6
votes
1answer
69 views

Congruences and prime powers

I have just a small question that probably is not hard to answer, but I could not find and elegant solution to this question. Let $p$ and $q$ be prime numbers. $$5^q\equiv 2^q \pmod p$$ $$5^p\equiv ...
0
votes
2answers
16 views

On extracting primes from coprimes

Proof or disprove the following statement - There exists infinitely many $a$ and $b$ which are pair of co-prime integers , either $ab+1$ or $ab-1$ is prime. Motivation- Looking at some twin prime ...
0
votes
0answers
18 views

How to apply the generalised divisibility rule to numbers of the form $10^k+n$

This is kind of a long question but bear with me. There's actually a question mark at the end. I'm trying to apply the generalised (decimal) divisibility rules to numbers of the form $10^k+n$ where ...
3
votes
0answers
71 views

diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then diophantine equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers obviuosly, en, $x^2-py^2=-1$ has no solution in integers. Thanks a lot!
1
vote
1answer
53 views

Coprimality of $2^n + 3^n$ and $5^n + 7^n$

Prove or disprove that for all positive integers $n$, $2^n +3^n$ and $5^n + 7^n$ are always coprime. This is not a homework problem, it is merely a problem I set myself after doing some number theory ...
0
votes
1answer
77 views

Firoozbakht's conjecture solution?

Not so much an question as adding another level to the same question as Ratio of logarithmic primes. (See answers, same as here.) The Firoozbakht's conjecture (1982) is equal to: $$(p_{n+1})^{n} ...
0
votes
1answer
47 views

infinite primes $p\equiv1\pmod n$ without cyclotomic polynomial

Without cyclotomic polynomial, is there an elementary proof of the following: for each integer $n>1$, there are infinitely many primes $p$ such that $p\equiv1\pmod n$ ? please don't refer to ...
0
votes
1answer
49 views

Explain theorem in Number theory

can some one explain with a clear example this theorem for me, Let ($A_1$, $A_2$, $A_3$,..., $A_n$) be integars and $p$ a prime number. if $p|(A_1A_2A_3...A_n)$ then there exist some $1 \leq k \leq ...
2
votes
1answer
33 views

Concerning types of square-free numbers and comparing sizes of their subsets.

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer N > 230 such that the number of ...
1
vote
2answers
42 views

Concerning types of square-free numbers.

Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there ...
2
votes
1answer
24 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
3
votes
2answers
85 views

elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
4
votes
1answer
99 views

Primality of the number of obtained by concatenating the n consecutive digits

Let $f_n$ be the number obtained by concatenating the first $n$ numbers (in base 10). For example $f_1 = 1, f_3 = 123$ and $f_{13} = 12345678910111213.$ Now if $n$ is even or divisible by $5$ then ...
1
vote
1answer
54 views

Is it true that $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$?

Let $p$ be a prime number, are the following statements true? 1.Quadratics of the form $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$. And all such primes ...
2
votes
1answer
52 views

Difference between sum of first n primes and prime(prime(n))

The seq is: -1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, ... http://oeis.org/A239731 Is there's a formula looks like $$a(n) =n^2logn/2$$ for this seq?
1
vote
3answers
73 views

First 10-digit prime in consecutive digits of e

Problem. What is the first 10-digit prime in consecutive digits of e. For those of you who don't know, in 2004 the answer produced a URL to a Google employment page (sort of). I just found about this ...
0
votes
1answer
23 views

Truncatable primes

Why only 11 ? The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. ...
1
vote
2answers
186 views

Distribution of prime numbers. Can one find all prime numbers?

