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

I awoke with the following puzzle and 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 ...
5answers
25k views

### 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 ...
6answers
27k views

### 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 ...
5answers
21k views

### 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?
6answers
6k views

### 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 ...
3answers
15k views

### 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 ...
15answers
11k views

### 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 ...
14answers
58k views

### 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 $1$?
7answers
25k views

### Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
1answer
5k views

3answers
5k views

### How did Euler prove the Mersenne number $2^{31}-1$ is a prime so early in history?

I read that Euler proved $2^{31} -1$ is prime. What techniques did he use to prove this so early on in history? Isn't very large number stuff done with computers? Do you know if Euler had a team of ...
5answers
7k views

### Are there an infinite number of prime numbers where removing any number of digits leaves a prime?

Suppose for the purpose of this question that number $1$ is a prime number. Consider the prime number $311$. If we remove one $1$ from the number we arrive at the number $31$ which is also prime. If ...
5answers
10k views

### What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ ...
10answers
9k views

### Why are primes considered to be the “building blocks” of the integers?

I watched the video of Terence Tao on Stephen Colbert the other day (here), and he stated, like many mathematicians do, that the primes are the building blocks of the integers. I've always had ...
5answers
7k 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 ...
4answers
4k views

### Does the string of prime numbers contain all natural numbers?

Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
11answers
17k views

### 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: ...
3answers
3k views

### $5^n+n$ is never prime?

In the comments to the question: If $(a^{n}+n ) \mid (b^{n}+n)$ for all $n$, then $a=b$, there was a claim that $5^n+n$ is never prime (for integer $n>0$). It does not look obvious to prove, nor ...
4answers
45k views

### 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?
5answers
7k views

### What is Trinity Hall Prime number?

It is a prime number with 1350 many digits. I did not get much information about this number on the internet. Question : What is Trinity Hall Prime number? I watched this video but did not get ...
1answer
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1answer
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### Palindromic Numbers - Pattern “inside” Prime Numbers?

EDIT: rewritten and reduced entire post to present things more clearly. I'm asking how to calculate the next element(s) in the sequence, located at the end of this post. Introduction - ...