Prime numbers are natural numbers not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

learn more… | top users | synonyms

2
votes
3answers
81 views

Why do we call primes, and not the number one, *the atoms of numbers*?

The fundamental theorem of arithmetic asserts that we can produce every composite number from a unique set of prime multiplicands, so long as none of those primes equals one. Consequently, some ...
3
votes
0answers
60 views

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

Prime $p\equiv3\pmod4$, then the 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!
6
votes
1answer
39 views

Order of groups and group elements? [duplicate]

Let G be a group and let p be a prime. Let g and h be elements of G with order p. I am wondering how I can use group theory to find the possible orders of the intersection between ...
0
votes
2answers
26 views

Composite Numbers with 1 Prime

What is the method for finding a long sequence of consecutive composite numbers that has only 1 prime? Specifically, how to find 2011 consecutive natural numbers, 1 of which is prime.
1
vote
1answer
20 views

Is a Mersenne prime always of the form $4n + 3$?

Is a Mersenne prime always of the form $4n + 3$? Examples: $M_3 = 7 = 4 \times 1 + 3$ $M_5 = 31 = 4 \times 7 + 3$ $M_7 = 127 = 4 \times 31 + 3$ $M_{13} = 8191 = 4 \times 2047 + 3$ ...
1
vote
3answers
46 views

Is a Mersenne-Prime always of the form $3n + 1$?

Examples are: $M_3 = 7 = 3\times 2 + 1$ $M_5 = 31 = 3\times 10 + 1$ $M_7 = 127 = 3\times 42 + 1$ $M_{13} = 8191 = 3\times2730 + 1$
1
vote
1answer
14 views

Prime Counting: Relationship between Chebyshev's function and the Prime counting function

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting ...
1
vote
1answer
87 views

Is $(1+2+3+…)=(1+2+2^2+2^3+…)(1+3+3^2+…)(1+5+5^2+…)…$?

Are these equal? $$(1+2+3+…)=(1+2+2^2+…)(1+3+3^2+…)(1+5+5^2+…)…$$ Where the RHS has a series for each prime. Looks like they are the same series by the fundamental theorem of arithmetic. Every number ...
2
votes
1answer
13 views

Using primes to create unique character mappings for scrambled substring searching

Problem: given a string needle, and a string haystack determine if there is there an anagram of needle present as a substring of haystack? (Assume case doesn't matter). One solution is to map the ...
1
vote
1answer
40 views

Is there any prize for proving conjecture on Fermat's prime ?-+

I know this site is for mathematical questions and answer places, but I need a little help from you in some other aspect. I have searched in google but didn't get any satisfactory answer for it. This ...
2
votes
1answer
42 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
0
votes
2answers
15 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 ...
1
vote
3answers
24 views

Factor factorials

How would you find the greatest prime factor of a factorial? For instance, 82! The 2 and 41 that are yielded when you prime-factor 82 seem to have no correlation to the prime factorization of 82!
1
vote
1answer
122 views

Consider the number $n= 2^{10^{33}} +1$ [on hold]

Consider the number $$n= 2^{10^{33}}+1$$ Suppose that it is known that none of the numbers $1 < k < 10^{6}$ divide $n$. Does it follow that n is a prime number? I know that the answer is a ...
2
votes
1answer
27 views

ON types of squarefree numbers and comparing their amounts < a given integer N.

Let an m-prime be a square-free number with m prime divisors. Also let the number of t-primes < N be represented as #(t-prime){N} (t and N being positive elements of integers). Is the following ...
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 ...
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 ...
1
vote
0answers
33 views

About primes and counting them. [on hold]

Are there bounds to the prime counting function that do not involve logarithms? Considering the best bounds use logarithms why is the natural logarithm so closely related to the prime counting ...
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} ...
1
vote
1answer
78 views

Why there are no other known Fermat primes.

Fermat primes are prime numbers of the form $2^{2^n} + 1$: $$3,~5,~17,~257,~65537$$ There are no other known Fermat primes. But why?
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 ...
12
votes
1answer
345 views

Prove that $\sum\limits_{i=0}^{k} p^{2i}$ ($p$ is prime) is never a perfect square

Prove that $$ \sum_{i=0}^{k} p^{2i} $$ where $k > 0$ and $p$ is an arbitrary prime, is never a perfect square. I think you can prove it by letting $q = \sum\limits_{i=0}^k a_ip^i$, then expanding ...
0
votes
1answer
41 views

How does this algorithm find the largest prime factor?

This question on math.stackexchange details an algorithm that can be used to find the largest prime factor of a number. I used it to solve Project Euler #3. Here's a short description of the ...
0
votes
3answers
79 views

A question on consecutive prime numbers

Prime numbers: 2 3 5 7 11 13 17 19 23 29 .... Difference between to consecutive primes: 1 2 2 4 2 4 2 4 6 .... We know that there are infinite prime numbers. This is Ok. But does the difference ...
14
votes
2answers
176 views

Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes

Starting from 2, the sequence of sums of all consecutive primes is: $$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
0
votes
1answer
60 views

Riemann Zeta circularity?