I want to know if it is possible to find a formula that gives all the prime numbers? or can one find the distribution of prime numbers? I know that there is a set of ongoing research on prime ...
4
votes
3answers
65 views

Connections between prime numbers and geometry

This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which ...
1
vote
1answer
28 views

Prime values of binomial

Does there always exist $x$, such that $x>b$ $x>a$ and $a+bx^n$ is prime? Of course, $a$, $b$ are given relatively prime numbers. I know that is true for n=1 in general, and I understand that it ...
0
votes
1answer
25 views

Why $x \le (1+\frac {ln(x)} {ln(2)})^{\pi(x)}$ imply $\pi(x)\ge \frac {ln(x)} {2lnln(x)}$ for $x \ge 8$?

Let $\pi(x) = |\{ p \le x : p \in P\}|$ denote the prime counting function $\pi:\mathbb R \rightarrow \mathbb N$ and $P$ the set of primes. The equality $$x \le \lfloor \prod_{p_i\le x} 1+\frac ...
1
vote
3answers
37 views

Prove that for all $a\in \mathbb{Z}$ and all primes $p$, $p^2$ does not divide $a^2-p$

What would be a method to start, or some can prove useful theorem for this problem Prove that for all $a\in \mathbb{Z}$ and all primes $p$, $p^2$ does not divide $a^2-p$
2
votes
1answer
42 views

How to do partial summation?

I don't understand the following step in a proof: gcd(k,l)=1. Then we have the following formula (p is a prime number): $\sum_{p\leq x, p\equiv l( k)}{\frac{log^2(p)}{p}}\leq ...
0
votes
2answers
139 views

Twin Prime Conjecture's Proof [closed]

I've found this article that claims to have a proof of the Twin Prime Conjecture. Can you find any error? (I have some doubts about the last page of the paper...)
0
votes
1answer
41 views

A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
0
votes
0answers
18 views

Reduced residue systems and prime k-tuple bijection

First off, the terminology: Primorials: the products of the first $n$ primes, written as $P_n \#$. Reduced residue system modulo a positive integer $K$: Those numbers smaller than $K$ that are ...
1
vote
1answer
38 views

What is the reasoning behind ways of splitting up this summation sign?

Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von ...
2
votes
0answers
68 views

On Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...
0
votes
0answers
42 views

Lehmer's Totient Problem

Recently I have been trying to prove the famous Lehmer's Totient Problem by Elementary Methods and surprisingly enough I have found success to some extent. While researching, I have deduced the very ...
-3
votes
1answer
77 views

Infinite and Finite Sets of Primes [closed]

The set of all primes is infinite. Here is the list of some sets of primes with additional structure: Real Eisenstein primes: $3x + 2$ Pythagorean primes: $4x + 1$ Real Gaussian primes: $4x + 3$ ...
0
votes
1answer
94 views

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions in $\mathbb{Z}^+$, if $y\ge 3$.
4
votes
2answers
106 views

What would be the consequences of proving Riemann's hypothesis for Legendre's conjecture?

I've heard somewhere that Riemann's hypothesis doesn't imply Legendre's conjecture. But if Riemann's hypothesis is true, would an interval maybe a bit larger than $[n^2,(n+1)^2]$ contain always at ...
0
votes
1answer
129 views

A question about prime gaps

Recently, I have been reading the Wikipedia article about prime gaps (http://en.wikipedia.org/wiki/Prime_gap) and I came across the following: Hoheisel was the first to show that there exists a ...
0
votes
1answer
152 views

Maximum Number with this condition satisified

Given an array $A$ of $N$ elements $A[1],A[2],A[3]...A[n]$, I need to find maximum element in the array such that $GCD(G,A[i]) > 1$ for given $G$ and $1\leq i\leq N$. Example : Let we have $N=6$ ...
1
vote
1answer
34 views

Finding infinite sequences with pairwise relatively prime outputs.

I am looking for a formula which for every element in $\mathbb{Z}$ as an input, gives pairwise relatively prime outputs. That is for example thanks to Greg Martin's suggestion the positive outputs of ...
0
votes
1answer
78 views

Modular exponentiation with Chinese Remainder Theorem

I'm learning modular exponentiation with Chinese remainder theorem. I found a great answer from below ...
5
votes
1answer
111 views

Does there exist some $k$ such that $2^n+k$ is never prime?

