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.

A prime number (or a prime) is an element of the greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in :

Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Here you get the first 50 millions of primes.


The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.

One can easily see that this extends the definition of prime numbers in the natural numbers.

11585 questions
284
votes
14 answers

Help with a prime number spiral which turns 90 degrees at each prime

I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the…
Karl
  • 4,633
  • 4
  • 17
  • 23
279
votes
5 answers

Is $7$ the only prime followed by a cube?

I discovered this site which claims that "$7$ is the only prime followed by a cube". I find this statement rather surprising. Is this true? Where might I find a proof that shows this? In my searching, I found this question, which is similar but…
David Starkey
  • 2,283
  • 2
  • 16
  • 18
219
votes
2 answers

Meaning of Rays in Polar Plot of Prime Numbers

I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, $(r,\theta) = (p,p)$. I was not expecting anything in…
dwymark
  • 2,203
  • 2
  • 10
  • 7
174
votes
7 answers

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
user103028
171
votes
6 answers

Deleting any digit yields a prime... is there a name for this?

My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100: But today he wanted the prime 719, which I obliged. When…
Fixee
  • 11,199
  • 6
  • 39
  • 64
153
votes
3 answers

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 elementary algebra and also using the theory of field…
user8465
  • 1,633
  • 3
  • 12
  • 6
152
votes
0 answers

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$ For example, for $n =…
daniel
  • 9,714
  • 5
  • 30
  • 66
149
votes
5 answers

What is the Riemann-Zeta function?

In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
144
votes
6 answers

Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

I was looking at a list of primes. I noticed that $ \frac{AM (p_1, p_2, \ldots, p_n)}{p_n}$ seemed to converge. This led me to try $ \frac{GM (p_1, p_2, \ldots, p_n)}{p_n}$ which also seemed to converge. I did a quick Excel graph and regression and…
Soham
  • 1,493
  • 1
  • 10
  • 11
133
votes
15 answers

Why is $1$ not a prime number?

Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
bryn
  • 8,471
  • 12
  • 37
  • 33
118
votes
15 answers

Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is…
Tomas Wolf
  • 1,321
  • 2
  • 9
  • 5
114
votes
1 answer

$n!$ 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 $2n-2$. A proof can be found here. Two weeks and…
112
votes
25 answers

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 you what proofs I have and I'd like to know more…
Patrick Da Silva
  • 39,395
  • 5
  • 76
  • 126
112
votes
5 answers

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
user27617
109
votes
4 answers

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 prime, it need not be a proven prime.
Peter
  • 78,494
  • 15
  • 63
  • 194
1
2 3
99 100