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

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 […]

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, […]

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

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?

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 $a/p^{\beta}$ […]

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

I want to find all primes $p$ for which $14$ is a quadratic residue modulo $p$. I referred to an example that was already posted for finding all odd primes $p$ for which $15$ is a quadratic residue modulo $p$, but I am getting stuck. This is what I have done: $$\left(\frac{14}{p}\right)=\left(\frac{7}{p}\right)\left(\frac{2}{p}\right)=(-1)^{(p-1)/2}\left(\frac{p}{7}\right)\left(\frac{p}{2}\right). $$ There are […]

This question already has an answer here: Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$? 6 answers

Let $p \equiv q \equiv 3 \pmod 4$ for distinct odd primes $p$ and $q$. Show that $x^2 – qy^2 = p$ has no integer solutions $x,y$. My solution is as follows. Firstly we know that as $p \equiv q \pmod 4$ then $\big(\frac{p}{q}\big) = -\big(\frac{q}{p}\big)$ Assume that a solution $(x,y)$ does exist and reduce […]

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