Questions tagged [prime-numbers]
Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.
13,080 questions
2 votes
0 answers
46 views
Since $\pi(x) - \pi(\sqrt{x}) + 1 = \sum_{d \mid \sqrt{x}\#}\mu(d)\lfloor\frac{x}{d}\rfloor$ is discontinuous, is it everywhere-discontinuous?
Let $0 \in \Bbb{N}$. Prime number theorists will be familiar with the formula: $$\rho(x) = \pi(x) - \pi(\sqrt{x}) + 1 = \sum_{d \ \mid\ \sqrt{x}\#}\mu(d)\lfloor\frac{x}{d}\rfloor$$ It is usually taken ...
- 22.7k
5 votes
0 answers
117 views
Prove that the number of sign changes $\sim x^{3/4}$ (square-free integers)
Let $Q(x)$ be the number of square-free integers up to $x$. The asymptotic formula is well known, namely $$Q(x) = \frac{x}{\zeta(2)} + \Delta(x)$$ with error term $\Delta(x)$. It is also known that $\...
user1295974
5 votes
1 answer
246 views
+50
Primes for which sum of sine ratios is nearly always an integer
Let $p$ be a prime, $1\le b \le p-1$ be an integer, and $\text{ord}(b)$ be the order of $b$ mod $p$. I am interested in the sum $$S_p(b) = \sum_{k=1}^{\text{ord}(b)}\frac{\sin\left(b^{k+1}\cdot\frac{p-...
- 7,744
2 votes
0 answers
49 views
An issue with the definition of regular primes in Marcus's Number fields ??
I began my journey of Algebraic number theory with "Marcus's Number fields ." At page 3 , he defined a relation $\sim$ on the set of ideals of $\mathbb{Z}[w]$ ,where $w=e^{\frac{2\pi i}{p}},...
7 votes
1 answer
337 views
Are twin primes of form $10n+9$ and $10n+11$ rarer than the other forms?
For primes $p>7$, the smallest twin primes are $(11,13)$, $(17,19)$ and $(29,31)$. We then have three classes: $$A: (10n_1+1,\,10n_1+3)\\ B: (10n_2+7,\,10n_2+9)\\ \,C:(10n_3+9,\,10n_3+11)$$ A ...
- 61.3k
0 votes
0 answers
43 views
preimage of an interval overlap under multiplication
$\newcommand\Z{\mathbb Z}$Let $M=pq$ for some distinct prime numbers $p$ and $q$. Consider the homomorphism $\phi: \Z/M\Z \to \Z/M\Z$ given by $\phi(x):= qx \bmod M$. We have $$|\phi(\Z/M\Z) \cap (-p,...
- 1,809
1 vote
0 answers
61 views
A conjecture on representing composite Fibonacci numbers
I've been exploring a question about composite Fibonacci numbers and I'm not sure if my findings are new or if this is a known problem. Definitions Let $F_n$ be the $n$-th Fibonacci number ($F_1=1, ...
5 votes
1 answer
248 views
A $3$-adic valuation regarding $ C_n = \frac{1}{3+\frac{2}{5+\frac{3}{7+\frac{4}{11+\frac{5}{13+\dots\frac{n}{p_n}}}}}} $ is always $1$?
My previously asked question motivated me to ask this question. For $n^{th}$ odd prime $p_n$, We define the following fraction: $$ C_n = \frac{1}{3+\frac{2}{5+\frac{3}{7+\frac{4}{11+\frac{5}{13+\dots\...
- 341
9 votes
1 answer
168 views
The numerator of $ C_{n>4} = \frac{1}{3+\frac{5}{7+\frac{11}{13+\frac{17}{19+\frac{23}{29+\dots p_n}}}}} $ is always divisible by $17$?
For $n^{th}$ odd prime $p_n$, We define the following fraction: $$ C_n = \frac{1}{3+\frac{5}{7+\frac{11}{13+\frac{17}{19+\frac{23}{29+\dots p_n}}}}} $$ We also define $N_n$ as the numerator of the $\...
- 341
2 votes
0 answers
81 views
Proof of Nagell's Theorem
There's this theorem by T. Nagell which states: Given two non-constant polynomials $P,Q \in \mathbb{Z}[X]$, there exists infinitely many primes $p$ which divide $P(a), Q(b)$ for some integers $a,b$. ...
- 513
11 votes
3 answers
516 views
Formula for the number of divisors of an integer
I recently derived the following formula for the number of divisors of an integer $n$ $$ D(n)=\lim_{h\longrightarrow 0}h\cdot \pi\cdot \sum_{i=1}^{\infty}\frac{\cot\left( \pi\cdot\frac{n+h}{i} \right)}...
- 333
13 votes
1 answer
1k views
Why do these valleys in $LCM(P_n - 1, P_n + 1) \bmod n$ exist and are they random?
I wanted to see what kind of properties the numbers just one more or one less than primes would have. I supposed $P_n$ was the $n^{th}$ prime and obtained two numbers from it, $(P_n-1)$ and $(P_n+1)$. ...
- 149
1 vote
0 answers
72 views
How many consecutive integers can be multiples of at least one prime from a set $P$ of size $k$? [closed]
Let $P$ be a finite set of primes of cardinality $k$. Consider the set $A(P)=\{n\ge 1: n \text{ is divided by at least one prime in }P\}$. Let $L(P)$ be the largest number of consecutive integers that ...
- 11
3 votes
0 answers
132 views
Are there infinitely many numerator primes?
Let $p_1, p_2, p_3, \cdots$ denote the sequence of prime numbers. Consider the partial sums of their reciprocals $$H_k := \sum_{i=1}^k \frac{1}{p_i}$$ Now if $H_k = N_k/D_k$ is fully reduced (coprime),...
user1295974
1 vote
1 answer
71 views
Interpret the halving of entropy
I am trying to understand and explain entropy intuitively. I think of entropy as ambiguity, and also, as the expected value of the knowledge gained. The doubling of entropy is straightforward to ...
