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Mathematics

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.

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2 votes
0 answers
75 views

There's this theorem by T. Nagell which states: Given two non-constant polynomials $P,Q \in \mathbb{Z}[X],ドル there exists infinitely many primes $p$ which divide $P(a), Q(b)$ for some integers $a,b$. ...
6 votes
1 answer
287 views
+50

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)}...
-2 votes
0 answers
53 views

I say that because I am not a mathematician, but I do know that mathematicians like the Sieve of Eratosthenes. As a non-mathematician, I view primes slightly different. For example, my first 3 ...
11 votes
1 answer
969 views

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 $(P_n+1)$. ...
1 vote
0 answers
52 views

Let $P$ be a finite set of primes of cardinality $k$. Consider the set $A(P) = \{ n \ge 1 : \text{every prime dividing } n \text{ is in } P \}$. Let $L(P)$ be the largest number of consecutive ...
3 votes
0 answers
126 views

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),...
1 vote
1 answer
67 views

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 ...
3 votes
1 answer
192 views

For integers $n,ドル the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1,ドル for any prime $p$. $D(mn) = D(m)n + mD(n),ドル for any $m,n\in\mathbb{N}$ (Leibniz rule). $D(-n) = -D(n)$. The ...
18 votes
0 answers
333 views

For integers $n,ドル the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1,ドル for any prime $p$. $D(mn) = D(m)n + mD(n),ドル for any $m,n\in\mathbb{N}$ (Leibniz rule). $D(-n) = -D(n)$. The ...
3 votes
0 answers
116 views

Puzzle 1159 The prime 1709,2843ドル...$ can be decomposed into the sum of 7 other 5th powers of unlike primes: 1709ドル^5=1567^5+1373^5+719^5+503^5+431^5+367^5+349^5$ (Michael Lau, 08/20/2002) 2843ドル^5=2731^...
7 votes
2 answers
461 views

I am trying to prove that $(0,0,0)$ is the only integer solution of $$a^p+ p b^p+ (10p+1) c^p=0$$ I found this question on an Italian forum for the particular case in which $p=11$. The question was ...
0 votes
0 answers
62 views

Is it possible to use the sieve method to solve problems like these? Count a number of $p_1p_2\cdots p_r,ドル a product of $r\geqslant 1$ primes such that $p_i\in [P,2P]$ for all $i=1,2,\ldots, r$ in ...
0 votes
0 answers
96 views

Erdős problem 409 ("How many iterations of $n↦\phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?...
14 votes
2 answers
710 views

Consider the nonlinear recurrence $$x_n=1+\frac{\phi(x_{n-1})+\phi(x_{n-2})}2,\;\;\;x_0,x_1\gt2$$ where $\phi$ is Euler's totient function. The only possible fixed points $C$ of the sequence satisfy $$...
2 votes
1 answer
71 views

I am checking an intersection of two properties: is a square-free Carmichael number, i.e. composite, square-free, and for every prime we have (Korselt). has two distinct representations as a sum of ...

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