Questions tagged [prime-gaps]

The difference of two prime consecutive prime numbers is the prime gap. $g_i := p_{i+1} - p_i$.

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2answers
56 views

Is there an infinite amount of primes in base n made from an equal amount of even and odd digits.

Is there an infinite amount of primes in base n made from an equal amount of even and odd digits? A list of primes that have this property is this sequence $$23,29,41,43,47,61,67,83,89,1009,1021,1049,...
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77 views

Existence of a prime in $(\phi(n), n]$

The question is: for any $n\geq2$, is there always a prime $p$ satifying $\varphi(n)<p\leq n$? Here $\varphi(n)$ is the Euler totient function. We know that there is always a prime between $n-O(n^\...
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2answers
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Prove by elementary means that $n\#\geq 3n$ for $n\geq 5$, where $n\#$ is the primorial function.

Prove by elementary means that $$n\#\geq 3n$$ for $n\geq 5$, where $n\#$ is the primorial function. update: I have found an elementary proof, see my answer to my question. The remainder of this post ...
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1answer
29 views

Counting consecutive integers that are divisible by primes relatively prime to an arbitrary $n$

Let: $c > 0, n, m, x > 0$ be an integers $p\#$ be the primorial of $p$ $D_n(m,x)$ be the count of integers $i$ where: $m-x \le i < m$ There exists a prime $p$ that $p \nmid n$ but $p | i$ ...
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2answers
86 views

Reasoning about relatively prime factors of consecutive integers

Let: $n,m,x$ be any integer with $n$ being even $D_n(m,x)$ be the count of integers $i$ where: $m-x \le i < m$ There exists a prime $p \le x$ such that $p \nmid n$ but $p | i$ Examples: $D_6(0,5)...
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1answer
130 views

Does a counter example exist where no prime is found given the following conditions…

Let: $x>1$ be an integer $y$ be an even integer with $2x \le y \le x(x+1)$ gcd$(a,b)$ be the greatest common divisor of $a$ and $b$ $U(x,y)$ be the set of integers $u$ such that $0 < u \le x$ ...
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1answer
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Understanding Wittgenstein's proof of Infinitude of prime

Can someone please tell me why the last claim "It is thus the case..." is true? I tried considering negation of the last claim. But it didn't help. Any help would be appreciated. Thanks in ...
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1answer
72 views

Conjecture on prime gaps [duplicate]

I am presenting a conjecture i really like. Statement Consider $p_n$ the $n$th prime. Find wether there exist any $n$ such that $p_n-p_{n-1}=n$ and if there are any, fond wether there are infinitely ...
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1answer
129 views

L. Gegenbauer's proof of Infinitude of Primes

I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that L. Gegenbauer proved Infinitude of Primes by ...
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1answer
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A claim regarding Perott's proof of infinitude of primes

The following is a picture from 'History of the theory of numbers, volume l Divisibility and Primality' by L. E. Dickson. Dickson wrote J. perott's proof of 'Infinitude of primes'. The first line does ...
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2answers
71 views

Infinite primes history

I am little confused with who was the first to modify Euclid's argument of infinitude of primes from $p_{1}p_{2}...p_{r}+1$ to $p_{1}p_{2}...p_{r}-1$? Some writers say it was E.E. Kummer ,($1878$) (...
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1answer
105 views

Understanding Sylvester' s $1871$ paper of primes in arithmetic progression of the forms $4n+3$ and $6n+5$

The following is the proof of infinitude of primes in arithmetic progression of the form $4n+3$ and $ 6n+5$ done by Sylvester in $1871$ in his paper "On the theorem that an arithmetical progression ...
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1answer
90 views

Infinite Primes in Arithmetic progression $10n+9$

Can anyone provide How J. A. Serret proved infinitude of primes in the arithmetic progression $10n+9$? I know there are many general proofs available now. But I want this one. Any help would be ...
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1answer
97 views

Proof of prime in $(p,p^2)$?

Let $p$ be any prime. Let $S$ be the range of natural numbers in $[1, p^2]$. Suppose that there are no primes in $(p,p^2)$, which means that all prime factors of every number in $S$ must be $p$ or ...
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27 views

An inequality that involves consecutive primes, prime gaps and roots of prime numbers as a weak form of Firoozbakht's conjecture

In this post for integers $n\geq 1$ we denote the $n$-th prime number as $p_n$. When we consider that $k>1$ runs over integers, from the theory of the Stolarsky mean we can deduce that as $k\to \...
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2answers
126 views

Is the expected number of primes in a specific interval $[p_n^2,p_{n+1}^2]$ approximately $p_n$?

