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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Proof for relation between divisor functions

For start lets define $\sigma(d)$: the sum of divisors of d, with $d \in \mathbb{Z}$ $\tau(d)$: the number of divisors of d, with $d \in \mathbb{Z}$ I want to prove that, $\sum_{d|n} \frac{n}{d}\sigma(...
DontWorry's user avatar
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Show that if $n$ is a Fermat pseudoprime to the base $a$, then $n$ and $a$ are relatively prime.

I am trying to prove the following theorem: If $n$ is a Fermat Pseudoprime to the base $a$ then $n$ and $a$ are relatively prime. As a recap a Fermat Pseudoprime $n$ has the property that for $n$ not ...
eeqesri's user avatar
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-3 votes
1 answer
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How to find the digits a, b and c?

I am trying to solve the following mathematical expression: $\frac{1}{a+b+c}=0,abc$ Where a, b and c represent digits. My goal is to find the values ​​of the digits a, b and c. I have tried to ...
golub001's user avatar
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Using FTA to prove exponential relation between integers [duplicate]

Question: Show that if $x$ and $y$ are non-zero integers and $x^2 = y^3$ then $x=a^3$ and $y=b^2$ for some integers $a$ and $b$ My attempt: If $y^3 = x^2$ then this means $y$ is a natural number, ...
altayir1's user avatar
4 votes
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68 views

Is it known whether or not OEIS sequence A091308 is finite?

OEIS sequence A038395 is defined as the concatenation of the first $n$ odd numbers in reverse order: $1,$ $31,$ $531,$ $7531,$ $97531,$ $1197531,$ etc., and A091308 is the sequence of primes in ...
Ted Hopp's user avatar
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For which n is p(n)# + 1 prime? [duplicate]

The primorial $\color{blue}{p_{11}\text{#} = 200560490130}$, and $\color{blue}{p_{11}\text{#} + 1 = 200560490131}$ is prime. There are other times when $\color{blue}{p_{n}\text{#} + 1}$ is not prime; ...
user3134725's user avatar
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2 answers
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Proof that the $(n+1)$-th prime is less than or equal to the $n$-th primorial.

Assuming that $p_{n}$ is the $n^{ th}$ prime and $p_{n}\text{#}$ is the $n^{ th}$ primorial, what is a proof in elementary number theory that, for all $n \ge 2$, $p_{n+1} \le p_{n}\text{#}$ ?
user3134725's user avatar
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Seeking Efficient Prime Identification Methods for Constrained Numerical Ranges [closed]

With precision in my inquiry, I am interested to learn whether there is any mathematical framework, method, or algorithm capable of efficiently identifying all prime numbers around a specific ...
Dood's user avatar
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1 vote
1 answer
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a small doubt in the proof of the quantitative form of the prime number theorem

I have been studying the proof of the prime number theorem in the quantitative form as in Theorem 6.9 of Montgomery & Vaughan's book "Multiplicative Number Theory, which focuses on proving ...
Josh's user avatar
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upper- and lower bounds of primes in intervals

Let's define two intervals $A =\{1,\dots,\frac{1}{4}N\}$ and $B =\{\frac{3}{4}N,\dots,N\}$. Let $\pi(A)$ be the counting function which counts the prime numbers in interval $A$. And similar, $\pi(B)$ ...
Pim Dumans's user avatar
3 votes
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27 views

Does a set with no divisibility pairs necessarily have arbitrarily large gaps? [duplicate]

The set of prime numbers has the following properties: No element is divisible by any other element. We can find arbitrarily large gaps between consecutive elements. Does (1) imply (2) for arbitrary ...
Karl's user avatar
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Comparing the density of primes in intervals

Let $\pi(N)$ be the counting function of the primes equal to or smaller than $N$. Three statements (conjectures): For any $N \geq T_1$ holds $\quad E_1 = \Big(\pi(N) - \pi(\frac{2}{3}N)\Big) - \frac{...
Pim Dumans's user avatar
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Prime counting in some specific interval

I am interested in the ratio (R) of the number of prime numbers in certain intervals. Let N be a sufficient large positive integer. Let's define two intervals $$I_a = \{\frac{1}{3}N,...,\frac{1}{2}N\}$...
Pim Dumans's user avatar
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1 answer
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If $\varphi(n)=2p$ then $p$ is a Sophie Germain Prime [closed]

