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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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Remainders of primes

Maybe an idiot question but I can't find any info! We divide successive prime numbers by some fixed prime number $n$ (e.g. 7 or 17). We'll get some remainders $r[i]= 1..n-1$ Is there any law or theorem about their distribution? It seems Fermat's…
lesobrod
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Why does this work? "If a & b are relatively prime, any integer can be written as ma + nb for m, n ∈ ℤ."

I don't understand how that theorem is true. For example, "If a & b are relatively prime", 3 and 4 are relatively prime. "any integer", let's choose 5. There's no m and n (∈ ℤ) combination for 3 and 4 that gives 5. What am I thinking wrong here?
dsfdf
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Prime Postulate

Why is it the case that for every prime number $p_i$ there exist unique positive integers $m$ and $n$ such that $$ m\, p_i^2 + 1 = n \, p_{i + 1} $$ where $m\ne n$ and $m \le p_i$? That is, why is it that there is some positive integer $m$, less…
Dolphin
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prime prove: how to prove if $n! + n^2 + 1$ is prime then $n^2 + 1$ is also prime

how to prove this: $$ \text{if } n! + n^2 + 1 \text{ is prime then } n^2 + 1 \text { is also prime}$$ I was thinking that n! is definitely not prime since it can be written as $n\times (n-1)....2\times 1$. So $n^2 + 1$ is not prime. In other…
Zhenqing Xu
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What is the following prime number after this huge $87$ digit number?

So I know there are infinitely many prime numbers, for Euclid proved there were in $300$ BCE. However, I cannot find the following prime number after $293703234068022590158723766104419463425709075574811762098588798217895728858676728143227$ Let this…
Mr Pie
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Do all Mersenne Primes come from Powers of 2 that end in either 2 or 8?

I was working on he Collatz Conjecture and noticed that the reverse of 3x+1 is x-1/3 This is possible for Powers of 2 that end in 6 and 4. Where the result will have in the case of 6 a factor of 5 and in the case of 4 no factor of five. From…
Ernst D. Berg
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Proof of Wilsons theorem

How to understand the grouping of integers from $2$ to $p-2$ in the proof af Wilsons theorem? I don't understand how you can group the integers $2$ to $p−2$ into $(p-3)/2$ and next how you become the $2*3*...=1 (mod p)$
WinstonCherf
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prime quadruplets differing by more than 8

There's http://oeis.org/A007530 with the least prime being 8 less than the fourth prime. Then there's one with the difference being greater, as in 211,223,227,229. Do you think there are more of the former than of the latter?
J. M. Bergot
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How many numbers between a prime $p$ and $p^2$ have some prime factor $>p$?

For $p=2$ the answer is all of them (given that between here can only reasonably mean strictly between). For $p=3$ or 5 or 7 the answer is just under half of them. But for $p=11$, unless I slipped up it is just 29 out of 110. I am afraid the…
Colin McLarty
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Is there a name for the distinct lines in a plot of primes: $p_n/p_{n-1}$ vs $n$

If you plot the ratios of consecutive primes against the number of preceding primes there is a whole bunch of distinct lines. Do they have a name? Do the individual lines have a formula? Here's a plot of $p_n/p_{n-1}$ against $n$ where $p_n$ is the…
Lucas
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Interpreting big O for Sum of Primes < n

The sum of primes < n has been answered (though not asked, some book/paper references of this would be nice too) by @Eric Naslund here: What is the sum of the prime numbers up to a prime number $n$? as $$\sum_{p\leq x }…
onepound
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Median of prime sequence

Let $S_n$ be a sequence of consecutive $2n-1$ primes starting from $2$. For example $S_3=(2,3,5,7,11)$. Denote $p_{2n-1}$ as the ${2n-1} ^{th}$ (last) number in this sequence, then $p_n$ denotes median value in this growing sequence. In the example…
Widawensen
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Is there a prime for every starting sequence of digits?

Suppose, I have a random finite sequence of digits (that doesn't start from 0). Is it true that there is a prime number (no matter how big) that starts with this sequence of digits? Examples: 3 => 31 10 => 101 123 => 1237 1000 => 100007
sesm
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set of primes plus 1 contain multiples of every integer

Saw a claim that for every natural number $j$, the set $\{ nj-1: n \ge 1 \} $ must contain at least one prime. Is that correct? If so, is there a simple proof? I can see that at least one can be easily extended to infinitely many, if the statement…
Co-ordinator
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What's the largest number of prime divisors that a number $n$ could have?

A natural number $n$ has 20 divisors. What's the largest number of different prime divisors that a number $n$ could have? What I have is $$(\alpha_1+1)(\alpha_2+1)(\alpha_3+1)\cdots(\alpha_k+1)=20$$ where $$n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots…
Karagum
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