# 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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### Multiplicative order of $2$ modulo $p$.

When calculating the multiplicative order of $2$ modulo a prime $p$ you often get $p-1$ or $\frac{p-1}{2}$ as a result, but there are cases where this does not hold, is there a general form for those ...
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### Is $f(x)$ worse than $\text{Li(x)}$ at counting primes?

This is the Gram series: $$G(x)=1+\sum_{k=1}^\infty\frac{(\ln x)^k}{kk!\zeta(k+1)}$$ It is equivalent to the Riemann prime counting function: $$R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n})$$ I ...
73 views

### A function asymptotical equivalent with the prime counting function?

Let $p_n$ be the $n$-th prime number and $Q_a(N)$ be the number of primes of the form $p_n^2+a$ where $1\leq n\leq N$ and $a$ is positive and even. For some $a$ like $26,56$ it seems that no solutions ...
1k views

### Longest geometric progression of primes

There are arbitrarily long arithmetic progressions of primes e.g. $5, 11, 17, 23, 29$ for a $5$-length progression, but no (infinite) arithmetic sequence of primes with common difference $d\neq 0$, as ...
1 vote
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### Bertrands Postulate generalization

Bertrands postulate states that there's always a prime number in [N,2N] and I was thinking... Considering that N=1*N and that (1,2) are the first prime numbers maybe this is just a particular case and ...
1 vote
38 views

### There are at most $n$ primes between $1$ and $2n$

This question originates from one of my tasks: Choose $n+1$ whole numbers $a_1 \le a_2 \le ... \le a_{n+1}$ between $1$ and $2n$ inclusive. Prove that among those $n+1$ number there exist 2 indexes $i$...
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### Goldbach Conjecture and Triangular Number

Goldbach conjecture states that every even number greater than 2 is sum of two prime numbers. We know that every positive integer can be represented as a sum of three triangular numbers. Is it ...
229 views

### (Novel?) sieve that contains all primes. [closed]

Question Per some individuals request I have to phrase this as a focused question for this Q&A site. So the main question is: Is the following sieve and conclusion novel? All prime numbers are ...
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### question about GIMPS (Finding Mersenne Prime) on MacBook Terminal [closed]

This is about GIMPS – Great Internet Mersenne Prime Search, which some mathematicians use to find big Mersenne Prime numbers. When I first opened the mprime folder to join GIMPS, I had all the ...
91 views

### How to expand $x(x-1)(x-2)...(x-k)$

I got $$P(x)=1+\prod_{i=0}^{2021} (x-i)$$ and need to use Eisensteins's Criteria to solve the irreducibility of $P(x)$ but I found a problem how to elaborate the coefficient and choosing prime $p$. ...
218 views
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### "Multiply everything so far, plug into polynomial" - can these always yield primes?

EDIT: I forgot how open number theory is! (I think that gets me put on mathematician probation or something.) For this question, I will accept any answer which assumes "standard conjectures" ...
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1 vote
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### Miller-Rabin primality test final decision. [closed]

I feel like this question is more about math than, programming but it will include some simple code for the Miller-Rabin test (in scheme-lisp). ...
### Does $\{(f)：ℙ↣ℙ\}$ contain any analytic members $f$?, and if so What is the simplest such injective $f$? Are all $f$ necessarily monotonic? [duplicate]
(Above, $ℙ≔\{\text{all primes}_ℤ\}$.) Are there any analytic functions that will give a unique prime output for every distinct prime input? Analyticity should preclude cheap reiterating upon a prime-...