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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$a_n$ is relatively prime to $a_k$ for $k<n$

Let the sequence $\{a_n\}_{n=0}^\infty$ be defined by $a_n=|n(n+1)-19|$. Show that for $n\neq 4$, if $a_n$ is relatively prime to $a_k$ for all $k<n$, then $a_n$ is prime. The first few terms are $...
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Even natural numbers are sums of two primes with twins or of two primes without twins

I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins. That is, suppose that $\mathbb P'$ is the set of the primes not ...
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1answer
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Small primes congruent to $a$ mod $p$.

Let $p$ be a prime and $a$ be an integer such that $0 \lt a \lt p$. Is there a prime number, $q$, congruent to $a$ mod $p$ such that $q\lt p^2$? I have checked that this is true for the first $3000$...
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1answer
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Euclid Mullin Sequence

Consider the Sequence as follows. Let $a_1 = 2$, $a_n$ be the largest prime divisor of $P_n = 1 + {\prod_{i = 1}^{n - 1} a_{i}} $ Then we obtain a sequence of prime numbers How do you show that 5 ...
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1answer
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Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
4
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Which primes $p$ divide $q^q-1$ for a prime divisor $q$ of $p-1$

I am looking for (a formula) for all the primes $p$ less than or equal to $X$ with the following criteria: There is at least one prime $q$ dividing $p-1$ such that $p$ divides $q^q-1$. $7$, for ...
2
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2answers
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Can this function be a new test for primality?

The following function returns always 0 only if a number is not prime. $$ H(x)=\prod_{i=2}^{x-1}\left\{\left[\sum_{k=1}^{x/i}(-i)\right]+x\right\} $$ what do you think? Bye!
2
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1answer
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For each prime $p>3$ there are non twin primes $q,r$ with $p^3=2q+r$

Define $\mathbb P'=\{n\in\mathbb P|n-2,n+2\notin \mathbb P\}$. Conjecture: Given a prime $p>3$, then $\exists q,r\in\mathbb P':p^3=2q+r.$ Tested for the first 10000 primes. The solutions ...
3
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1answer
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Maximum length of a string that has no substring divisible by a prime number $p$ is $p-1$?

What is the maximum length of a string of nonzero digits that has no substring that is divisible by a given prime number? I want to find a string of length n which has no substring divisible by the ...
3
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1answer
93 views

multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
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1answer
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Conjecture: Every prime number is the difference between a prime number and a power of $2$

Conjecture: $\forall p\in\mathbb P\exists q\in\mathbb P\exists n\in \mathbb N: q-p=2^n$ Verified for the 100 first primes.
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5answers
240 views

A statement about divisibility of relatively prime integers

I'm solving a problem, and the solution makes the following statement: "The common difference of the arithmetic sequence 106, 116, 126, ..., 996 is relatively prime to 3. Therefore, given any three ...
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2answers
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Dirichlet inverse of $(-1)^n$

I was tinkering around and noticed the Dirichlet inverse of $\,f(n) = (-1)^n$ seems to be $$ f^{-1}(n) = -\mu\!\left(n\,/\,2^{\nu_2(n)}\right)\left\lceil 2^{\nu_2(n)-1} \right\rceil, $$ where $\nu_p(n)...
4
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2answers
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Growth of $\pi(2x) - 2\pi(x)$

In Hardy & Wright's Theory of Numbers (p. 494f in 6th ed.) there's a little discussion following the proof of the prime number theorem. We have $$ \pi(2x) - \pi(x) = \frac{x}{\log x} + o\...
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7answers
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How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?

It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian ...
5
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1answer
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Are the extremas of $h(x)$ global?

It is well known that $li(x)$, the integral logarithm is a very good approximation of $\pi(x)$, the nunmber of primes not exceeding $x$. So, a very good approximation for the probability, that a ...
4
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1answer
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What is the computational complexity of calculating $\pi(x)$ exactly?

