Outline of Twin Prime Conjecture
Introduction
- History of the Twin Prime Conjecture
- French mathematician known as Alphonse de Polignac gave the first statement on the Twin Prime Conjecture in 1846 (Cohen 399). He wrote that it is possible to express any number in infinite ways.
- Alphonse asserted that twin primes are infinitely many, but they become less frequent as the numbers get larger.
- The Twin Prime Conjecture introduced by Alphonse is known as Euclid’s twin prime conjecture since he proved the existence of the infinite number of prime numbers but failed to prove the existence of an infinity of the twin prime conjectures.
- The most significant progress on twin prime conjecture occurred in 1919 when Viggo Brun, a Norwegian mathematician, asserted that the convergence of the sum of the reciprocals to the amount. The theory is famous as Brun constant (Cohen 399).
- Other developments on the twin prime conjecture occurred in 1994, 2003, 2005, 2010, and 2014 (Cohen 399).
- Applications
The twin prime conjecture has several applications in the computing industry. Thomas Nicely, an American mathematician working equipped with Intel Corporation’s new Pentiumchip, found a computer flaw that produced inconsistent results associated with Brun constant. He later gave the Brun constant value equivalent to 1.902160583209 0.000000000781 for the twin primes that do not exceed 2
- Failed Proofs
- Using an Infinite Series
Dirichlet’s theorem was one of the possible strategies adopted initially to prove the twin primes. The series looks as shown below.
= + + + + + + + + …. (Maynard 175)
The divergence of the reciprocals of the twin primes indicates that there is an infinite number of twin prime pairs.
Theory, however, failed to prove the twin primes as the mathematicians later found that it converges to a constant B known as Brun’s constant.
- Yitang Zhang Approach
Zhang proved a weaker version of twin primes that claimed that the difference separating the infinite twine primes is a finite number. Yitang Zhang used a boundary number of 7,000,000. However, Terence Tao later reduced the number to 246. Zhang’s approach has remained the most relevant attempt on the proof for twin primes (BREITZMAN SR 1).
- c) Elementary Proof
Let s be prime number such that Ps is the multiplication of the first s prime numbers and Ps as the sth prime number, and as the set of prime numbers that are less than Ps but relatively prime toPs, the following equations stand:
a in As, a1 + 2 = a2 and O ≤ m ≤ Ps occurs from two arithmetic progression confirms the existence of a twin prime such that (m1Ps + a1) and (m2Ps + a2) occurs for m1 = m2. The least difference between the twin primes above 3 was 2.
Conclusion
Zhang made the greatest attempt in expressing the twin primes after the works of many of his predecessors. He showed how the prime numbers looked random in a different manner. Zhang succeeded together with other mathematicians such as Green and Tao. It led to the discovery of the differences in the twin primes, which was almost equal to 2. There is a need to further the studies on the twin primes to bring structures to it. Such works will improve the understanding of the twin primes’ theory.
References
BREITZMAN SR, A. N. T. H. O. N. Y. “MAJOR MILESTONES IN THE TWIN PRIME CONJECTURE.” Mathematical Scientist 41.1 (2016).
Cohen, Joel E. “Statistics of primes (and probably twin primes) satisfy Taylor’s law from ecology.” The American Statistician 70.4 (2016): 399-404.
Maynard, James. “The twin prime conjecture.” Japanese Journal of Mathematics 14.2 (2019): 175-206.