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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.

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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 searching, I found this question, which is similar but…
David Starkey
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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 '-' character and every prime number by a '|'. The output begins like that: - || -|- |--- |-|--…
Bruno Henrique
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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 people to follow algorithms for him, dubbed…
SwimBikeRun
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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 known prime numbers exist. The website…
f.ardelian
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How far is the list of known primes known to be complete?

So there is always the search for the next "biggest known prime number". The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twenty million digits. Wikipedia also lists the twenty highest known prime numbers, only the four…
Arthur
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Is a prime number still a prime when in a different base?

Is a prime number in the decimal system still a prime when converted to a different base?
Pram
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What is an odd prime?

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?
user221304
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Is the number $-1$ prime?

From my understanding it's not prime because it's not greater than $0$. So my followup question is why did mathematicians exclude $-1$? The definition of prime is having only two factors. $-1 \cdot 1 = -1$
Chris W
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Is this proposed method of finding primes valid? If so, would it be effective?

(See question towards the end) Suppose we take the first four consecutive primes, $$2, 3, 5, 7$$ Since these are prime numbers, the greatest common divisor will be 1. In other words, they will be co-prime. Knowing this, this also means their lowest…
Tauist
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Proving that if $p^2 = a^2 + 2b^2$ then also $p$ can be written in form $a^2 +2b^2$

I'm high school student, and this problem has bothered me for about 2 weeks now. I don't necessarily need a solution, but for example mentioning a helpful theorem or property that could help me to prove this would be nice. If $p^2$ can be written…
Derris
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Proof that there are infinitely many prime numbers starting with a given digit string

To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they start with 314159265358979323. I think using a…
user848465
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Is there any prime number of the form $0^1+1^2+2^3+3^4+4^5+....+(n-2)^{n-1}+(n-1)^n$?

Let A(n)= $0^1+1^2+2^3+3^4$....+$(n-2)^{n-1}+(n-1)^n$. So: A($1$)= $0^1$ A($2$)= $0^1+1^2$ A($3$)= $0^1+1^2+2^3$ A($4$)= $0^1+1^2+2^3+3^4$ and so on.... Is there a prime number of such form?, because I checked quite plenty numbers of that form and…
Die Farbe meiner Haus
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Tests for prime numbers

I'm told to list the prime numbers from 7 to 150 . I know how to do it the slow way by one by one checking the numbers till 150. But in an exam condition I can't possibility do that . Is there any way to test for prime numbers ? Or to prove whether…
user307640
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Finding a better approximation to a prime number relation

The basis of this problem, and that which allows for the approximations to be made here, can be summarised in one approximation: $$\Biggl(\frac{n^k -{\lfloor n^{\frac{1}{k}} \rfloor}^{k-1}\gcd({\lfloor n^{\frac{1}{k}} \rfloor}^{k-1},\Bigl\lfloor…
Adam Ledger
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How was the 506-digit prime number 999...9998999...999 found?

I was surprised to encounter a claim made on the internet that the following number is…
geometrian
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