Articles of number theory

Multiples of 4 as sum or difference of 2 squares

Is it true that for $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} – y^{2}$ for $x,y \in \mathbb{N} \cup (0)$. I was just working out a proof and this turns out to be true from $n=1$ to $n=20$. After that I didn’t try, but I would like […]

Raising a rational integer to a $p$-adic power

Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$? Under what conditions would this be an element of $\mathbb{Z}_p$? Any help would be very much appreciated.

How to calculate the index between two complex lattices?

Let $\Lambda$,$\Lambda’$ be two complex lattices and $m\neq 0\in\mathbb{C}$ satisfying $m\Lambda\subset\Lambda’$. Suppose $\omega_1,\omega_2$ are the basis of $\Lambda$, $\omega’_1,\omega’_2$ are the basis of $\Lambda’$. So we have $$ \begin{bmatrix}m\omega_1\\m\omega_2\end{bmatrix}=\alpha\begin{bmatrix}\omega_1’\\\omega’_2\end{bmatrix}\text{ for some }\alpha\in M_2(\mathbb{Z}) $$ Then it is said that $[\Lambda’,m\Lambda]=\det\alpha$. Why?

Can the determinant of an integer matrix with $k$ given rows equal the gcd of the $k\times k$ minors of those rows?

I’m interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the equation $$\left|\begin{array}{c}A\\\hline X\end{array}\right|=\gcd(A_1,\ldots,A_N)$$ always has a solution $X\in\mathbb Z^{n-k\times n}$. This is a generalisation of Can the determinant of an integer matrix with a given […]

$\frac{x^5-y^5}{x-y}=p$,give what p ,the diophantine equation is solvable

for$$\frac{x^3-y^3}{x-y}=x^2+xy+y^2=p$$$p=6k+1$give p prime, On what conditions,the diophantine equation $$\frac{x^5-y^5}{x-y}=p$$ is solvable in integers.does it have a linear expression.for $$\frac{x^n-y^n}{x-y}=p$$n =7,11,13…etc,is it easy to solve.and in what conditons ,it’s solvable in $Z[i]$

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + c y^2 – d z^2 = (a m^2 + b m n + c […]

Prove or refute that $\frac{t^a-1}{t^b-1}$ has more than 100 digits if $a \mod b \neq 0$

I’m a computer science student from Mexico and I have been training for the ICPC-ACM. So one of this problems called division sounds simple at first. The problem is straight for you have and 3 integers $t$, $a$ and $b$, greater or equal than $0$ and less or equal than $2^{31} – 1$. Your job […]

Is there an explanation for these gaps?

Here : there is a complete list of the integral points of the Mordell-curves for $-10^7\le n\le 10^7$. I searched for large solutions (in particular for large solutions on curves with only $2$ integral points) and accidently found the following unusual gaps : 779996 2 [170,2386] [170,-2386] 787577 4 [256074644,4097791586581] [256074644,-4097791586581] [1778,74977] [1778,-74977] 794610 […]

Good applications of modular forms on $SL_2(\mathbb{Z})$

I’ve just read some materials of modular forms on $SL_2(\mathbb{Z})$, and find some interesting application. Deal with Ramanujan $\tau$ function. I saw it in Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?. And I think is really a good example. Prove Ramanujan conjecture. I have read the proof of Ramanujan conjecture in Ahlgren S, Boylan M. Arithmetic […]

Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $

This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (=”$\eta$”) in terms of matrixoperations I asked myself, what we get, if we generalize the idea of the alternating signs to cofactors from the complex unit-circle. $$ \zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $$ With this the usual […]