The prime counting function $\pi(x)$ has been determined for $x=10^{26}$. The list of the $10^n$-th primes , however , ends at $n=18$. The $10^{18}$-th prime has $20$ digits. Apparantly, the ...
25
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1answer
874 views

Primes approximated by eigenvalues?

Consider the infinite matrix starting: $$\displaystyle T = -\left( \begin{array}{ccccccc} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\...
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Division/Remain by a Mersenne Prime

Is it possible to compute the integer division and remainder of an integer $x$ by a Mersenne prime $p$ using only bitwise operations?
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How many digits of the googol-th prime can we calculate (or were calculated)?

Here, a lower and upper bound for the $n$-th prime are given. Applying the given bounds $$n(\ln(n\cdot\ln(n))-1)<p_n<n\cdot\ln(n\cdot\ln(n))$$ and the approximation $$p_n\approx n(\ln(n\...
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1answer
41 views

Primes that are neither twin, cousin or sexy [on hold]

I'm reading up on prime pairs, and I had a question... I can't seem to find an answer to this anywhere, and the wikipedia list of prime types is enormous! Afraid I missed it when going through it. I ...
3
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2answers
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Why can the sieve of eratosthenes not be used to confirm the twin primes conjecture?

I have been having fun thinking about sieves and more particularly the twin prime conjecture. As I am fairly new to this type of mathematics, I am wondering, if we use the sieve of erastothenes, aka ...
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1answer
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Strange results in mersenne.org database

I am interested in GIMPS project. I was browsing through known Mersenne prime numbers when I discovered strange records in their database. For example, M6972593 is the 38th Mersenne prime. However, ...
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Is there a way to translate the Sacks Spiral into a triangle?

The sacks spiral is our natural number system written in the form of a spiral and it highlights the primes which seem to fall on certain curves within the spiral. I am interested to know if there is a ...
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A function can provide the complete set of Euler primes via a Mill's-like constant. Is it useful or just a curiosity?

The following $f(m,n)$ function provides the complete set of Euler primes (OEIS A196230): $$f(m,n)=m^2-m+[\lfloor E^{2^n} \rfloor - {\lfloor E^{2^{n-1}} \rfloor}^2 +\frac{\lvert n-(\frac{1}{2}) \...
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Generalization of Mill's theorem

Mill's theorem states that there exists a positive real number A such that the floor of the double exponential function $A^{3^n}$ are primes for all positive integers n. The value of A is ...
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0answers
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A conjecture relating to Goldbach

I have a conjecture related to the strong Goldbach conjecture and the Goldbach function. It is that: for any $g(E)$, there are a finite number of even numbers which can be expressed as a sum of two ...
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2answers
115 views

Primes in the binomial transform of $ [1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …]$.

This question is related to this sequence A139482. A commentator gives the following formula for $a_m$ $$a_m = {3m^2-9m+10 \above 1.5pt 2}$$ I have that you should consider the sequence $b_n =3n+2$ ...
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Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$.

Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$. I'm having a difficult time proving this problem. I was able to verify that it works for prime $n$ up to ...
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7answers
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Can a complex number be prime?

I've been pondering over this question since a very long time. If a complex number can be prime then which parts of the complex number needs to be prime for the whole complex number to be prime.
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Can do you repeat these calculations combining the explicit formula and Nicolas criterion, on assumption of the Riemann Hypothesis?

I did easy calculations to get for $x=N_k=\prod_{n=1}^k p_k$ the kth primorial, combining the so-called explicit formula$\dagger$ for the second Chebyshev function and Nicolas criterion for the ...
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3answers
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$p\in\mathbb P\iff\Big(2\leq k<\sqrt p\implies\gcd(k^2,p-k^2)=1\Big ),\;p>3$

This is sharper variant of A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$ It seems enough to test that for some sums: $p=m+n\implies\gcd(m,n)=1$, namely ...
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1answer
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Common generator of units in finite prime fields

It is well known that the unit group of a finite field is cyclic. What can we say about the generators? Specifically I am interested in the following question: Suppose we fix a positive integer $a$, ...
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2answers
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Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
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2answers
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Two randomly chosen coprime integers

This is a twist on the problem commonly known to have solution $6/\pi^2$. Suppose when choosing from all natural numbers $\mathbb{N}$, the probability of choosing $n \in \mathbb{N}$ is given by $P(n)=...
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3answers
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Find $n$ with $100<n<2000$ such that $2^n+2$ is divisible by $n$?

