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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7
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3answers
285 views

What does it mean to count a group of numbers with their multiplicity?

In this question someone previously asked They presented the problem: Given that the number 8881 is not a prime number, prove by contradiction that it has a prime factor that is at most 89. One ...
0
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1answer
92 views

Primes of the form $a^2+b^2+c^2$, $0<a<b<c$ [closed]

Are there results showing which prime numbers that can be expressed as the sum of three different integers greater than zero? By the three square theorem of Legendre a natural number can be written ...
4
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0answers
195 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
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3answers
79 views

Division of prime numbers

a) Let p and q be different odd primes. Is there any such numbers for which $$(p-1)(q-1) \mid (pq)^2 +3$$? b) Let q be an odd prime number. Is there any q for which $$6(q-1) \mid 81q^2 + 3$$? (I ...
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2answers
64 views

Is the largest gap between consecutive primes less than the first $27,000$ integers equal to $52?$

Is the largest gap between consecutive primes below the first $27,000$ integers equal to $52?$ At what point does a gap greater than $52$ occur? I tried analyzing a formula due to Maynard, Tao, and ...
0
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3answers
110 views

Prove these two statements are logically equivalent

Consider the following two statements about non-negative integers: Goldbach's conjecture: Every integer > 1 is the average of two primes SumOfThreePrimes: Every integer > 5 is the sum of three primes....
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1answer
176 views

A Simple Proof of the FTA using only elementary theory?

UPDATE/WARNING: DO NOT READ (WASTE OF TIME) The effort I put in here is now an embarrassment as it goes nowhere. I can't delete this posting since there is an answer, but if a moderator could delete ...
3
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4answers
174 views

What is the remainder of $65!$ divided by $67$?

What is the remainder of $65!$ divided by $67$? Attempt: By Wilson's theorem, we have $66! = -1\mod(67) $. $$66! = -1\mod(67) \implies 66 (65!) = -1 \mod(67)$$ and we also know that $66 = -1 \mod(...
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0answers
49 views

What about distribution of primes if the Riemann hypothesis is false?

I have read many papers related to the distribution of primes number if RH is true like GUE ( Guaussian unitary matrix ) and GOE (Gaussian orthogonaly matrix) both of them occurs in Gaussian ...
1
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1answer
48 views

There is no set of exactly five elements $A \subset \mathbb{N}$ such that sum of any distinct three is prime.

Each the five elements belong to any of the modulo classes of $3$, i.e. $P=\{3k+1:k\in\mathbb{N} \cup \{0\}\}\bigcap A$ $ \ Q=\{3k:k\in\mathbb{N} \setminus\{ 0\}\}\bigcap A$ $R=\{3k+2:k\in\mathbb{N} ...
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1answer
47 views

The pattern of $n$-th prime minus its reversal in even and odd bases

I recently saw the sequence A265326 on OEIS and also in Brady's Numberphile video Amazing Graphs ft. Neil Sloane The sequence is such: Start from $2$, given the $x$-th prime number and convert it ...
2
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1answer
60 views

Remainders of powers of 10 when divided by prime

While doing a number theory question, I observed that the repeating digits of decimal expansions of $\frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ form a pattern. They are $...
3
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0answers
541 views

Weak k-Tuple conjecture form and what we should prove

Let $x \in \mathbb{R}_{+}$. For $q \in \mathbb{P}$, let : $\mathcal{B}_q = \{b \in \mathbb{N}^{*} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p premier}}} {\normalsize p}})...
3
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1answer
117 views

The maximum of real function with 4 prime parameters and $\lfloor \ \rfloor$

Let $a$,$b$,$c$ and $d$ be prime numbers such that $a>b>c>d$. Let $x$ be an integer greater than $a$. Let $f(x) = \left(\dfrac{x}{a}\right) – \left(\left(\dfrac{x}{ab}\right) + \left(\dfrac{...
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0answers
42 views

To prove $\gcd\{p^2-1: p $ is a prime $ \geq 7 \}=24$ [duplicate]

Let $p$ be a prime $\geq 7$. $p^2-1=(p+1)(p-1)$. Both $(p-1)$ and $(p+1)$ are divisible by $2$ and they are two consecutive even numbers. So, either one of the two must be divisible by $4$. So, $8 \...
3
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1answer
107 views

Project Euler #134

The problem statement is here: Consider the consecutive primes $p_{1}$ = 19 and $p_{2}$ = 23. It can be verified that $1219$ is the smallest number such that the last digits are formed by $p_{1}...
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1answer
83 views

Strong Lucas Test; as soon as any condition is met does it stop?

