# 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 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 ...
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### 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 ...
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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 ...
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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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### Can we remove any prime number with this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the first ...
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### If $x$, $y$, $x+y$, and $x-y$ are prime numbers, what is their sum?

Suppose that $x$, $y$, $x−y$, and $x+y$ are all positive prime numbers. What is the sum of the four numbers? Well, I just guessed some values and I got the answer. $x=5$, $y=2$, $x-y=3$, $x+y=7$. All ...
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### Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?

Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form. For example, last ...
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### For any prime $p > 3$, why is $p^2-1$ always divisible by 24?

Let $p>3$ be a prime. Prove that $24 \mid p^2-1$. 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 ...
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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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### Mathematicians shocked(?) to find pattern in prime numbers

There is an interesting recent article "Mathematicians shocked to find pattern in "random" prime numbers" in New Scientist. (Don't you love math titles in the popular press? Compare to the source ...
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### 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 ...
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### 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 ...
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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?
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### 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 $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ ...
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### 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 ...
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### 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?
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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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### 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 ...
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### Why are the last two numbers of this sequence never prime?

I had the idea to make a script that generates a pattern like this: 1 2 3 4 5 6 7 8 9 10 ... and so on. After that, I replaced every non-prime by a '-' ...
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### A question about divisibility of sum of two consecutive primes

I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least natural ...
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### Is zero a prime number?

Is zero a prime number? When talking about prime numbers, it seems like the examples given $(2,3,5,7,11,13,...)$ have the property that they have no factors less than themselves and greater than one. ...
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### How many prime numbers are known?

Wikipedia says that the largest known prime number is $2^{43,112,609}-1$ and it has 12,978,189 digits. I keep running into this question/answer over and over, but I haven't been able to find how many ...
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### Pattern “inside” prime numbers

Update $(2020)$ I've observed a possible characterization and a possible parametrization of the pattern, and I've additionally rewritten the entire post with more details and better definitions. It ...
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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 ...
I heard the term "odd prime" often. Isn't it redundant? If $n$ is even then $2$ divides $n$ so it's not prime. Why is "odd" emphasized?
This is a variant of Prime number building game. Player $A$ begins by choosing a single-digit prime number. Player $B$ then appends any digit to that number such that the result is still prime, and ...