Define $n,p\in\mathbb{N}$ with p prime. I'm struggling to show that if $\varphi(n)=2p$, then $2p+1$ is prime.
Donald fischer's user avatar
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Consecutive numbers

Are there ever more consecutive composite numbers than there are primes up to that point? I imagine not, because the primes are the ones which cancel out multiples, so will inevitably have to fill in ...
Talon Eaglefeathers's user avatar
5 votes
0 answers
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Is this a legitimate method of finding another prime number? [duplicate]

My motivation for asking this question stems from Euclid's elegantly simple proof of the infinitude of prime numbers. I am not suggesting an alternative proof, since my method, even if is valid, ...
user1153980's user avatar
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6 votes
3 answers
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Prime powers; are there more powers of 2, or powers of higher primes, smaller than N?

Given the set https://oeis.org/A025475 with the prime powers, excluding the primes themselves. Let's define two subsets of A025475. $A=\{4,8,16,32,\dots\}$, with all powers of 2 ($2^k, k>1$) ...
Pim Dumans's user avatar
3 votes
1 answer
108 views

Diophantine equation involving prime number

Given prime number $p$ and $n \in\mathbb{N}^*$ such that $n>p$, such that:$$(3\sqrt{p+n}+3p^2-n)(\frac{3n+1}{p}+\frac{1}{n})=3p(3n+1)+8(\frac{n}{p}+1)$$ identify the numbers $n$ and $p$. Here's how ...
fikooo's user avatar
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Has it been proven that $x^6+y^6≠z^6+ w^6$, when all variables are distinct primes?

I was told that R. Balasubramanian and N. D. Hegde in 1986 proved $x^6+ y^6≠z^6 + w^6$ for distinct primes, but I couldn't find any literature. Before I continue, I will say that all variables are ...
The T's user avatar
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Splitting the set of primes dividing an integer polynomial

Let $f \in \mathbf{Z}[x]$ and let $\mathbb{P}$ denote the set of positive integer primes. Let $\mathcal{P}(f)$ denote the set of primes $p$ which solve $f(n) \equiv 0 \text{ (mod } p)$ for some $n \in ...
Antosha's user avatar
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Is there connection between digits of nth prime and natural logarithm? [closed]

I started with a small list of primes less than 100 and checked counted its digits then I wonder what will happen for larger values I made a plot for values up to 500,000 for this range the closest ...
d4shm1r's user avatar
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Verify a result from the paper "The Periodic Table of Primes"

Someone sent me this link to the paper "The Periodic Table of Primes". I decided to see if I can verify one of their examples, so chose Example 8. I used Mathematica with the command: ...
Moo's user avatar
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Relative distribution of primes among consecutive squares

For each prime $p,$ define $n_p>0; {n_p}^2$ is the greatest square integer $\leq p,$ so that ${n_p}^2 \leq p < (n_p + 1)^2.$ [Now note that $(n_p + 1)^2-1-{n_p}^2 = 2n_p $]. Is $S:= \left\{ \...
Adam Rubinson's user avatar
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0 answers
361 views

Consequence (if true) of $\gcd(n!+1,n^2+1)=1$ for $n>1$ an integer . The Buniakowsky's conjecture.

It's a follow up of Do we have $\sum_{n=1}^{\infty}\frac{\gcd\left(1+n!,1+n^{2}\right)}{n!}\stackrel{?}{=}e$? Context/ I want to find a consequence of this conjecture which could resists perhaps for ...
Miss and Mister cassoulet char's user avatar
4 votes
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How do we measure the "holes" (zeros) of the set of $\Bbb{Z}$-linear combinations of $f_i : G \to G$ where $G$ is an abelian group?

Question: Let $G$ be a normed abelian group. Namely the triangle inequality holds $|g + h|\leq |g| + |h|$. An example would be $G = \Bbb{Z}$ together with $|\cdot| =$ absolute value. Now suppose ...
Daniel Donnelly's user avatar
2 votes
1 answer
255 views

How to prove no solution exists for this equation?

How can I prove that there are no positive integer solutions $(x, y)$ and prime $p$ satisfying the equation $(x+y)(x^2+y^2) = p(p + 2xy)$? I made a c++ program to find out that there are no solutions ...
Abdelrahman Yousf's user avatar
5 votes
0 answers
125 views

How do you find constants in an k-Tuple conjectures?