I am wondering something about the strong Lucas Pseudoprime test. Wikipedia states: (click for larger picture) Does the normal test (i.e. the not strong one) still need to be done, where $U_{\...
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0answers
40 views

Best possible estimate for $n$th prime [duplicate]

We all know the $n$th prime can be approximated by. $nln(n)$ by Prime number theorem. We also got inequalities for the estimate . But my question is what is the best possible estimate for $n$th ...
0
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4answers
80 views

Last non zero digit in 20! [duplicate]

So I have a question where it says to find the last non zero digit of $20!$ I proceeded in the following way: Found the prime factorization of $20!$ by calculation the greatest powers of $2,3,5,7,11,...
1
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1answer
38 views

What is the rough upper bound to find nth prime? Also give the maximum error.

At first please don't mark this as duplicate. I couldn't get a satisfactory answer in previous questions. I want a simple upper bound calculating formula for n-th prime which should not have ...
6
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1answer
276 views

show that $n\Upsilon_{n-1} \equiv -1 \pmod{n}$ iff $n$ is prime

The Bernoulli numbers $B_n$. where all numbers $B_n$ are zero with odd index $n>1$. first values are given by $B_{0} = 1$ , $B_{1} = -1/2$, $B_{2} = 1/6$, $B_{3} = -1/30$. Agoh conjecture: let $...
3
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1answer
55 views

$n^{th}$ Dimensional Prime Nexus Conjecture (Hamilton's Path & Cycle)

The system and proposition of Prime Nexus - We have to arrange $n>1$ Distinct natural numbers in a sequence such that the adjacent elements sums upto any Prime number. Convenience is when we set ...
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2answers
32 views

Algorithm to decompose a number into the product of an integer and a base two exponencial

So i`ve been asked to code an algorithm that decomposes an integer into the product of a base two exponencial and some integer.Something like number = k.(2^n) , k and n being random integers with k ...
7
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1answer
144 views

What is known about the counting function of Gaussian primes"

The counting function of primes among $\Bbb{N}$, describing the asymptotic density of the primes, is well known (the Prime Number theorem). Let's define a mild generalization of the counting function ...
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0answers
41 views

time complexity of some specific system of $(n-3)/2$ linear equations with $(n-3)/2$ unknowns.

let take n a odd integer greater than 3 if I have to solve this system of $(n-3)/2$ linear equations with $(n-3)/2$ unknowns to find some propriety about the number n, here is formula for every ...
1
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1answer
72 views

Is it true that each number $2p>6, \ p \ is \ prime$ has a Goldbach partition which contains two twin primes?

For instance, $$2\cdot 5 = 10 = 7+3$$ $$2\cdot 7 = 14 = 3+11$$ $$2\cdot 11 = 22 = 5+17$$ $$2\cdot 13 = 26 = 7+19$$ $$2\cdot 17 = 34 = 3+31=5+29$$ So, Is it true that each number $2p>6, \ p \ is \...
4
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1answer
70 views

Can we rearrange primes to make this ratio converge to any real?

Let $p_n$ be the $n$-th prime and let $q_1, q_2, \ldots, q_n$ be any rearrangement of the first $n$ primes. Using the rearrangement inequality and the solution to this problem, we can prove that $$ 1 ...
1
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1answer
104 views

Limiting value of the sum of two sequences of prime numbers

Let $p_n$ be the $n$-th prime. Is the following true? $$ \lim_{n \to \infty}\frac{p_1 + 2p_2 + 3p_3 + \cdots + np_n}{p_n + 2p_{n-1} + 3p_{n-2} + \cdots + np_{1}} = 2 $$ Note: The rearrangement ...
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4answers
125 views

Trying to understand why “long blocks between primes” must exist

I'm currently going through "An Introduction to the Theory of Numbers" by Hardy and Wright and at one point, they discuss why the distance from one prime to the next must have a long chunk of ...
2
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2answers
53 views

Are all numbers representable as the sum or difference of the products of prime powers of a finite set of primes?

If we take the first $k$ primes, are there always going to be natural numbers not representable as: $$p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \pm p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}\ ?$$ Assume non-...
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0answers
46 views

Correct primality certification?

With pari/gp , I tried to prove the primality of the number $$63638089207891963635134908^{1024}+1$$ I applied the Pocklington test and the output was ...
0
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1answer
60 views

Sophie Germain-like primes and Cunningham-like chains

A Sophie Germain prime is a prime number $p$ such that $2p+1$ is also prime. What do we know about other similar "Sophie Germain-like" primes, such that for instance $4p+1$ is prime (or generally, ...
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0answers
60 views

$\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$ [duplicate]

Show that $\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$ for $0\le k\le p^{a}-1$. Solution: $\binom{p^a-1}{k}=\frac{(p^a-1)(p^a-2)\cdots(p^a-k)}{k!}$. Idea: Then power of $p$ in $k!$ must be less than ...
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1answer
74 views

