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General Knowledge about Primes....

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

\(\bullet \quad\) We call a positive integer \(p\) to be a \(\color{Green}{\textbf{prime number}}\) if out of the set of all the positive integers, \(\color{Red}{\textbf{only}}\) \(1\) and \(p\) are the divisors of \(p\).

\(\bullet \quad\) We call a pair of \(\color{Green}{\textbf{prime numbers}}\) \((p_1,p_2)\) to be \(\color{Blue}{\textbf{Twin Primes}}\) iff \[\color{Purple}{\left| p_1 - p_2 \right| = 2}\]

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

\(\bullet \quad\) But what's more interesting, is the \(\textbf{BIGGEST KNOWN}\) pair of \(\color{Blue}{\textbf{Twin Primes}}\) and it is \[\Huge{65516468355 \cdot 2^{333333} \pm 1}\]

Both of them are \(\color{DarkRed}{100355}\) digits long.

\(\bullet \quad\)There are \(\color{Green}{152891}\) pairs of \(\color{Blue}{\textbf{Twin Primes}}\) which are less than \(\color{Green}{3\times 10^7}\) .

\(\bullet \quad\)There are \(\color{Green}{\textbf{only 20 pairs}}\) of \(\color{Blue}{\textbf{Twin Primes}}\) between \(\color{Green}{10^{12}}\) and \(\color{Green}{10^{12}+10000}\).

\(\quad\) This shows the scarcity of \(\color{Blue}{\textbf{Twin Primes}}\) as the numbers increase.

\(\bullet \quad\) The smallest gap between \(2\) consecutive prime numbers is \(1\) and it is for the pair \((2,3)\).

\(\bullet \quad\) The largest (known till now) gap between \(2\) consecutive prime numbers is \(\color{Purple}{1442}\), and it is seen just after the prime \(\color{Purple}{804212830686677669}\). (There are 1441 consecutive composite numbers after this prime).

Source of this information:- \(\color{Red}{\text{Elementary Number Theory }} , \color{Blue}{\text{Author- David M. Burton}}\)

Note by Aditya Raut
3 years, 2 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 - 3 years, 2 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 - 3 years, 2 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.

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

Mursalin Habib - 3 years, 2 months ago

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I think the gap has been lowered to 600

Bogdan Simeonov - 3 years, 2 months ago

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

Aditya Raut - 3 years, 2 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 - 3 years, 2 months ago

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@Sagnik Saha AWESOME ! Thanks for telling... \(\color{DarkRed}{T}\color{Red}{H}\color{Orange}{A}\color{LimeGreen}{N}\color{Green}{K}\) \(\color{Blue}{Y}\color{Blue}{O}\color{Purple}{U}\color{Violet}{!!!}\)

Aditya Raut - 3 years, 2 months ago

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