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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Every prime number divide some sum of the first $k$ primes.

Let $S_n=\Sigma^n_{k=1}p_k$, where $p_k$ is the $k$-th prime number. Conjecture: $$\forall p\in\mathbb P\exists n\in\mathbb N: p|S_n$$ Verified for the $1000$ first primes. Is there a proof for ...
3
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1answer
101 views

How to prove that $\lim_{x \to \infty}\frac{\pi\left(\frac{4x}{3}\right)}{\frac{x}{3\ln x}}=4$?

I'm a member of a Facebook-based mathematics group. Recently, one of the members made a post detailing an observation he made in his free time, namely that $\pi(4x/3)-\pi(x)$ (here, $\pi(x)$ denotes ...
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Can we conclude $p_n < n^2$ from $\zeta(2)$ and Euler's prime theorem?

We know that $$\sum_{n=1}^{\infty}\cfrac{1}{p_n}$$ diverges. And we know too that $$\sum_{n=1}^{\infty}\cfrac{1}{n^2}$$ converges (to $\frac{\pi^2}{6}$). That means that $$\sum_{n=1}^{\infty}\cfrac{1}{...
3
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1answer
58 views

Smallest integer greater than or equal to another integer but with prime factors less than or equal to 7

More clearly stated than title, let $n \in \mathbb{N}$ and $A = \{m \in \mathbb{N} : m \geq n \text{ and } m = 2^a 3^b 5^c 7^d \text{ for some } a,b,c,d \in \mathbb{N}\}$. Find $\min(A)$. This can be ...
3
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2answers
894 views

How many 8 digit palindromes are prime?

Find the number of primes that are 8 digit palindromes. I got this question in a entrance paper. The only thing I know is the definition of a palindrome. Also, is there any method/formula to count or ...
3
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1answer
39 views

Is there a “nice” formula for the product $\prod_{n\neq i}(x-z_n)$ which contains all roots of unity except exactly one?

Let $Q_p(x)=x^p-1$, $p$ is an odd prime. I was wondering if there is some nice formula for the product $\prod_{n\neq i}(x-z_n)$ which contains all except one (let's say $z_i$) $p$-th roots of unity. ...
5
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1answer
72 views

Find Sylow $p$-subgroup within subgroup

Let $p$ be a prime number, $G$ a group with subgroup $H$ and $S$ a Sylow $p$-subgroup of $G$. Show that there exists $g\in G$ such that $H\cap gSg^{-1}$ is a Sylow $p$-subgroup of $H$. Moreover, come ...
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1answer
35 views

the only solution of prime $q|(p^2-1)$ with $p<q$ is $p=2$ and $q=3$. [duplicate]

I am trying to prove a lemma that the only solution of $q|(p^2-1)$ with $p<q$ is $p=2$ and $q=3$. i.e if $p<q$ are primes and $q|(p^2-1)$, then $p=2$ and $q=3$. Since $q|(p^2-1)$, then $q|(p+1)$ ...
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1answer
142 views

Error Prime Prediction With Prime Triangles (Q: growth and symmetry).

Error in Prime Prediction. A method is given to estimate the position of the next prime based on the previous two primes. The error in the estimation is determined. I would like to now if this error ...
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3answers
42 views

Primality of $2^n - 1$ [duplicate]

Is it true that $2^n - 1$ is a prime if and only if n is a prime?
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45 views

Sieve for primes

In the sieve of Eratosthenes the erased composite numbers are identified by $ 5 $ equations: $nc=2(a+1)$ , $nc=3(2a+1)$ , $nc=−1+6(6ab+a−b)$ , $nc=1+6(6ab+a+b)$ , $nc=1+6(6ab−a−b)$ with $ a\geq 1 $ ...
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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
101 views

Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$

$\textbf{Question:}$Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$ I could easily see that the given is equivalent to showing that there are infinitely many primes $p$ ...
4
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3answers
933 views

Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$

These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1: Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=...
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1answer
50 views

Prime numbers in an arithmetic progression

Let $k>3$ be an integer. Show that it's not possible fir k prime numbers each greater than $k$ to be in arithmetic progression with a common difference less than or equal to $k+1$. The process I'm ...
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35 views

Does a basis $b$ exist such that only a finite number of palindromes in base $b$ are primes?

Can we prove that a base $b$ must necessarily exist so that only a finite number of palindromes in base $b$ are primes? My friend came up with this, I hope it isn't one of those ubiquitous "innocent" ...
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1answer
17 views

A question on the Dedekind Psi function

This question is inspired by this question https://mathoverflow.net/questions/370921/additive-number-theory-hilbert-spaces-and-polynomial-rings by changing the sum of divisors function $\sigma$ with ...
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Attempting to prove the claim: “Every prime greater than $3$ can be written in the form $6n + 1$ or $6n + 5$” by induction.