Does there exist some odd positive integer $k$ such that, for all integers $n>0$, $2^n+k$ is never prime? Extension: If yes, for any $a$, does there always exist some $b$ such that $a^n+b$ is ...
4
votes
1answer
125 views

The Sum $\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$

$$\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$$ Does this sum converge or does it diverge? Are there any results related to this? ($\pi(n)$ is the number of primes less than or equal to $n$)
4
votes
2answers
100 views

Primes $p$ such that $p^2$ divides $x^2 + y^2 + 1$

Call a prime $p$ awesome if there exist positive integers $x$ and $y$ such that $p^2$ divides $x^2+y^2+1$. Observation: $2$ is not awesome, because $x^2+y^2+1\not\equiv 0$ (mod $4$). But $3$ is ...
1
vote
0answers
52 views

Primality of $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , when $q$ is prime, $j\ge0$?

Let $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , $q$ prime and $j\ge0$. $P_{2,j}$ is a Fermat number, $P_{q,0}$ is a Mersenne number. Apart from Fermat primes and Mersenne primes, and apart from ...
0
votes
2answers
100 views

${p^km \choose p^k} \equiv m \pmod p$

Let $p$ be a prime number and $m, k$ two positive integers. Then ${p^km \choose p^k} \equiv m \pmod p$. I've been trying to demonstrate this lemme all the day. Have you got any suggestion? Thank you ...
2
votes
1answer
42 views

Second part of Eloi's Conjecture

We know that "There exist some real k such that ∀ integer n>1 the integer part of k∗nln(n) is always prime?" is false (prove here Is there a $k$ for which $k\cdot n\ln n$ takes only prime values? ) ...
7
votes
2answers
136 views

Is there a $k$ for which $k\cdot n\ln n$ takes only prime values?

There exist some real $k$ such that $\forall $ integer $ n > 1$ the integer part of $ k *n\ln(n)$ is always prime?
0
votes
1answer
84 views

a ≡ b (mod n1) and a ≡ b (mod n2), then a ≡ b (mod n)

Verify that if a ≡ b (mod n1) and a ≡ b (mod n2), then a ≡ b (mod n), where the integer n = lcm(n1 , n2). Hence, whenever n1 and n2 are relatively prime, a ≡ b (mod n1*n2). So I know that if a ≡ b ...
4
votes
1answer
59 views

Prime number that are recursively made up of other prime number — what is this called

I've noticed that some prime number are composed entirely of other prime numbers for example -- some have parents on the left hand side (all the numbers below are prime): ...
4
votes
2answers
148 views

Proving the falsity of the Riemann Hypothesis

The Riemann Hypothesis is equivalent to the statement: $$|\pi(x)-{\rm li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657,\text{ (Schoenfeld, 1976)} $$ Which can be visually ...
37
votes
5answers
3k views

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
2
votes
1answer
109 views

What numbers can be expressed with the following expression?

Given $\displaystyle \frac{a^3 - b^3}{c^3 - d^3}$, where a,b,c and d are distinct prime numbers, which integers can be expressed? Somebody asked this elsewhere online and it is beyond my abilities. I ...
2
votes
0answers
53 views

Riemann prime counting function / Log Integral

I include the beginnings of an investigation: $$\text{A plot of R}(x)\text{ against }\pi(x):$$ $$\text{A plot of li}(x)\text{ against }\sum_{n=1}^{x}\frac{\pi(x^{1/n})}{n}:$$ It seems as though ...
1
vote
1answer
50 views

$a$ modulo ${\prod_{i}p_i}$ where $p_i$ are primes.

This may be a very simple question for many of you. But somehow I can not see how to find a good way to answer this. The question is that if it is given that $$a\equiv k_i\mod{p_i},\quad ...