Articles of number theory

Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$

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 $xy$.

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.

Minimum set of US coins to count each prime number less than 100

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 […]

Asymptotic divisor function / primorials

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 […]

Is this division theorem already a proven idea?

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 […]

Representations of integers by a binary quadratic form

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 $(f^\alpha)^\beta$ = […]

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 $a^4-b^4=c^2$.

How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}$

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 […]

a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$

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 for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

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 = […]