By introducing modular objects associated to the sequel of rings $$(Z/2,Z/6,Z/30,Z/210,Z/2310,..., Z/p_n\#Z)$$ a sequence of coefficients is updated$$(2;\color{green}{\frac83}; 3.2;...$$ (see my ...
Stéphane Jaouen's user avatar
3 votes
0 answers
139 views

A sequence about sums of prime numbers.

I was playing with primes lately and thought about terms of a possible sequence. Let's define $p_n$ as the $n$th prime number from the smallest prime number. A sum of prime numbers from 1st to the $n$...
mokrodo923's user avatar
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0 answers
62 views

Highly composite numbers which are the middle of a twin prime

From the first $10\ 000$ highly composite numbers listed in OEIS , the following $20$ are the middle of a twin-prime that is we have a highly composite number $N$ such that both $N-1$ and $N+1$ are ...
Peter's user avatar
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-2 votes
0 answers
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If p prime and p∣ab , then p∣a or p∣b [duplicate]

Proposition: if a and b are integers, p is prime and p|ab then p|a or p|b I'm getting started on math proofs, I want to know how can I write a simple valid proof of this without using the Bézout's ...
Peco's user avatar
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3 votes
2 answers
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Why is the difference of consecutive primes from Fibonacci sequence divisible by $4$?

The primes represented in the Fibonacci sequence are written in the form $6n + 1$ and $6n -1$, respectively. $$5=6\times1-1$$ $$13=6\times2+1$$ $$89=6\times15-1$$ $$233=6\times39-1$$ $$1597=6\times266+...
Polona Čuk Kozoderc's user avatar
4 votes
0 answers
53 views

Is it possible to construct a sequence using the first $n$ prime numbers such that each segment has a unique sum?

For example, consider the sequence $2,7,3,5$. The sums of the segments of this sequence are as follows, and they are all unique: $$2, 2+7, 2+7+3, 2+7+3+5, 7, 7+3, 7+3+5, 3, 3+5, 5$$ Can we generate ...
dodicta's user avatar
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3 votes
1 answer
58 views

Finding all prime powers of the form $\frac{n(n-1)}{10}$

I was trying to come up with a problem where the solutions would be the numbers 5, 6, 10 and 11. It seems like "finding all integers n, where $\frac{n(n-1)}{10}$ is a prime power", is a good ...
Sebi19's user avatar
  • 31
0 votes
0 answers
67 views

The Consecutive Composite Conjecture (Proof required)

I have been working on a conjecture the past couple days and would like assistance in determining whether or not it is true or not. The question is as follows. Given a distinct set of ascending prime ...
Michael Franklin's user avatar
1 vote
1 answer
37 views

Estimating the upper bound for $\prod\limits_{p \le x}{p^{\frac{1}{p}}}$

An upper bound for the primorial can be found based on the first chebyshev function. From $\vartheta(x) < 1.00028x$, it is clear that: $$\prod\limits_{p \le x}p \le e^{1.00028x} < (2.72)^x$$ I ...
Larry Freeman's user avatar
3 votes
0 answers
72 views

Magical trigonometric inequality (feat. number theory)

A couple of days ago I found an interesting problem on the Art of Problem Solving website. It says: For odd and coprime positive integers $p$ and $q$, the following inequality holds $$ \sum_{m=1}^{p} ...
Terry's user avatar
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3 votes
0 answers
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Does euclid's proof of infinitude of primes generate all the primes if you keep adding the new primes to the list?

If I start with a set of consecutive primes, let's say $\{2,3,5,7\}$. Consider $2\times 3\times5\times7+ 1 = 211$. Now, $211$ is prime, so I add it to the set: $\{2,3,5,7,211\}$. If the resulting ...
Tora's user avatar
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1 vote
0 answers
42 views

How to correct the error between $\log(x!)\approx x\sum_{n\leq x}\delta(n)\frac{\log(n)}{n}$, where $\delta(n)$ is the density of primes near x?

Well assuming that the Prime Number Theorem is true, when substituting $\delta(x)$, the density of primes near $x$—which I am being vague of what it means 'cause I don't have enough foundation about ...
Mina Basilious's user avatar
3 votes
0 answers
68 views

Can we show that $\frac{\sum_{j=1}^n j^2\cdot j!}{99}$ generates only finite many primes?

