Articles of number theory

Binary Expansion of 2^n-1

Show that the positive integral multiples of $2^{n}-1$ when expressed in binary will have at least $n$ ones’. My work so far: Let $b(k)$ be the number of ones when k is expressed in binary and let $e(n)$ be the number of factors of 2 dividing $n$. I found that $$b(n-1)=b(n)+e(n)-1.$$ So i know that […]

How large can the smallest integral solution of a Mordell curve be?

Suppose, a Mordell curve $$y^2=x^3+k$$ has at least one integral solution. Denote $d$ to be the smallest absolute value of $x$, such that $x^3+k$ is a square, in other words, $d$ is the the smallest possible absolute $x$-coordinate of an integral solution. Is any reasonable upper bound for $d$ depending on $k$ known ? In […]

An application of Mobius Inversion $\sum_{d \mid n} \mu(\frac{n}{d})\nu(d) = 1$

Show that $\sum_{d \mid n} \mu(\frac{n}{d})\nu(d) = 1$, for any positive integer n. Where $\mu$ denotes the Mobius function defined by $\mu(n)=(-1)^{s}$ if $n=p_{1} \dotsc p_{s}$ for distinct primes $p_{1} \dotsc p_{s}$ and $\mu(n)=0$ otherwise, and $\nu(n)$ denotes the number of divisors of $n$. I think I have to apply the Mobius Inversion Theorem somehow […]

Prove an inequality of Prime Numbers

The problem on which I am currently stuck is, Is it true that, $$x+y< \dfrac{p_{\pi(x)}+p_{\pi(y)}+p_{\pi(x)+1}+p_{\pi(y)+1}-2}{2}$$ for all sufficiently large $x$ and $y$, $x+y$ is a prime and $\pi(x)$ is the prime counting function? My Atempt I tried to prove the problem by showing that $$x<\dfrac{p_{\pi(x)}+p_{\pi(x)+1}-1}{2}$$and $$y<\dfrac{p_{\pi(y)}+p_{\pi(y)+1}-1}{2}$$ But I cannot prove or disprove it by any […]

Show that Fermat number $F_n$ and its index $n$ are coprime.

I want to show that $\gcd(F_n,n)=1$, where $F_n=2^{2^n}+1$. How to prove this? I can show that that $\gcd(F_n, F_m)=1$ for any natural $n$ and $m$, and that $F_{n+1}=(F_n)^2-2F_n+2=F_0\dots F_{n-1}+2$, but I can’t see how I can apply this to my problem. What am I missing?

Remarks on a Previous Post: Elementary proof of $n>\frac{p_n}{\ln p_n}$

Recently I have been reading this post and I have noted something significant in the fake argument. As one can easily see that the basic idea behind the argument had been to show that the sequence $x_n=\dfrac{\pi(n)\ln n}{n}$ is strictly decreasing for all $n$. But, notice that the Prime Number Theorem is equivalent to the […]

For which small $n$ is unknown whether the Mordell-curve has an integral point?

Since it is not easy to determine the integral points of a Mordell curve $$y^2=x^3+n$$ with integer $n\ne 0$, I came to the following questions : $1)$ What is the smallest (in absolute value) integer $n$ , such that it is unknown whether the Mordell-curve $y^2=y^3+n$ has an integral point ? $2)$ What is the […]

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 […]