General Knowledge about Primes....

\quad \quad Here's some information that you'll surely like to know about " prime numbers "\textbf{" prime numbers "}

\bullet \quad We call a positive integer pp to be a prime number\color{#20A900}{\textbf{prime number}} if out of the set of all the positive integers, only\color{#D61F06}{\textbf{only}} 11 and pp are the divisors of pp.

\bullet \quad We call a pair of prime numbers\color{#20A900}{\textbf{prime numbers}} (p1,p2)(p_1,p_2) to be Twin Primes\color{#3D99F6}{\textbf{Twin Primes}} iff p1p2=2\color{#69047E}{\left| p_1 - p_2 \right| = 2}

\bullet \quad The smallest known pair of Twin Primes\color{#3D99F6}{\textbf{Twin Primes}} is (3,5)(3,5).

\bullet \quad But what's more interesting, is the BIGGEST KNOWN\textbf{BIGGEST KNOWN} pair of Twin Primes\color{#3D99F6}{\textbf{Twin Primes}} and it is 655164683552333333±1\Huge{65516468355 \cdot 2^{333333} \pm 1}

Both of them are 100355\color{darkred}{100355} digits long.

\bullet \quadThere are 152891\color{#20A900}{152891} pairs of Twin Primes\color{#3D99F6}{\textbf{Twin Primes}} which are less than 3×107\color{#20A900}{3\times 10^7} .

\bullet \quadThere are only 20 pairs\color{#20A900}{\textbf{only 20 pairs}} of Twin Primes\color{#3D99F6}{\textbf{Twin Primes}} between 1012\color{#20A900}{10^{12}} and 1012+10000\color{#20A900}{10^{12}+10000}.

\quad This shows the scarcity of Twin Primes\color{#3D99F6}{\textbf{Twin Primes}} as the numbers increase.

\bullet \quad The smallest gap between 22 consecutive prime numbers is 11 and it is for the pair (2,3)(2,3).

\bullet \quad The largest (known till now) gap between 22 consecutive prime numbers is 1442\color{#69047E}{1442}, and it is seen just after the prime 804212830686677669\color{#69047E}{804212830686677669}. (There are 1441 consecutive composite numbers after this prime).

Source of this information:- Elementary Number Theory ,Author- David M. Burton\color{#D61F06}{\text{Elementary Number Theory }} , \color{#3D99F6}{\text{Author- David M. Burton}}

Note by Aditya Raut
6 years, 11 months ago

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"Prime numbers" is a big topic and you have included mostly Twin Primes in your note.

I would love to see their special groups and properties even including Mersenne Primes, Weifrich Primes etc. etc. etc. and never ending....

Kartik Sharma - 6 years, 11 months ago

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Another thing: Recent studies by Yitang Zhang showed that the gaps between primes is at most 70,000,000...

John Ashley Capellan - 6 years, 11 months ago

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@John Ashley Capellan : Your statement is misleading. I can make the gap between two consecutive primes as large as I want. What Yitang Zhang did was prove that there were an infinite number of prime-pairs which differ by less than 70 million.

[(n+1)!+2,(n+1)!+3,(n+1)!+n,(n+1)!+(n+1)(n+1)!+2, (n+1)!+3, \cdots (n+1)!+n, (n+1)!+(n+1) are all composite for any n1n\geq 1. So, prime gaps can be arbitrarily large.]

Mursalin Habib - 6 years, 11 months ago

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That's great, I will include it... please tell more about it

Aditya Raut - 6 years, 11 months ago

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On April 17, 2013, Zhang announced a proof that there are infinitely many pairs of prime numbers which differ by 70 million or less. This proof is the first to establish the existence of a finite bound for prime gaps, resolving a weak form of the twin prime conjecture. Zhang's paper was accepted by Annals of Mathematics in early May 2013. If P(N) stands for the proposition that there is an infinitude of pairs of prime numbers (not necessarily consecutive primes) that differ by exactly N, then Zhang's result is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P(k) is true. The classical form of the twin prime conjecture is equivalent to P(2); and in fact it has been conjectured that P(k) for all even integers k. While these stronger conjectures remain unproven, a recent result due to James Maynard, employing a different technique, has shown that P(k) for some k ≤ 600. See this Article for more.

Sagnik Saha - 6 years, 11 months ago

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@Sagnik Saha AWESOME ! Thanks for telling... THANK\color{darkred}{T}\color{#D61F06}{H}\color{#EC7300}{A}\color{limegreen}{N}\color{#20A900}{K} YOU!!!\color{#3D99F6}{Y}\color{#3D99F6}{O}\color{#69047E}{U}\color{#BA33D6}{!!!}

Aditya Raut - 6 years, 11 months ago

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I think the gap has been lowered to 600

Bogdan Simeonov - 6 years, 11 months ago

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