Articles of number theory

Legendre's Proof (continued fractions) from Hardy's Book

I’m tryng to understand the proof given in Hardys book A Theory of Numbers from the chapter on continued fractions. It states that If $$\left|\frac{p}{q}-x\right| < \frac{1}{2q^2}$$ then $\frac{p}{q}$ is a convegent to $x$. Proof: I the above inequality holds then $$\frac{p}{q} – x = \frac{\epsilon \alpha}{q^2},$$ $\epsilon = \pm 1$ and $ 0 < […]

Simultaneous Diophantine approximation: multiple solutions required

Suppose we have $n$ linearly independent (over $\mathbb{Q}$) irrational numbers $\{ \alpha_i | 1\leq i \leq n \}$. For the simultaneous Diophantine approximation problem $$ |q \alpha_i – p_i | < \epsilon , $$ where $q$ and the $p$’s are all integers, we have the LLL algorithm. The problem is, by this algorithm, for each […]

Solving the Diophantine Equation $2 \cdot 5^n = 3^{2m} + 1$ over $\mathbb{Z}^+$.

Prove that $(m, n) = (1, 1)$ is the only solution for the Diophantine Equation $$2 \cdot 5^n = 3^{2m} + 1$$ where $(m, n) \in (\mathbb{Z}^+)^2$. I’ve managed to prove that both $m$ and $n$ are odd seeing $\bmod 3\text{ and } 10$ respectively. Also, $\forall n \ge 1$, $10$ divides the LHS. I […]

Show $\vert G \vert = \vert HK \vert$ given that $H \trianglelefteq G$, $G$ finite and $K \leq G$.

My problem is a variation of one in Dummit and Foote: Let $G$ be a group and $H \trianglelefteq G$. Prove: If $G$ is finite and $[G:H] = p$, a prime number, then for any $K \leq G$, either $K \leq H$ or $G = HK$ and $[K : H \cap K] = p$. Ok, […]

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood for 40 years as a “numerical curiosity” until Bremner and Delorme discovered it had the highly structured form, $$\small(u + 9)^k + (u + 14)^k + (u + 19)^k + […]

General term of this interesting sequence

A sequence of positive integer is defined as follows The first term is $1$. The next two terms are the next two even numbers $2$, $4$. The next three terms are the next three odd numbers $5$, $7$, $9$. The next $n$ terms are the next $n$ even numbers if $n$ is even or the […]

Proof about fibonacci numbers by induction

Let $u_1,u_2,….$ be the fibonacci sequence. a) Prove by induction or otherwise thar for n>0, $$u_{n-1}+u_{n-3}+u_{n-5}+…<u_n$$ the sum on the left continuing so long as the subscript remains larger than 1 b) Show that every positive integer can be represented in a unique way in the form $u_{n_1}+u_{n_2}+….+u_{n_k}$, where $k \geq 1$, $n_{j-1} \geq n_j+2$ […]

Proving $\mathbb{Z}$, $\mathbb{Z}$, $\mathbb{Z}$, and $\mathbb{Z}$ are euclidean.

I have this short class note from my graduate number theory: THEOREM: Assume that $\vert N(x + y \sqrt d)\vert < 1$ for any two rational numbers $x$ and $y$ with $\vert x \vert \leq 1/2$ and $\vert y \vert \leq 1/2$. Define $\delta : \mathbb Z[\sqrt d] \setminus \{0\} \to \mathbb N$ by $z […]

Number of solutions to $x+y+z = n$

If $\beta(n)$ is the number of triples $(x, y, z)$ such that $x + y + z = n$ and $0 \le z \le y \le x$, find $\beta(n)$. Attempt: I think there are many cases to look at to find $\beta(n)$. We know that the number of solutions without restriction is $\binom{n+2}{2}$, but we […]

Alternative form to express the second derivative of $\zeta (2) $

It is known that the first derivative of the Riemann zeta function $\zeta’ (x)$ can be espressed, for $x=2$, as $$\zeta'(2) = -\frac {\pi^2}{6} [ 12 \log(A)- \gamma -\log(2 \pi)]$$ where $A $ is the Glaisher-Kinkelin constant and $\gamma $ is the Euler-Mascheroni constant. I would be interested to know whether it is possible to […]