Claim: Every prime greater than $3$ can be written in the form $6n + 1$ or $6n + 5$ for some $n\in \mathbb Z^+$. Proof (my attempt): Base case: $n=0$. $6n + 5 = 6*0 + 5 = 5$, which is prime. ...
3
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1answer
135 views

Palindromes with distinct palindromic prime factors

Imagine the sequence of palindromic numbers where each term is defined as the smallest square-free palindromic number with no other prime factors but the n distinct palindromic prime factors. The ...
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90 views

Conjecture or theorem on addition of prime numbers and a constant which sum is another prime

I am looking for theorems or conjectures that may exist regarding adding prime numbers. In particular, about "adding two or more primes, plus a fixed value, that results in a prime number". ...
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174 views

Is every natural number the truncation of a prime?

E.g. $30$ can be "extended" to a prime, namely 30$19$, and $575$ can be extended to a prime, namely 575$4853343$. Is this true for every natural number? To make things precise: Let $n\in\...
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524 views

Origins of the conjecture on the existence of infinitely many palindromic primes

A palindromic prime with respect to a base $b \geq 2$ is a prime number such that, when you reverse its sequence of digits in base $b$, you get the same prime. For example in base $10$, the prime $...
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1answer
145 views

How does GIMPS work and what are these iterations?

I downloaded GIMPS today just out of curiosity and have been running it. On my machine it is checking $M_{52898149}=2^{52898149}-1$. From what I could find on Wikipedia I suppose that GIMPS uses ...
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133 views

$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\ln(y)$ for every integers $x,y > 1$. I read that experts ...
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2answers
88 views

If a prime natural number $p\neq 3$ divides $a^3-3a+1$ for some integer $a$, then $p\equiv \pm1\pmod{9}$.

$\textbf{Problem:}$ Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is an integer. I tried to complete ...
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4answers
83 views

If $5|7b^2$, can I deduce that $5|b$? [duplicate]

I've encountered an example in the proof of irrationality of square root of $35$ where I arrived to a situation where I got $5|7b^2$. Is it possible to deduce that if $5|7b^2$, then $5|b$?. If yes, ...
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0answers
52 views

Prime-aligned spiral

A while ago, I came up with an algorithm that I called the arbitrary spiral. It lets me align whatever integers I'd like to lay on the positive $x$-axis in a spiral. This is what I got when I had it ...
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1answer
212 views

How many massively palindromic primes exist?

A massively palindromic prime is when for in all bases (in which it has more than $2$ digits) it is a palindrome: Edit : My example was broken because of a silly miscalculation, I haven't even found ...
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25 views

Zeta variant $ \zeta_m(s) = \prod_{i = 1}^{\infty} \frac{p_i^s}{p_i^s - 1} $ has only 2 simple poles?

Is there an infinite subset of primes $p_i$ known such that $$ \zeta_m(s) = \prod_{i = 1}^{\infty} \frac{p_i^s}{p_i^s - 1} $$ has an analytic continuation to $Re(s)> -2$ and has simple poles only ...
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26 views

The amount of invertible matrices and sequences with given properties in field

Let $p$ be a prime number, and $F_p$ be a field which consists of $p$ elements. With this knowledge I have to show that $k$-dimensional linear space upon the field $F_p$ has $p^k$ elements. Evaluate ...
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2answers
66 views

Proof that composite numbers have unique prime factors [duplicate]

I was trying to prove this little fact that all composite numbers have unique prime factors, Let $P$ be a composite number and $$P=a\times b\times c\times d\cdots$$ And let us assume that it dosen't ...
5
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1answer
133 views

What percentage of positive integers, written in base 10, are composite regardless of what base they are interpreted in?

There is a sequence of numbers (OEIS A121719) with the following defenition: If the string of base-$10$ digits corresponding to the positive integer $k$ is composite when interpreted in any possible ...
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1answer
86 views

Is the Euler prime of an odd perfect number a palindrome (in base $10$), or otherwise?

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$). (That is, $2N=\sigma(N)$ where $\sigma$ is the ...
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1answer
84 views

Prove that $n$ and $n + k$ are both primes if and only if $(k!)^2[(n - 1)! + 1] + n(k! - 1)(k - 1)! \equiv 0 \mod n(n + k)$.

