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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The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
69
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12answers
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Why 1 is not considered to be a prime number?

Why $1$ is not considered to be a prime number? Or why definition of prime numbers is given for integers greater than $1$?
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6answers
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Efficiently finding two squares which sum to a prime

The web is littered with any number of pages (example) giving an existence and uniqueness proof that a pair of squares can be found summing to primes congruent to 1 mod 4 (and also that there are no ...
26
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9answers
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Is there possibly a largest prime number?

Prime numbers are numbers with no factors other than one and itself. Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "...
21
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4answers
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Simple explanation and examples of the Miller-Rabin Primality Test

Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test. Specifically: I understand that for some reason, having non-trivial ...
69
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19answers
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Different ways to prove there are infinitely many primes?

This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show ...
53
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16answers
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For any prime $p > 3$, why is $p^2-1$ always divisible by 24?

I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
2
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1answer
652 views

If the order divides a prime P then the order is P (or 1)

I've just come up with this question as I'm studying for a number theory midterm. If $p$ and $q$ are different prime numbers, and it's known that $2^p \equiv 1 \bmod{q}$, then $q\equiv 1 \bmod{p}$. I'...
43
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Finding a primitive root of a prime number

How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
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Two Representations of the Prime Counting Function

The bounty for the best work out of Greg's answer, especially the "solving for $\pi^*(x;q,a)$ in terms of all $\Pi^*$ functions (tedious but possible)" part is over. Since Raymond's ...
88
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1answer
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$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $...
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4answers
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Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?

This question relates to a discussion on another message board. Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My ...
13
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2answers
351 views

Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$

Let $p$ be prime and $(\frac{-3}p)=1$, where $(\frac{-3}p)$ is Legendre symbol. Prove that $p$ is of the form $p=a^2+3b^2$. My progress: $(\frac{-3}p)=1 \Rightarrow$ $(\frac{-3}p)=(\frac{-1}p)(\...
7
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2answers
350 views

Compositeness of $n^4+4^n$ [duplicate]

My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number. So we memorized this for future use in math competition. But I don't understand why is it?
88
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3answers
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Least prime of the form $38^n+31$

I search the least n such that $$38^n+31$$ is prime. I checked the $n$ upto $3000$ and found none, so the least prime of that form must have more than $4000$ digits. I am content with a probable ...
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4answers
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How can we prove that among positive integers any number can have only one prime factorization?

I have read right from school that prime factorization is unique, but have never found proof for this. Can someone show me the proof?
37
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6answers
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Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
15
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4answers
848 views

Prove that $n^2+n+41$ is prime for $n<40$

Here's a problem that showed up on an exam I took, I'm interested in seeing if there are other ways to approach it. Let $n\in\{0,1,...,39\}$. Prove that $n^2+n+41$ is prime. I shall provide my ...
10
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2answers
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Help understand the proof of infinitely many primes of the form $4n+3$

This is the proof from the book: Theorem. There are infinitely many primes of the form $4n+3$. Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in ...
4
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3answers
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$p=4n+3$ never has a Decomposition into $2$ Squares, right?

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma. Is it correct to say that, primes of the form $p=4n+3$, never have ...
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Why are very large prime numbers important in cryptography?

Firstly, you guys are awesome, and I learn quite a bit just from reading the questions of others. Secondly, a friend asked me recently why large primes are important for data security, and I was ...
17
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2answers
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Can every even integer be expressed as the difference of two primes?

Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof?
2
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2answers
2k views

Prime powers that divide a factorial [duplicate]

If we have some prime $p$ and a natural number $k$, is there a formula for the largest natural number $n_k$ such that $p^{n_k} | k!$. This came up while doing an unrelated homework problem, but it is ...
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3answers
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Does the sum of reciprocals of primes converge?

Is this series known to converge, and if so, what does it converge to (if known)? Where $p_n$ is prime number $n$, and $p_1 = 2$, $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
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8answers
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The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n^\text{th}$ square number – it is $n^2$ – but we do not have a (useful) exact formula for the $n^\text{th}$ ...
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6answers
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Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
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3answers
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How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an information-...
26
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4answers
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Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
12
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3answers
439 views

Are there any Combinatoric proofs of Bertrand's postulate?

I feel like there must exist a combinatoric proof of a theorem like: There is a prime between $n$ and $2n$, or $p$ and $p^2$ or anything similar to this stronger than there is a prime between $p$ and $...
11
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2answers
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Is this formula for the $n^{th}$ prime number useful?