Interesting problem regarding prime numbers [closed]

Let $p_{i}$ be the prime number corresponding to the integer $1\leq i \leq n$ $(n\geq 1)$ and $\alpha: \{i \in \mathbb{Z}: 1\leq i \leq n\} \rightarrow \mathbb{N}$ and $\beta: \{i \in \mathbb{Z}: 1\...
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2answers
73 views

Remainder of Fermat's little theorem sum

With p being a prime number, what is the remainder of $$\sum_{k=1}^{p-1} {k^{p-1}}$$ divided by p ? I know that Fermat's little theorem states that for a prime number p, and a number A that is not ...
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0answers
916 views

Can we remove any prime number with this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the ...
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1answer
164 views

What is $\frac{1}{p}$ mod p?

In a multiplicative group $0$ doesn't have an inverse. But is $1/p \bmod p = 0$ or is $1/p \bmod p$ undefined? I wondered if you can argue that $$\begin{align}1/p &\equiv p^{-1} \\ &\equiv p^...
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1answer
71 views

Evaluating this sum. Is there a closed form?

Trying to evaluate this sum: $$ S=\sum_{n=1}^\infty \ln(p_n^2)K_1(\ln(p_n^2)). $$ Here $p_n$ is the nth twin prime and $K_1$ is the modified bessel function of the second kind. So one should be ...
4
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3answers
122 views

Prime counting function; is it true that $\pi(n - m) \geq \pi(n) -\pi(m)$?

Let $\pi$ be the prime counting function, and $m, n$ positive integers, $n > m > 1$. Is it proven that $\pi(n - m) \geq \pi(n) -\pi(m)$ ?
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1answer
63 views

Find all integers $x$ such that $2^p + 3^p = x^2$ where $p$ is prime

Find all integers $x$ such that $2^p + 3^p = x^2$ where $p$ an arbitrary prime number. I think that this equation has no solution. Thank all for help!
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0answers
66 views

Conjecture on Prime Numbers and Exponential Equipment

How to generate a rigorous proof for - The specific conjecture which states that we can always frame every Prime number in an exponentially framed equation with perfect cubes & squares. I'll ...
1
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1answer
29 views

Continuation of the Riemann Prime Counting function

Let $f(x)=\sum_{n=1}^{\lfloor \ln(x)\rfloor} \frac {\pi(x^{\frac 1n})}n$ ($\pi(x)$ is the Riemann prime counting function, here are some more informations). It is thanks to Riemann well known that $\...
1
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1answer
42 views

Generalization of digit-wise divisibility lemmas [duplicate]

Background In undergraduate Abstract Algebra homework, for an integer $n$ with decimal representation $a_m a_{m-1} ... a_1 a_0$, I proved that $3$ divides $n \iff 3$ divides $\sum_{i = 0}^{m} a_i$, ...
3
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1answer
68 views

Confusing thing at the proof of proth theorem

$Proth$ $theorem$ simply depends on a result which proved by pocklington ; The result says : Let $N-1=q^nR$ where $q$ is prime, $n\ge1$ , and $q$ doesn't divide $R$. Assume that there exists an ...
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1answer
46 views

Are Wieferich primes Wieferich numbers?

In the article 'On a conjecture of Crandall concerning the $qx+1$ problem' by Franco and Pomerance, they define Wieferich primes to be prime numbers $p$ for which $p^2|2^{p-1}-1$ and Wieferich numbers ...
1
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1answer
49 views

Is there a set of primes such that the union of their multiples occurs with frequency $1/n$, where $n$ is an integer?

Are there two or more primes, $p_1, p_2, \ldots$ such that the union of their multiples occurs with frequency $1/n$, where $n$ is an integer? If so, can that n be prime? I put it into a computer and ...
16
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1answer
457 views

An interesting algorithm about prime numbers that I thought today

I thought up the following algorithm today: Choose $a_1\in\mathbb{Z}^+\setminus\{1\}$. Then let $a_{n+1}=a_n+p_n$, where $p_n$ is the largest prime factor of $a_n$. The algorithm is easy, but I ...
7
votes
3answers
181 views

Why such an interest for the error term in the Prime Number Theorem

I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough ...
2
votes
1answer
90 views

Error term of the prime number theorem on GRH?

I know that under GRH we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a (q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} (\log X^2)). $$ From this I would like to deuce a abound for $E$ ...
3
votes
2answers
46 views

Limit involving prime and composite numbers

Can someone help me to figure out what $$\lim_{n\to\infty} \frac {c_n}n-\frac{c_n}{p_n}-\frac {c_n}{n^2}$$ is equal to? I am pretty sure its $1$ and i tried many different things but i couldnt figure ...