \mathbb{Z}.$ The congruence $x^2 \equiv a \mod m$ has a solution if and only if for each prime $p$ dividing $m,$ one of the following conditions holds, where $p^{\alpha} \mid \mid m$ and $p^{\beta}\mid \mid a$: $\beta \geqslant \alpha$; $\beta < \alpha$, $\beta$ is even, and $a/p^{\beta}$ […]

If $a^m-1 \mid a^n -1$ then $m \mid n$

This question already has an answer here: Showing that $a^n – 1 \mid a^m – 1 \iff n \mid m$ 3 answers