For any $\ell > 0$ can you find $M, N$ such that $ \boxed{\mathrm{gcd}(x,y) > 1}$ for all $x \in [M, M+\ell]$ and all $y \in [N, N+\ell]$ ? This is related to the statement that the set of integers $(x,y)$ such that $x, y$ are relatively prime has natural density $\frac{6}{\pi^2} \approx \frac{2}{3}$. I […]

Prove or disprove: if $x$ and $y$ are representable as the sum of three squares, then so is $xy$. How to prove or disprove it? I am unable to get any idea on it. It would be of great help if any one could help.

Say I wanted to be able to carry enough coins in my pocket such that at any time, I could count out exact change totaling any of the prime numbers less than 100. How would I determine the minimum set of coins I would need to carry? I don’t care about being able to count […]

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Could someone please tell me what the general asymptotic of $\dfrac{\sigma(p_n\#)}{p_n\#}$ is? It looks like $O \log(n)$. Is there anything more precise? Update Thank you to Daniel Fischer for his answer. Out of interest, I include the plot of $\dfrac{\sigma(p_n\#)}{p_n\#}$ against $\dfrac{6 e^{\gamma}}{\pi^2} \log p_n$. The primorial curve […]

Ok so my theorem goes like this For any Z = X/Y Where X,Y are positive integers & Y > 0 A1,A2…. are factors of X & B1,B2…. are factors of Y There exists a Bn of Y such that Ai = Bn * Z If Bn < Y then X must be composite If […]

Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$. Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$. Let $\alpha = \left( \begin{array}{ccc} p & q \\ r & s \end{array} \right)$ be an element of $SL_2(\mathbb{Z})$. We write $f^\alpha(x, y) = f(px + qy, rx + sy)$. Since $(f^\alpha)^\beta$ = […]

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

How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}[\alpha]$. ($\mathcal{O}$ is ring of algebraic integers) $\alpha$ is a root of $f(x)=x^3+2x^2+4$ which is irreducible in $\mathbb{Q}(x).$ $\alpha^2$ is a root of $g(x)=x^3-4x^2-16x-16$ which is irreducible in $\mathbb{Q}(x).$ I found the discriminant as $disc(\alpha)=-16.5.7$ Also, I know $\alpha^2/2 \in R$ but $\alpha^2/4 \notin […]

Let $p$ a prime number, ${q_{_1}}$,…, ${q_{_r}}$ are the distinct primes dividing $p-1$, ${\mu}$ is the Möbius function, ${\varphi}$ is Euler’s phi function, ${\chi}$ is Dirichlet character $\bmod{p}$ and ${o(\chi)}$ is the order of ${\chi}$. How can I show that: $$\sum\limits_{d|p – 1} {\frac{{\mu (d)}}{{\varphi (d)}}} \sum\limits_{o(\chi ) = d} {\chi (n)} = \prod\limits_{j = […]

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ then, $$a^k + b^k + (a+2c)^k + (b+2c)^k + (-c+d+e)^k = \\(a+b-c)^k + (a+b+c)^k + d^k + e^k + (3c)^k \tag{2}$$ for $k = […]

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