Prime numbers are natural numbers 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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Question regardles primes and the fundamental theorem of arithmetic

I have been reading through my book of practice proofs and came across this particular question which has stumped me. $p$ and $q$ are primes. Prove $\forall p \in \mathbb{Z}, \forall k \in ...
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0answers
61 views

Resources about infinite primes of form $n^2 + 1$

Where can one find existing work on the following problem? Prove there are infinitely many primes of the form $n^2 + 1$. Resources about related work would also be appreciated.
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5answers
665 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
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4answers
836 views

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
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0answers
57 views

Birthday problem & primes

Let $\pi_k(n)$ be the almost prime counting function, then $\pi_k(2^kn)$ reaches a max value, since $\pi_k(2^kn)=\pi_{k+1}(2^{k+1}n)$ for large enough $k$. (eg, ...
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1answer
68 views

$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$ for $n\geq 7$

I can prove that $\text{lcm}(1,2,3,\ldots,n)\geq 2^{n-1}$. Newly, i read in a paper that for $n\geq 7$ we have: $$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$$ Can you prove it? (this inequality is an ...
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0answers
81 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
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1answer
42 views

Can a Mersenne number be a power (with exponent > 1) of a prime?

Let $n \geq 1$ and consider the (Mersenne) number $M_n = 2^n-1$. Is it possible that $M_n = p^k$ for some prime $p$ and some (necessarily odd) $k > 1$? Thanks in advance.
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1answer
55 views

AP term multiple of prime number

I am having this equation : (a+(n-1)d)%p=0 Here a and d can go upto 10^18 and p is prime number upto 10^9 . How to find the least value of n here? Example : If ...
3
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0answers
33 views

Prime Triangle:: How to find the position(row and column) of prime number in a triangular arrangement

I was working on problem which asks the position of a prime number in a triangular arrangement. If we arrange the all prime up to 10^8 as shown in image we can find the row and column number of a ...
3
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1answer
59 views

Improving Bertrand's postulate

Recall that Bertrand's postulate states that for $n \ge 2$ there always exists a prime between $n$ and $2n$. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series $$ 1 + ...
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1answer
21 views

How to combine three integers from three sequences such that all the numbers in the triplet be coprime.

So there are three sequences of integers denoted $s_1, s_2$ and $s_3$. Each of them starts at $2$ and goes up to an integer $N$. In other words, they're all identical sequences of numbers of the form ...
3
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4answers
111 views

Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.

The title says it. I thought of the following: we want $$\Bbb N = \dot {\bigcup_{n \geq 1} }A_n$$ We pick multiples of primes. I'll add $1$ in the first subset. For each set, we take multiples of some ...
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0answers
43 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
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1answer
33 views

Physical reflections of prime-number distribution

Not a purely mathematical question: I have read somewhere that Atomic Orbital is closely related to the distribution of prime numbers, but I am unable to find any reference to that. Can someone ...
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1answer
44 views

Proof for Carmichael theorem

if $n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\dots p_r^{a_r}$ and $\lambda(n) = lcm[(p_1-1)(p_1^{a_1-1}),(p_2-1)(p_2^{a_2-1}),(p_3-1)(p_3^{a_3-1}),\dots,(p_r-1)(p_r^{a_r-1})]$ then $k^{\lambda{n}} \equiv ...
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1answer
30 views

Extrema of the Ratio of Consecutive Primes

Let $p_i$ denote the $i$th prime number. We know that $\frac{p_{n+1}}{p_n}\rightarrow 1$ as $n\rightarrow\infty$. Therefore, if we pick some real number $c>1$, there should be some positive integer ...
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1answer
56 views

The binomial coefficients $\binom{n}{ p}$ are divisible by a prime $p$ only if $n$ is a power of $p.$

I'm looking for a "high school / undergraduate" demonstration for the: All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ ...
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1answer
42 views

How many $\overline{a}\in\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$ pass the Fermat and Miller-Rabin primability tests?

Let $$\text{F}_{91}:=\left\{\overline{a}\in\left(\mathbb{Z}/n\mathbb{Z}\right)^\times:91\text { passes the Fermat primality test to base }a\right\}$$ and ...
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1answer
32 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
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0answers
39 views

Nondiagonalizable matrix with successive primes?

The 4x4-matrix filled from left to right and from top to bottom with successive primes, starting with the prime $21005627$ , is not diagonizable. Is there a 3x3-matrix with the same property ? I ...
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1answer
42 views

(p-1)! number theory problem

I was working on this little problem: Let $\frac{a}{(p-1)!} = 1 + 1/2 + 1/3 + ... + 1/(p-1)$, where p is a positive prime. (a) Prove $p\mid a$. (b) Can $p^2 \mid a$? I thought (a) was pretty ...
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1answer
73 views

Combining primes for getting primes?

I was thinking about what would happen when we combine two prime numbers $p$ and $q$ into one number $:pq:$ . Like if $p=5$ and $q=3$ , then $:pq:=53$ . Then if $p=7$ and $q= 11$ then $:pq:=711$ and ...
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4answers
559 views

Is every prime the average of two other primes?

$\forall {p_1\in\mathbb{P}, p_1>3},\ \exists {p_2\in\mathbb{P},\ p_3\in\mathbb{P}};\ (p_1 \neq p_2) \land (p_1\neq p_3) \land (p_1 = \frac{p_2+p_3}{2})$ Now I'm not a 100% sure about this, but I ...
5
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1answer
178 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
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5answers
771 views

Proof of infinitely many primes, clarification

Proof: The proof is by contradiction. Suppose there are only finitely many primes. Let the complete list be $p_1,p_2,\dots,p_n$. Let $N = p_1p_2 \dots p_n+1$. According to the Fundamental Theorem of ...
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1answer
20 views

Primes in a concave sequence

Are there infinitely many consecutive tuples of primes $(p_{n-1},p_n,p_{n+1})$ satisfying $$\frac{1}{2}(p_{n-1}+p_{n+1})\le p_n$$ I don't know how to do this problem. Maybe we use the prime number ...
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2answers
82 views

How can I solve this problem without having to do it by hand?

