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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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What is the sum of the following series with a prime denominator

$$\sum_{j=1}^{546}\left[ \frac{(5j)}{(1093)}\right ]$$ Where the brackets are the floor function. I'm not even sure how to start this question besides just figuring out for how many numbers the floor of the function equates to 0, 1, 2, etc by trial…
Ralph B
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Numbers from 2 - 31

Let's say we have whole numbers from 2 to 31 (imagine them as blocks). We also have a number that is not divisible by two of those numbers that are next to each other on the number line but that number is divisible by the other numbers. For…
Pavol Komlos
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$n$-digit prime mid point for maximum pairs of $n$-digit primes

Has anyone found which $n$-digit prime (or primes) is the midpoint for the maximum number of pairs of other $n$-digit primes? For four pairs $47$ is the midpoint for $(5,89), (11,83), (23,71)$, and $(41,53)$. I searched "balanced primes" with no…
J. M. Bergot
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Is there a way to find the prime numbers up to 1000 with less than 200 calculations?

By using a sieve created by Prime Number Tables set up by the formula PN+(PNx6) for numbers generated by 6n+or-1, takes 182 calculations to identify 170 composite numbers. Using the Sieve of Eratosthenes would take around 1600 calculations. The…
Orlando Vidads
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Has anyone based a Sieve on: every prime number greater than 3 is a multiple of 6 + or - 1? I think I may have discovered one.

I am not a mathematician. But I think I may have stumbled into something interesting. I wanted to try to make a number Prime Number table where multiples of 5 would line up. I wanted to line up the sequence of numbers of 5 6 7. The next one occurs…
Orlando Vidads
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what is the smallest prime where we are unsure what the next biggest prime is

We know some pretty large primes thanks to projects like GIMPS, but for most of the large primes we know there are undiscovered primes inbetween those primes. To what number are we certain we know of all prime numbers within that range.
Ace shinigami
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To check if a number is prime or not.

To check whether or not a naturel number, say $n$, is a prime number, we only need to check it with prime numbers less than or equal to its half($\frac{n}{2}$) rather than all odd numbers till it's half ($\frac{n}{2}$).Why so??
Pratyush Kumar
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Generate a prime which divides a prime - 1

I generated a random prime $p$. Now I want to find a random prime with a fixed size of $256$ Bits which divides $p - 1$. $p$ has $3072$ Bits, so I can't just try it out.
91378246
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Capturing all prime numbers and only prime numbers greater than 3

Please advice whether below is correct, anything new or useful? where n, a and b are integers ≥ 1 For $6n - 1$ all numbers will be prime numbers except where $n= 6ab + b - a$ For $6n + 1$ all will be prime numbers except where $n= 6ab ±(b + a)$ I…
K. Hussain
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For $a>0$ prime $p=a+b$ and $p\cdot a+b$ and $p\cdot b+a$ gives all primes

How many primes will not be found from prime $p=a+b$ and prime producing formulae $a\cdot p+b$ and $b\cdot p+a$? An example for $11=2+9$ gives $2\cdot 11+9=31$ and $9\cdot 11+2=101$. Is there a maximum prime that will not be found? The larger the…
J. M. Bergot
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Congruences and Primes

Show that if $p$ is an odd prime, with $p = 3 \pmod{4}$, then $$ (\mathbb{Z}_{p}^{*})^4 = (\mathbb{Z}_{p}^{*})^2 $$ More generally, show that if $n$ is an odd positive integer, where $p = 3 \pmod{4}$ for each prime $p \mid n$,…
Steven
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Subset of the Set of Prime Gaps whose elements are always even

It is straight forward to see that: $$\tag0p_{\pi(n)+1}-p_{\pi(n+1)}=0 \operatorname{iff} \quad n \in {\{p_k-1:k \in \mathbb N}\}$$ however what is not as straightforward firstly, is to show that $n$ must be the lesser of a pair of twin primes if…
Adam Ledger
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A relation of prime numbers : $p_m + p_n ± 1, m,n ≠ 2$ is a prime where $p_r$ is a prime.

Using a table of primes I noted the following pattern: $$p_2 + p_3 - 1 = p_4$$ $$p_3 + p_4 - 1 = p_5$$ $$p_3 + p_4 + 1 = p_6$$ $$p_4 + p_5 - 1 = p_7$$ $$.……………$$ But the pattern failed when, $$p_8 + p_9 + 1 = p_{14}$$ $$p_9 + p_{10} + 1 =…
Awe Kumar Jha
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What is the first 21-digit prime number after the decimal point of Pi

We know the recent computers are 64-bit, and the maximum integer number is 18446744073709551615, whether you can find the first 21-digit prime number after the decimal point of $\pi$? Please show me.
hjin15
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Property of Prime Counting Function

For $ n $ be a positive integer and $ f(n) = \dfrac{\pi^2(n)}{n} $, where $ \pi(x) $ is the number of the prime numbers less than $x$ . Then find the value of $\displaystyle S = \lim_{n \to \infty} \dfrac{1}{n} \left(\sum_{k=1}^{n}…
Makoto Kohno
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