Find a number $n$ with $100<n<2000$ such that $2^n+2$ is divisible by $n$ ? Its can easily be seen that $n=6$ is possible case but it does not satisfy the main constraint of being greater than $...
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It is possible to use the Zeta Function as primality test? [closed]

It is possible to use the Zeta Function as primality test? $$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s} = 1+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+ ... $$ Where can I find the non-trivial zeros ...
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1answer
442 views

Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
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1answer
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Are there infinitely many primes $p$ such that $\frac{(p-1)! +1}{p}$ is prime?

Here I have the following conjecture: Let $$S_1(n)= \frac{(n-1)! +1}{n}.$$ Then there exist infinite prime numbers $p$ for which $S_1(p)$ is prime. And I don't know how to prove it. EDIT Let $...
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1answer
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Does any sum of twin primes, where the sum is greater than 12, also represents the sum of 2 other distinct primes?

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows: Any given sum of twin primes (...
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1answer
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Prime conjecture containing primorial: the difference between the primorial $n\#$ and the smallest prime $p > n\# + 1$ is always a prime

Help me find the exact conjecture statement. What I roughly remember is that it stated that the difference between primorial $n\#$ (product of first $n$ primes) and "some" larger number than the ...
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2answers
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A conjecture about the prime function $p_n$: $p_m \cdot p_n >p_{m \cdot n}$

While testing my system Zet for computational mathematics I find possible relations now and then. The latest is: Conjecture: For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } (4,4)$...
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1answer
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Is this a known conjecture? Given odd primes $p,q$ with $p + q$ sufficiently large, must there exist a different pair $p',q'$ with $p+q = p'+q'$?

Conjecture: There is a natural number $N\in\mathbb N$ such that given odd primes $p,q$ with $p+q>N$ there are primes $p',q'$ where $p' \notin \{p,q\}$ such that $p+q=p'+q'$. Is this known?
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For every prime $p > 3$ that is $3$ mod $4$, does $q+1 \mid p-q$ for some other prime $q$?

Yet another random conjecture about primes: Given a prime $p>3$ of the form $4n+3$. Then there exist a prime $q<p$ such that $q+1\mid p-q$. Verified for all $p<100000$.
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Does every power of two arise as the difference of two primes?

Conjecture: For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$. Verified for $n\leq 26$: ...
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0answers
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A conjecture about primes: if $a,b$ are coprime and not both odd, is $A(a,b,m)$ finite for some $m$?

Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$: $p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far... For some coprime numbers $a,b$, ...
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Does this conjecture about prime numbers exist? If $n$ is a prime, then there is exist at least one prime between $n^2$ and $n^2+n$.

I made an observation on prime numbers, want to check if any conjecture already exist or not? I am a computer programmer by profession and I am interested in number theory. As like many others I am ...
4
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3answers
238 views

Does the sequence $q(n)=3n+1+\frac{1-(-1)^n}{2}$ generate all the prime numbers?

Define $$q(n)=3n+1+\frac{1-(-1)^n}{2} \quad, \quad n\in \mathbb N.$$ $$1,5,7,11,13,17,19,23,25,29,31,35,\dots$$ It seems like this formula gives all primes $>3$ (although not just primes of ...
4
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3answers
44 views

The sum of more than two consecutive natural numbers cannot be prime.

The sum of more than two consecutive natural numbers cannot be prime. Is the statement true and is there any way to prove it? I was able to prove that the sum of an odd amount of consecutive ...
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0answers
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Asymptotic Growth of Function of Prime Counting Function

Consider $f(x)$ defined by $$f(x)=\sum_{k=1}^\infty \pi\Big{(}\frac{x}{k}\Big{)}$$ How may one another function $g(x)$ be defined such that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$I have tried $g(x)=c\...