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
6
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1answer
378 views

Infinite sum of prime reciprocals

Let $\mathbb D$ be the set of all real numbers that can be expressed as a sum of distinct prime reciprocals, i.e. $\mathbb D = \{ d \in \mathbb R \mid d = \sum_{k \in \mathbb K} \frac 1k $ for some ...
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1answer
68 views

How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
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2answers
66 views

Is there any way to solve this problem without having to do it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement. Is there any way to group ...
0
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2answers
32 views

$\log$ approximation for $\pi(x)$

It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where $f(y, x) := \log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor ...
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1answer
56 views

Finding Prime triples with $p_{n} +p_{n+1} −p_{n+2} = 1$

I was just looking at a sequence of primes and suddenly I got this thought that $p_2 +p_3 −p_4 = 1$ since $p_2 = 3, p_3 = 5, p_4 = 7$. Also for $p_3 = 5, p_4 = 7, p_5 = 11$ one has $p_3 +p_4 −p_5 = ...
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2answers
53 views

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 ...
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0answers
32 views

On non-divising primes of an integer $x$

We know more about divisor than non-divisors, If we consider the sets : $$P^{1}_{x} =\left \{ p \leq x : \ p \in \mathbb{P} \right \}$$ ($\mathbb{P} $ is the primes set) $$ P^{2}_{x} =\left \{ ...
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2answers
28 views

Relative error in prime number fitting

I got curious about what it would look like if I made a scatter plot of the n-th prime number as a function of n (lets call it p(n), so that p(1)=2, for instance ). Not being an expert in the area I ...
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1answer
55 views

How many natural value of n such that $n^5+2n^4+n-1$ is prime number?

From above polynomial, I can only get one value to make it prime. The value, I guess, is only one. For $n=1$, we got: $$(n^5+2n^4+n-1)= 1+2+1-1= 3 \quad\text{(prime)}$$ But, I cannot find the ...
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0answers
46 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
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3answers
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Conjecture: If $ N $ and $ N + 10 $ are prime, then $ N + 20 $ is composite.

Conjecture: If $ N $ and $ N + 10 $ are prime, then $ N + 20 $ is composite. Here are some examples: $ 19 $ and $ 29 $ are prime; $ 39 $ is composite. $ 241 $ and $ 251 $ are prime; $ 261 $ is ...
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1answer
23 views

Showing $f\in\mathbb{F}_{p^d}[X]:f'=0\Rightarrow\exists g\in\mathbb{F}_{p^d}[X]:f=g^p$

Let $\mathbb{F}_{p^d}$ denote the final field with $p^d$ elements and $\mathbb{F}_{p^d}[X]$ denote the polynomial ring in $X$ over $\mathbb{F}_{p^d}$. How can we show ...
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1answer
47 views

The most efficient method for generating new prime numbers

What is the most efficient method for generating a prime number larger than the largest known prime number, and what is the complexity of this method? Techniques considered: Mills' Constant - ...
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1answer
32 views

References for Legendre's prime-counting function

This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number ...
5
votes
2answers
168 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
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1answer
21 views

Find all $n$ so that $c_n$ $>$ $\pi(n^2)$

Find all $n$ $\in$ $\mathbb{N}$ so that $p_{c_n}$ $>$ $n^2$ where $p_n$ denotes the $n$-th prime and $c_n$, the $n$-th composite. I have tried doing the problem using The stronger version of ...
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7answers
222 views

Alternative Proof of Infinitely Many Primes? [duplicate]

I've seen Euclid's proof of infinitely many primes, what are other approaches to proving there are infinitely many primes?
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3answers
47 views

If there is an $a\in\mathbb{Z}$ with $a^{n-1}\equiv 1\mod n$ but $a^{\frac{n-1}p}\not\equiv 1$ for all primes $p\mid n-1$, then $n$ is a prime

Let $n\in\mathbb{N}$ with $n\ge 3$ and $a\in\mathbb{Z}$ such that $$a^{n-1}\equiv1\text{ mod } n\;\;\;\wedge\;\;\;a^{\frac{n-1}{p}}\not\equiv1\text{ mod }n\;\;\;\forall p\in\mathbb{P}:p\mid n-1$$ ...
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1answer
51 views

Conjecture about prime numbers

$$\forall k\in\mathbb{N},k\ge1,\exists p:k^3\lt p\lt (k+1)^3$$ with $p$ prime number. In other words is it possible to prove that for every $k\gt1$, with $k$ integer number it exists a prime number ...
2
votes
0answers
58 views

Is this a conjecture or an already existing one??

Does the following inequality hold? $p(n)\leq 2^n,$ where $p(n)$ is the $n$th prime. If this is true then it follows that: If $p(n)=p(m)^x+p(o)^y$, then $\max[x,y] \le n$.
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1answer
38 views

Euler totient of a number

If $n= \prod_{i=1}^{m} p_i$, all $p_i$ pairwise distinct, then number of coprimes below $n$ is $\prod_{i=1}^{m} (p_i-1)$. For example with $m=2$, there are $p_2-1$ multiples of $p_1$ below $n$ and ...
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1answer
67 views

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...