Mathematics

What are Twin-Primes?

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22 September 2020                

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Introduction

Twin Primes are the set of two numbers that have exactly one composite number between them. They can also be defined as the pair of numbers with a difference of \(2.\) The name twin Prime was coined by Stackel in 1916. The word twin prime is also used to describe one of the twin primes i.e. twin prime is a prime with a prime gap of \(2.\)

The first twin Prime Numbers are \(\text{{3,5}, {5,7}, {11,13}}\) and \(\text{{17,19}.}\) It has been conjured in the famous conjecture of twin primes that there are infinite twin primes. Further using sieve techniques it has been proved that the sum of the reciprocals of twin primes converge.

Other than the first pairs, all pairs of twin primes have the form \(\text{{6n-1, 6n+1}.}\)

Theorem: (Clement 1949)

The integers \(\text{n, n+2,}\) form a pair of twin primes if and only if

\(\rm{}4[(n-1)!+1] = -n\) (mod \(\rm {}n(n+2)\)).


Properties of Twin-Primes

  1. \(5\) is the only prime number that has a positive as well as negative prime gap of two and hence is the only prime to occur in two twin primes pair.
  2. Every twin primes pair other than \(\text{{3, 5}}\) is of the form \(\text{{6n-1, 6n+1}}\)
  3. \(\text{{2, 3}}\) is not considered a twin prime pair as there is no composite number between them.
  4. The sum of each twin prime pair except \(\text{{3,5}}\) is divisive by \(12\) as

\(\text{(6n-1) + (6n+1) = 12n}\)


What is Twin-Prime Conjecture?

Twin prime conjecture, also known as Polignac’s conjecture asserts that there are infinitely many twin primes. For example, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are twin primes. As numbers get larger, primes become less frequent and twin primes rarer still.

The first statement of the twin prime conjecture was given in 1846 by French mathematician Alphonse de Polignac, who wrote that any even number can be expressed in infinite ways as the difference between two consecutive primes.

When the even number is 2, this is the twin prime conjecture; that is, \(2 = 5 − 3 = 7 − 5 = 13 − 11 = ….\) (Although the conjecture is sometimes called Euclid’s twin prime conjecture, he gave the oldest known proof that there exist an infinite number of primes but did not conjecture that there are an infinite number of twin primes.)

Very little progress was made on this conjecture until 1919 when Norwegian mathematician Viggo Brun showed that the sum of the reciprocals of the twin primes converges to a sum, now known as Brun’s constant. (In contrast, the sum of the reciprocals of the primes diverges to infinity.) Brun’s constant was calculated in 1976 as approximately \(1.90216054\) using the twin primes up to 100 billion.

In 1994 American mathematician Thomas Nicely was using a personal computer equipped with the then new Pentium chip from the Intel Corporation when he discovered a flaw in the chip that was producing inconsistent results in his calculations of Brun’s constant. Negative publicity from the mathematics community led Intel to offer free replacement chips that had been modified to correct the problem. In 2010 Nicely gave a value for Brun’s constant of \(1.902160583209 ± 0.000000000781\) based on all twin primes less than \(2 × 10^{16}.\)

The next big breakthrough occurred in 2003, when American mathematician Daniel Goldston and Turkish mathematician Cem Yildirim published a paper, “Small Gaps Between Primes,” that established the existence of an infinite number of prime pairs within a small difference (16, with certain other assumptions, most notably that of the Elliott-Halberstam conjecture).

Although their proof was flawed, they corrected it with Hungarian mathematician János Pintz in 2005. American mathematician Yitang Zhang built on their work to show in 2013 that, without any assumptions, there were an infinite number differing by 70 million. This bound was improved to 246 in 2014, and by assuming either the Elliott-Halberstam conjecture or a generalized form of that conjecture, the difference was 12 and 6, respectively. These techniques may enable progress on the Riemann hypothesis, which is connected to the prime number theorem (a formula that gives an approximation of the number of primes less than any given value).


Other types of Primes:

1. Prime Triplet/ Quadruplet/ k-tuple

  • Prime Triplet: In mathematics, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by \(6.\) In particular, the sets must have the form \(\text{{p, p + 2, p + 6}}\) or \(\text{{p, p + 4, p + 6}}\) with the exceptions of \(\text{{2, 3, 5}}\) and \(\text{{3, 5, 7}.}\)
  • Prime Quadruplet: A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form \(\text{p, p+2, p+6, p+8}.\) This represents the closest possible grouping of four primes larger than \(3,\) and is the only prime constellation of length \(4.\)
  • Prime k-tuple: A prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple \(\text{(a, b, ...),}\) the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values \(\text{(n + a, n + b, ...)}\) are prime.

2. Cousin Prime

  • In mathematics, cousin primes are prime numbers that differ by four.

3. Sexy Prime

  • Sexy primes are prime numbers that differ from each other by \(6.\)

Similarly there also exists sexy prime triplets, sexy prime quadruplets, prime decade, quintuplet primes and sextuplet primes.


Frequently Asked Questions (FAQs)

What are Twin Primes?

Pair of prime numbers with a difference of two.

Examples of prime number pairs that have a difference of two

Examples of such pairs which are also called twin primes are \(\text{{3,5}, {5,7}, {11,13}, {17,19}}\) and so on.

What is the difference between co-prime and twin primes?

All twin primes are co-primes numbers but vice versa is not true. Co-primes need not be prime numbers, they can be any numbers with their \(\text{GCD=1.}\)

List of twin primes from 1-100

The list includes:

  1. \(\text{(3, 5)}\)
  2. \((5, 7)\)
  3. \((11, 13)\)
  4. \((17, 19)\)
  5. \((29, 31)\)
  6. \((41, 43)\)
  7. \((59, 61)\)
  8. \((71, 73)\)

 


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