Prove that positive integers $n$ and $n + k$, where $n > k$ and $2 \mid k$, are both primes iff $$(k!)^2[(n - 1)! + 1] + n(k! - 1)(k - 1)! \equiv 0 \mod n(n + k)$$ According to Wilson's theorem, ...
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1answer
46 views

If $\sigma : H \xrightarrow{\sim} G $ is a group isomorphism and $H = \langle S \rangle$, then does $G = \langle \sigma(S)\rangle$?

Let $\sigma : \Bbb{P} \xrightarrow{\sim} \Bbb{P}$ be a permutation of the primes $\Bbb{P} = \{ 2,3,5,7,11, \dots \}$. Then $\sigma$ extends uniquely to a surjective group hom $\Bbb{Q}^{\times} \to \...
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0answers
155 views

Least prime that removing the first and last digit in base $2$ for $n$ iterations is prime each time.

Let us consider a sequence, $a_n$, which represents the smallest prime number such that removing its first and last digits in base $2$ (including leading zeros) for $n$ iterations will stay prime for ...
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0answers
41 views

Palindromic divisibility and primes

Inspired by this question: Define odd-palindroming $X$ in base $b$, or $OP(X,b)$: Take an integer $X$ and write it in base $b$. Then, reverse its digits and concatenate the reversed digits to the end ...
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Pairs of palindromic primes without $1$ and have a palindromic product

While discussing about prime numbers with other users, I noticed that: $(1)$ There are very few pairs of palindromic prime numbers that do not contain the digit $1$ and that have products which are ...
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96 views

Are 2, 3 the only prime numbers that don't have the digit 1 and are palindromes whose squares are also palindromes?

While thinking about prime numbers, I noticed that: $(1)$ Very few prime numbers have squares that are palindromes. Ex: $2$, $3$, $11$, $101$, $307$ $(2)$ Even rarer are prime numbers that are ...
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1answer
58 views

Gaps between the primes [duplicate]

I had this question at the start of my number theory class so I think it is supposed to be an easy one but I did not receive a solution. Here is the problem: Let $n\in \mathbb{N}$ be arbitary. Prove ...
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879 views

Is either $n! + 1$ or $n! - 1$ not prime for all $n$?

I was looking at an article about factorial primes, and I noticed that both $n!+1$ and $n!-1$ were not prime. (As in, there are no numbers $n$ such that both $n!+1$ and $n!-1$ are prime). I think that ...
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1answer
30 views

Probabilistic argument why the numbers around a factorial cannot both be prime

This recent question asks whether it is possible for $n!-1, n!+1$ to both be prime when $n > 3$. According to the answers, this is an open problem. I am trying to figure out how you would justify ...
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2answers
100 views

Proving that a Gaussian integer $a+bi$ with $a,b\neq0$ is composite in $\mathbb{Z}[i]$ if its norm $a^2+b^2$ is composite in $\mathbb{N}$?

It is simple to show for any given composite sum of two squares, that there is at least one Gaussian composite with that norm, but more difficult to show that all Gaussian integers with that norm are ...
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2answers
107 views

If $f$ and $g$ are nonzero polynomials with $\deg f>\deg g$, and if $pf+g$ has a rational root for infinitely-many primes $p$, then …

An IMO shortlist polynomial problem: Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has ...
3
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1answer
60 views

Primality test for specific class of $N=k \cdot 2^n+1$

Can you prove or disprove the following claim: Let $N=k \cdot 2^n+1$ be a natural number that is not a perfect square such that $ 2 \nmid k$ , $n>2$ . Let $c$ be the smallest odd prime number such ...
12
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3answers
964 views

Where can I find the modern proof of the prime number theorem?

Terence Tao described a modern proof of the prime number theorem in a lecture in UCLA, which is stated in wiki(enter link description here). From wiki: In a lecture on prime numbers for a general ...
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1answer
75 views

Verify my proof that for any $n>1$, if $n^n+1$ is prime, then $n=2^{2^k}$ for some integer $k$.

I am solving a problem and I am respectfully asking someone to critique my work and offer suggestions on formatting or point out any glaring logical errors. Here is the problem: Prove that for any $n&...
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0answers
168 views

Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?

In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
3
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1answer
1k views

Super palindromes

Can anybody be kind enough to explain what exactly is a super palindrome? Also consider the following example : $923456781-123456789=799999992:9=88888888$ The largest prime factor of $88888888$ is ...
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1answer
90 views

Is there another palindrome-Carmichael-number?

$$101101$$ is a cute number indeed. It is the smallest palindrome-Carmichael-number. Furthermore, its square and its cube are also palindrome! And it is a "binary" number containing only the digits $0$...