Articles of number theory

Divergence of the Sum of the Reciprocal of the Primes

Since Mertens showed that $$\sum_{p\le x}\frac{1}{p} = \log\log x + M + O(1/\log x)$$is there a similar equation for $$x\sum_{p\le x}\frac{1}{p-1}$$? ($M$ is the Meissel-Mertens Constant, which is approximately $0.2614972$, and $p$ ranges over the primes less than $x$)

Possible divisors of $s(2s+1)$

I write $\psi(s) = s(2s+1)$ and let $d$ be the divisor function. If $s$ is prime then 4 divides $d(\psi(s))$. For example if $s=37$ then $d(\psi(s)) = d(2775) = 12$ and $4|12$. Is this trivial? I am not sure how to attack and prove this. Here is my approach: Now I am thinking $d$ is […]

Smallest prime p with N a quadratic residue mod p

Let $N$ be a square-free number equal to 2 or 3 $(\mod 4)$. Let $P(N)$ be the first odd prime, not a factor of $N$, for which $N$ is a quadratic residue. On average, $N$ would be a non-residue for all the first $k$ primes once in $2^k$, so I would expect $P(N)=O( \log^m N)$ […]

System of equations – modular arithmetic

I am asked to solve the following….. Let $n\in \mathbb{N}$ and suppose that $a,b,c,d,k,l\in\mathbb{Z}$. Consider the system $ax + by \equiv k$ mod $n$ and $cx+dy \equiv l$ mod $n$. Let $D=ad-bc$. Show that if $hcf(D,n)=1$ then the system has a solution. There were previous parts to the question, in which I showed $ax\equiv b$ […]

Infiniteness of twin prime powers

This question is a weaker version of the twin prime problem. Question: Are there infinitely many prime powers $p^a, q^b$ with $p^a-q^b=2$? Of course, we expect an answer easier than solving the twin prime problem. The examples for $p,q<20000$ and $a,b<10$ with $(a,b) \neq (1,1)$ are the following: $3^3- 5^2$, $3^2-7$, $3^4-79$, $3^5-241$, $3^6-727$, $3^9-19681$, […]

Proving consecutive quadratic residue modulo p

This question already has an answer here: Existence of Consecutive Quadratic residues 3 answers

Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many Mersenne primes but only finitely many Fermat primes. Is there an intuition that can help me appreciate the nature of […]

Root of multiplicity?

Show if a is a root of multiplicity $n\geq 2\ $, then $f(a) = 0$ and $f'(a)=0.$ I was trying to learn root of multiplicity and saw this question. My TA did not go over it yet but I was wondering how this proof would look like. I get root multiplicity under product solutions in […]

Solving a congruence with high numbers and computing the Legendre symbol

I am asked to compute $\left(\frac{77}{257}\right)$ specifically using Euler’s criterion. I have manged to get the following: $$\left(\frac{77}{257}\right)\equiv77^{256/2} \bmod257$$ $$\equiv77^{128}\bmod257$$ I don’t know where to go from here as I don’t know the trick to simplify congruences that are big like this without typing it in on an online calculator. I believe it may be […]

upper bound on partial sums of Dirichlet character

Let $\chi$ be a non-principal character modulo k. Then for two integers $m,n$ with $m<n$, prove that $$\left|\sum_{j=m}^n \chi(j)\right|\leq \frac{1}{2}\phi(k)$$ It’s easy to prove the trivial estimate which is $\phi(k)$. Also, I am able to prove the above estimate in case of odd characters. I don’t how to prove it for even characters.