The expected (average?) number of primes in the interval $[p_n^2,p_{n+1}^2]$ is approximately $p_n$. While thinking about a completely different problem, I noticed the above relationship, which I ...
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1answer
19 views

Prime gaps of the form 2+6n are equally as numerous as those of the form 4+6n, and exactly half as numerous as those of the form 2+4n

Has this conjecture been proved? Taking, for example all the prime numbers from 3 to 100,000,000 The total number of all the gaps equal to either 2 or 8 or 14 or 20...etc is 1,616,471 The total ...
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4answers
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Gap between two consecutive primes/. [duplicate]

Let $p_k$ be the $k$th prime number. Show that there are infinitely many $k$ such that $p_{k+1}-p_k>2.$ This question was asked in the entrance examination of the Indian Statistical Institute(ISI)....
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On integers $n\geq 1$ for which $n$ divides $\sum_{k=1}^n R_k$, where $R_k$ denotes the $k$-th Ramanujan prime

For integers $n\geq 1$ in this post we denote the Ramanujan primes as $R_n$, see for example the Wikipedia Ramanujan prime or [1]. I don't know if my question is in the literature but I think that it ...
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Inequality of the form $(R_{n+1})^{f(n)}<\prod_{k=1}^n R_k$, where $R_k$ is the $k$-th Ramanujan prime and $f(n)$ a suitable arithmetic function

I'm curious to know if it is in the literature a similar/analogous statement about Ramanujan primes (this Wikipedia Ramanujan prime or [1]), than the known as Bonse's inequality for prime numbers (see ...
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About the gap of consecutive prime numbers.

I recently read James Maynard's paper "Small gaps between primes". In his paper, he used the result $$p_{\pi (k)+k}-p_{\pi (k)+1}\ll k\log k.$$ Here $p_n$ denotes n-th prime number and $\pi (n)$ ...
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1answer
59 views

Ramanujan primes in short intervals

I'm curious to know if it is in the literature a similar/analogous statement about Ramanujan primes (this Wikipedia Ramanujan prime or [1]) in short intervals than those that refers the Wikipedia ...
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Does this sum $(1-\frac{1}{2^2})^{(1-\frac{1}{3^2})^{…^{(1-\frac{1}{p^2})}}}$ also related to Riemann zeta function?

I'm interesting for iterated exponention sum of the form $(z_1)^{z_2)^{...^{z_k}}}$ such that $z_1$ and $z_2$, $z_k$ are differents real exponents , This kind of sum was studied by many Authors such ...
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Help with basic combinatorics for alternative approach to Bertrand's postulate

Let $n\geq 4$ be some integer, and let $k$ be largest prime index such that $p_k \leq \sqrt{2n}$. We assert there is some integer $x$ in $\frac{n}{2} < x < n$ such that we have a set of ...
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1answer
75 views

Does the Euler constant allow to go from the prime realm to the zeta zeros realm?

On average the $n$-th critical zero of the Riemann zeta function has imaginary part around $\frac{2\pi n}{\log n}$, so that the average gap between the $n$-th and $n+1$-th zero is $\frac{2\pi}{\log n}$...
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1answer
113 views

Why do prime numbers have this pattern?

I wrote a fairly simple program that works basically like this: Take a natural number (starting at 2) and test if it is prime If it is prime, switch direction (from up to down, or from down to up) ...
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146 views

proof of prime in every interval $(p^2,p^2+p)$

Overview We'll introduce a sort of little hack called the missing modulo conjecture which can identify an integer's previous prime. We then show that this may not be perfectly reliable on account of ...
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1answer
78 views

The Rook Conjecture: arrangement of $p$ primes being distinct $\pmod{p}$ through $p^2$

For any prime $p$, divide $[1,p^2]$ into $p$ equal intervals of length $p$, so that the first interval is $[1,p]$, the next $[p+1,2p]$, and so on. It is definitely unproven but seems likely that there ...
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90 views

Small gaps between primes and arithmetic progressions

It was proved in Polignac Numbers Conjectures of Erdos on Gaps Between Primes Arithmetic Progressions in Primes and the Bounded Gap Conjecture for János Pintz, using Bounded gaps between primes for ...
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1answer
131 views

Do all elements of $[n+1,2n]$ have strictly higher gpf than elements of $[1,n]$ when sorted by gpf?

For any $n\in\mathbb N$, let $A=\{x\in\mathbb N \mid 1 \leq x \leq n\}$ and $B=\{x\in\mathbb N \mid n+1 \leq x \leq 2n\}$. Order the sets by greatest prime factor of each element, ascending. Let $...
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1answer
53 views

An inequality regarding the sum of primes

Let $p_n$ be the $n$th prime. For integers $k \ge 3, c \ge 1$, does it follow that: $$\sum_{k \le i < k+c}(p_i - 2c) > 0$$ If not, is there a minimum value of $k$ where it is true? I know ...
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1answer
94 views

Proof of infinitude of primes using ratio of n to its totient

A few preliminaries: A primorial is the product of the first primes. There are two notations for this ($n\#$ is the product of all primes under $n$, and $p_n\#$ is the product of the first $n$ primes;...
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1answer
34 views

Minimum difference between consecutive multiples of $k$ that are $k\text{-rough}$

A $k$-rough integer is any integer whose prime factors are all greater than or equal to $k$. Is there a known formula for the smallest possible difference between consecutive $k$-rough multiples of $...
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Techniques on finding consecutive primes with large gaps

I found two consecutive prime numbers $401!-3463$ and $401!+4021$. These have a difference of $7664$. Is there some kind of technique that is known in order to find consecutive prime numbers with ...
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Does there exist such a sequence $B$ when $p>5$?