In this post I show: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Wolfram Alpha shows an alternate form for the primes: $$\frac{p_n{}^2}{p_n{}^2-1}=\frac{\left(\sum _{k=1}^{2^n} ...
0
votes
0answers
24 views

A prime connection between two numbers with same prefix

If I know that the number n is prime, is there a fast algorithm to check if 10*n+k is prime, where k is 1, 3, 7 or 9? I mean, an algorithm based on the fact that n is prime. Thanks for help! P.S. : ...
0
votes
2answers
46 views

Is the following statement true

Is the following statement true and how to prove it? \begin{align} (a^2)^{3N} \equiv a^2 \mod{p} \end{align}
1
vote
4answers
38 views

solutions to linear equations involving prime numbers?

Suppose we have the two equations: $2Z - p = Xq$ $2Z - q = Yp$ where $X,Y,Z \in \mathbb{N} $ and $p,q \in \mathbb{P} - \left\{2\right\} $ Are there any solutions where $Z$ isn't prime?
0
votes
2answers
197 views

find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
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
22 views

Dirichlet prime counting function?

Let $a$ and $b$ be coprime (i.e. $a \perp b$). Let $f(a,b,x)$ denotes the number of the primes such that $p=ak+b$ and not greater than $x$. For example $f(4,1,10)= 1$. Is there an asymptotic formula ...
2
votes
1answer
32 views

Is there a asymptotic formula for product of primes? [duplicate]

$$P(x)=\prod_{p\leq x}p$$ As you can see P(x) represents the product of primes which are not greater than x. Is there a asymptotic formula for this?
0
votes
0answers
257 views

New method derived out of Fermat's factorization method

Let us take two examples: a) $N=943=41*23=(\frac{41+23}{2})^2-(\frac{41-23}{2})^2$ but if we take $B=\frac{N+1}{4}$ then we can represent it as $B={x}^2-({y}^2+y)$ and in our case: ...
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 ...
3
votes
2answers
66 views

Ratio of logarithmic primes

Any help is appreciated in proving/disproving the following inequality $$ \frac{\ln{p_{n+1}}}{\ln{p_{n}}} < \frac{n+1}{n} $$
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
213 views

is it possible to get the Riemann zeros

since we know that the number of Riemann zeros on the interval $ (0,E) $ is given by $ N(E) = \frac{1}{\pi}\operatorname{Arg}\xi(1/2+iE) $ is then possible to get the inverse function $ N(E)^{-1}$ ...
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 ...
3
votes
2answers
84 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 ...
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 ...
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 ...
0
votes
2answers
35 views
1
vote
0answers
13 views

Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
1
vote
3answers
30 views

Largest prime factor

Let $$ n = (1^2 - 0^2) * (2^2 - 1^2) * (3^2 - 2^2) * (4^2 - 3^2) * ... (100^2 - 99^2).$$ What is the largest prime that divides n? Please explain how to go about solving this, for I have never seen ...
4
votes
2answers
235 views

Is there a function that only generates primes?

The title sums it up: does there exist a "nice" injective function $f(n)$ such that $f(n)\in\mathbb P$ for all $n\in\mathbb N$? I'm having difficulty specifying exactly what I want "nice" to mean, ...
3
votes
2answers
76 views

Let $u_{n+3} = u_n + 2u_{n+1}$ . Show that $p$ divides $u_p$ for all $p$ prime number.

Let $(u_n)$ a sequence such that $u_0 = 3$, $u_1 = 0$, $u_2 = 4$ and $u_{n+3} = u_n + 2u_{n+1}$ Show that $p$ divides $u_p$ for all $p$ prime number. I'm really stuck on this exercise, ...
1
vote
0answers
26 views

About Abel Summation

http://arxiv.org/pdf/math/0504289v3.pdf Here i'm trying to understand page 5. Writer uses the abel sum to find the sum of the prime's reciprocals. So he founds the formula (2.2.1) Now here y=2 ...
-12
votes
2answers
194 views

How is 2 a prime number if you can divide it evenly?

From what I know about prime numbers is that a number is considered a prime number when it's not evenly divisible, such as any number that has decimal points after you divide it. But I can't figure ...
1
vote
1answer
38 views

Confused by a step in a proof that $a^x - b^y = c$ has at most two solutions in positive integers $x,y$

The theorem is Theorem 1.1 from Michael A. Bennett in his "On Some Exponential Equations of S.S. Pillai". Here is the statement of the theorem: Theorem 1.1. If $a,b,c$ are nonzero integers with $a,b ...