Is the below formula for the $n^{th}$ prime number in elementary functions useful somehow? $$p(n)=\sum _{a=2}^{2^n} \sin \left(\pi 2^{\left(n-\sum _{b=2}^a \frac{\sin ^2\left(\frac{\pi }{b}((b-1)!)^...
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1answer
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Summing over General Functions of Primes and an Application to Prime $\zeta$ Function

Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following: $$ \sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1} $$ and ...
11
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3answers
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A theorem about prime divisors of generalized Fermat numbers?

A theorem of Édouard Lucas related to the Fermat numbers states that : Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one. Does anyone ...
3
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2answers
874 views

Proof of lack of pure prime producing polynomials.

I recently encountered this following proposition: For every polynomial, there is some positive integer for which it is composite. What is the most elementary proof of this?
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Real-world applications of prime numbers?

I am going through the problems from Project Euler and I notice a strong insistence on Primes and efficient algorithms to compute large primes efficiently. The problems are interesting per se, but I ...
10
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3answers
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Infinitely number of primes in the form 4n+1 proof

Question: Are there infinitely many primes of the form 4n+3 and 4n-1? My attempt: Suppose the contrary that there exists finitely many primes of the form 4n+3, say k+1 of them: 3,$p_1,p_2,....,p_k$ ...
7
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1answer
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Bounds for $n$-th prime

In this Wikipedia page I have found that the bounds for $n$-th prime is given by, $$n(\ln n+\ln \ln n)>p_n>n(\ln n+\ln \ln n-1)$$ for all $n\ge6$. Are there even stronger bounds for the $n$-th ...
16
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2answers
897 views

Is there an infinite number of primes constructed as in Euclid's proof?

In Euclid's proof that there are infinitely many primes, the number $p_1 p_2 ... p_n + 1$ is constructed and proved to be either a prime, or a product of primes greater than $p_n$. Trivially, we ...
12
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1answer
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Sum of reciprocal prime numbers

How can the following equation be proven? $$ \forall n > 2 : \sum_{p \le n}{\frac1{p}} = C + \ln\ln n + O\left(\frac1{\ln n}\right), $$ where $p$ is a prime number. It's not homework; I just ...
7
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3answers
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When do the multiples of two primes span all large enough natural numbers?

It is well-known that given two primes $p$ and $q$, $pZ + qZ = Z$ where $Z$ stands for all integers. It seems to me that the set of natural number multiples, i.e. $pN + qN$ also span all natural ...
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4answers
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Alternate definition of prime number

I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works: A prime is a quantity $p$ such that whenever $p$ is a factor of ...
2
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1answer
220 views

Solutions to $y^2 = x^3 + k$?

As you know, the equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 count m, don't have any answer and its proof can ...
4
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2answers
802 views

Show for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime.

I need to show that for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2 \equiv-1\pmod p$. I then need to show this implies p isn't a gaussian prime. I have started to solve ...
61
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13answers
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Why is Euclid's proof on the infinitude of primes considered a proof?

I've expressed Euclid's proof on the infinitude of primes on Mathematica: ...
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12answers
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Proof that every number ≥ $8$ can be represented by a sum of fives and threes.

Can you check if my proof is right? Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$. Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 ...
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3answers
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Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$

This an interesting problem my friend has been working on for a while now (I just saw it an hour ago, but could not come up with anything substantial besides some PMI attempts). Here's the full ...
18
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7answers
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Prime dividing the binomial coefficients

It is quite easy to show that for every prime $p$ and $0<i<p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the ...
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5answers
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Why is Euler's Totient function always even?

I want to prove why $\phi(n)$ is even for $n>3$. So far I am attempting to split this into 2 cases. Case 1: $n$ is a power of $2$. Hence $n=2^k$. So $\phi(n)=2^k-2^{k-1}$. Clearly that will ...
29
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Percentage of primes among the natural numbers

How high is the percentage of primes in $\mathbb{N}$? ($\mathbb{N} := \lbrace { 1, 2, 3, \ldots \rbrace }$ ; a prime is only divisible by itself and 1 in $\mathbb{N}$) The percentage has to be lower ...
9
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1answer
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A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
8
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1answer
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Is the $n$-th prime smaller than $n(\log n + \log\log n-1+\frac{\log\log n}{\log n})$?

Let $p_n$ be the $n$-th prime. Wikipedia gives the following known bounds on $p_n/n$ when $n\geq6$: $$ \log n+\log\log n-1 \leq \frac{p_n}{n} \leq \log n+\log\log n. $$ If I take the first few terms ...