Let $A = (a_1, a_2, \ldots, a_n)$ be the sequence of odd primes are less than or equal to a prime number $p$. Let $C$ be the infinite ascending sequence of composite numbers that their factors are ...
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39 views

How one can show that this inequality holds true for infinitely many primes

Let $(g_{k})_{k≥1}$ be the sequence of primes gap. In 1938, Robert Rankin proved the existence of a constant $c > 0$ such that the inequality: $$g_{k}>c×(log k×loglog k×logloglog k)/((logloglog ...
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1answer
42 views

Finding a special sequence related to primes

Let $(g_{k})_{k≥1}$ be the sequence of primes gaps (https://en.wikipedia.org/wiki/Prime_gap#Lower_bounds). I am asking about the possibility of finding a real sequence $(x_{k})_{k≥1}$ with the ...
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1answer
52 views

Can we deduce that there is infinitely many indices $n$ such that the period length of $1/(2^{2^n}+1)$ is strictly less then $2^{2^n}$.

In this page (http://mathworld.wolfram.com/FermatPrime.html) we have the following result: $2^{2^n}+1$ is a Fermat prime if and only if the period length of $1/(2^{2^n}+1)$ is equal to $2^{2^n}$. In ...
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A weaker version of the Firoozbakht's conjecture

Firoozbakht's conjecture states that: $p_{k}^{1/k}$ is a strictly decreasing function of $k≥1$. Here $p_{k}$ is the sequence of primes. I know that is statement is not yet proved. But I am asking on ...
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257 views

A weaker version of the Andrica's conjecture

Andrica's conjecture states that: For every pair of consecutive prime numbers $p_{k}$ and $p_{k+1}$, we have : $$\sqrt{p_{k+1}}-\sqrt{p_{k}}<1\quad\quad \color{#2d0}{\text{(1.)}}$$ I know that is ...
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1answer
82 views

Interesting thing about $3$ consecutive prime numbers

Let $p_1,p_2,p_3$ are consecutive prime s.t. $p_3>p_2>p_1>3$ Then show that If $(\frac{p_2-p_1}{2})\equiv1\pmod3$ then $ (\frac{p_3-p_2}{2})\not\equiv1\pmod3$ If $(\frac{p_2-p_1}{2})\equiv2\...
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1answer
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Counterexample to a given claim about prime numbers [closed]

Let $(x_{k})_{k≥2}$ and $(y_{k})_{k≥2}$ be two non constant sequences of strictly increasing positive integers such that $x_{k}>1,y_{k}>1$ for all $k≥2$. I want to get a counterexample to the ...
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1answer
147 views

My Observation on Prime gap

Define $\sum_{q=1}^{u}q^{u-1}= S(u)$ Problem 1, show that Let $p$ be a odd prime, if $S(3p)\equiv t \pmod {2p}$ then $t\in\{0,p\}$ Update Can we show that $\frac{t}{p}=1-$ parity of $\frac{p(p+1)}{2}...
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3answers
2k views

A trivial proof of Bertrand's postulate

Write the integers from any $n$ through $0$ descending in a column, where $n \geq 2$, and begin a second column with the value $2n$. For each entry after that, if the two numbers on that line share a ...
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38 views

Alternative proof sketch of Bertrand's Postulate from GCD

Consider the following algorithm (Python, untested): ...
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1answer
64 views

Why isn't a simple sieve proof of Bertrand's Postulate?

I wish to show a prime in $(n,2n)$. I make a list of numbers $n+1$ through $2n-1$. I check $n+1$ for prime factors $p_i<\sqrt{2n}$. It has a factor of $p_1$. I cross off numbers $\equiv n+1 \pmod {...
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0answers
45 views

What 's the largest number of consecutive integers such that each is divisible by a prime $\le p_n$

What's the best upper bound for the largest number of consecutive integers such that each is divisible by a prime ≤$p_n$? Is it less than ${p_n}^2$?
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1answer
106 views

Asymptotic distance between $x^2+1$ primes?

As I recall, the most common difference between consecutive primes starts with $2$, moves on to $6$, then $30$, and is conjectured to progress like the primorials over a long time, without bound. Is ...
6
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1answer
79 views

The crosshatch conjecture, on primes in $(p,p^2)$

If the first $p^2$ integers are laid out in a $p\times p$ square, every row and column will have at least one prime. Easily visualized as so: I recognize this should maybe be packaged as two ...
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1answer
37 views

Is there any known configuration of primes through $n$ which covers $n^2$?

Is there any known initial arrangement of prime residues (apologies in advance, I'm going to play fast and loose with the nomenclature) through some $n$ such that for every value in $[n+1,n^2]$, some ...

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