Define $$f(n):=\frac{\sum_{j=1}^n j^2\cdot j!}{99}$$ Is $f(n)$ prime for only finite many positive integers $n\ge 10$ ? Approach : If we find a prime number $q>11$ with $q\mid f(q-1)$ , then for ...
Peter's user avatar
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4 votes
0 answers
173 views

Is $N=3\cdot 2^k+1$ prime if and only if $2^{N-1}\equiv 1 \pmod N$?

Is the following statement true? Let $k\geq 1$ be an integer and $N=3\cdot 2^k+1$. Then $N$ is a prime prime if and only if $2^{N-1}\equiv 1 \pmod {N}$ One implication is simply a Fermat's Little ...
Sil's user avatar
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7 votes
3 answers
234 views

An integer and its inverse modulo prime, both less than half of the prime

Question: A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $p/2$ such that $\frac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are ...
D S's user avatar
  • 5,025
5 votes
1 answer
271 views

Find prime numbers satisfying an equation

Find all triplets $(m, n, p)$, where $p$ is a prime number and $m, n ∈ \Bbb N$, such that $p=\frac{m}{4}\sqrt{{2n-m \over 2n+m}}$ My procedure is as follows: $p=\frac{m}{4}{\sqrt{(2n)^2-m^2}\over 2n+m}...
Pranav P J's user avatar
5 votes
2 answers
408 views

A prime is not a product of primes, so why is “every positive integer, except 1, […] a product of primes”?

I'm trying to understand why the fundamental theorem of arithmetic is phrased like this: Every positive integer, except 1, is a product of primes. A prime number is an integer but it is not a ...
zeynel's user avatar
  • 375
2 votes
2 answers
86 views

Find the number of functions $f : \{ 1,2,...,n \} \rightarrow \{p_1,p_2,p_3 \} $ for which the number $f(1)f(2)...f(n)$ is a perfect square.

Let $p_1,p_2,p_3$ be distinct prime numbers and consider $n \in \mathbb N$ . Find the number of functions $f : \{ 1,2,...,n \} \rightarrow \{p_1,p_2,p_3 \} $ for which the number $f(1)f(2)...f(n)$ is ...
Unknowduck's user avatar
-2 votes
1 answer
63 views

an arithmetic sequence contains infinite primes [duplicate]

Let $\{u_n\}_{n\geq 0}$ be an arithmetic sequence defined by $u_n=an+b \quad (a,b\in\mathbb{N}^*)$. a) Let $a=6$. Give 2 values of $b\leq 6$ s.t for each such value of $b$, $\{u_n\}$ has infinite ...
Alex Nguyen's user avatar
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0 votes
1 answer
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Proof of identity on sum of powers of primitive root.

Let $q = p^e$ for some prime $p$. Consider the trace function $\mathrm{Tr}_{\mathbb{F}_q/\mathbb{F}_p}:\mathbb{F}_q\to \mathbb{F}_p$ defined by $\mathrm{Tr}_{\mathbb{F}_q/\mathbb{F}_p}(x) = \sum_{i=0}^...
PTrivedi's user avatar
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0 votes
0 answers
36 views

Stationary sequence of orders in $(\mathbb{Z}/p^n\mathbb{Z})^\ast$

I am struggling to prove the following statement: Let $p, q$ be coprime numbers. For all $n \in \mathbb{N}^*$, we define $t_n$ as the order of $[q]$ in $(\mathbb{Z}/{p^n}\mathbb{Z})^\ast$. Show that $(...
Arthur Filippi's user avatar
2 votes
0 answers
96 views

Is another base-$2024$-Wieferich prime known?

Wieferich-prime question about the current year : I found only $4$ base-$2024$-Wieferich primes , in other words prime numbers $p$ satisfying $$2024^{p-1}\equiv 1\mod p^2$$ ...
Peter's user avatar
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3 votes
1 answer
86 views

Can you determine $n$ with oracle for primality of $n+m$ for given $m$?

I haven’t studied much number theory at all, but I got asked this question by a family member who is a fan of recreational math, and I have no clue how to answer it, or even the subfield it precisely ...
Joel Newman's user avatar
1 vote
0 answers
103 views

Can any of you do something relevant with this mathematical property I found? [closed]

I'm an amateur mathematician and I found a property that I've never seen anyone mention before, I think I managed to demonstrate it below. I confess that I don't know if it's new or not, as I haven't ...
Edu's user avatar
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