Articles of number theory

Efficient algorithm for factorizing Mersenne numbers?

Is there an efficient algorithm (maybe Polynomial-time?) for factorizing Mersenne numbers of the form $2^p – 1$? We can assume that $p$ is a prime because if it is not, then we can reduce the problem to factorizing smaller Mersenne numbers in polynomial time. I understand that there is a more efficient primality test for […]

Least common multiple of sequence of integers

Let $a_n$ be a strictly increasing sequence of natural numbers such that $$\lim_{n\to\infty}\frac{a_n}{2^n}=0.$$ Now, define $$ l_n=\frac{lcm(a_1,a_2,…,a_{n-1})}{a_n}.$$ My question is: From the definition of $a_n$, can we say that $l_n\to\infty$? While I haven’t been able to find any previous work answering this question, a similar question proves that the least common multiple of a random […]

Chebyshev function identity

given the Chebyshev function $$ \sum_{n \le x} \Lambda (n) = \Psi (x) $$ with $$ \Lambda (n) = \log p $$ for $ n=p^{k} $ and $ 0 $ otherwise is then true that (i think i saw it in apostol book) $$ \Psi(x) + \Psi(x/2) + \Psi (x/3)+ \ldots = \log\lfloor{x!}\rfloor $$ here […]

How can we show $1+ord_{p}n \le 2^{ord_{p}n} \le p^{(ord_{p}n)/x}$

$1+ord_{p}n \le 2^{ord_{p}n} \le p^{(ord_{p}n)/x}$ for $p> 2^{x},n\ge 2 \in \mathbb{N}$; $x>0 \in \mathbb{R}$ It was written here that this is clearly the case; I don’t see it. Does somebody see it and cares to explain?

Getting integer solutions for equation $x^{2}-y^{4}=336$

I need to get integer solutions for the next equation: $$x^{2}-y^{4}=336$$ I know equations that look like $x^{2}-y^{2}=n$ have solutions $x$ and $y$ where $x=\frac{a+b}{2}$ and $y=\frac{a-b}{2}$, $a$ and $b$ being both odd or even. But I can not figure out how to solve the equation.

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem – If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal V_p(n)=1-(n^{p-1}\mod p)$$ then$$\lim_{p\rightarrow \infty}\mathcal V_p(n) = \operatorname{sinc}(2\pi \,n)$$ and for the pair correlation of the non-trivial zeroes of the Riemann $\zeta$-function, […]

Why is the Jacobi symbol in this setting a unique homomorphism?

If $D \equiv 0,1$ mod $4$ a nonzero integer, why is the map given by $\chi: (\mathbb Z / D\mathbb Z)^* \rightarrow \{-1,+1\}, \chi([p]) = (D/p)$ for odd primes $p$ not dividing $D$, a unique homomorhpism? And why is $\chi([-1]) = 1$ if $D > 0$, and $\chi([-1]) = -1$ if $D < 0$? Why […]

sum squared möbius function

I would like to prove the following equality $$\sum_{n=1}^N \mu^2(n) = \sum_{k=1}^{\sqrt{N}} \mu(k) \cdot \lfloor N / k^2 \rfloor$$ with N a square number. Can anyone give me a hint? p.s. I know already that $$\frac{\zeta(s)}{\zeta(2s) } = \sum_{n=1}^{\infty}\frac{ \mu^2(n)}{n^{s}}$$ Perhaps this can help?

prove that there are infinitely many numbers of the form $x = 111…1$ such that $31|x$

I need to prove that there are infinitely many numbers of the form $x = 111….1$ such that $31|x$ what i tried – I wrote x as $\sum_0^{n-1} 10^i$ i know that $(10,31) = 1 $ now im stuck .. any help will be aprriciated

How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?

I would like to test whether or not the following product (or its complement) $$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n\, \gamma_n \, i} \right)$$ converges to fractions of known constants for e.g. $s=2n$. Note that $\gamma_n$ is the imaginary part of the n-th non-trivial zero of $\zeta(s)$. I am aware of Andrew Odlyzko’s tables of […]