| Tikalon Blog is now in archive mode.
An easily printed and saved version of this article, and a link
to a directory of all articles, can be found below: |
|
This article |
| Directory of all articles |
The Twin Prime Conjecture
June 3, 2013
One of my most prized
books as a
child was the Giant Golden Book of Mathematics, written by
mathematician,
Irving Adler.[1] This book was nicely illustrated by Lowell Hess, who illustrated quite a few children's books in his career.
One illustration I remember from that book was the
sieve of Eratosthenes, shown as an actual mechanism for filtering numbered
cubes to find the
prime numbers. The following figure illustrates such a mechanism.
The sieve of Eratosthenes, illustrated as a physical mechanism. The first platform has holes cut to stop all numbers divisible by two, and pass all others. The second platform catches all numbers divisible by three, etc. The prime numbers are the first number stopped on each level. Since the first platform catches all even numbers, platforms to stop numbers divisible by four, six, and all other even numbers, are not included. (Illustration by the author using Inkscape.)
A prime number is any
natural number greater than one with no
positive divisors aside from itself and one. All other natural numbers are
composite numbers, which are constructed by
multiplying prime numbers together. A
theorem with the impressive title,
the fundamental theorem of arithmetic, asserts that any natural number greater than one can be expressed as a product of primes in just one way; so,
54 is just
2 x 2 x 13, and it can't be expressed by any other combination of primes.
One thing you notice is that one is not a prime number. If it were, composite numbers could have an
infinite number of
prime factors, since we can just multiply by as many ones as we desire. Thus,
54 would be
1n x 2 x 2 x 13, where
n can be any integer. Also, two is prime, since that's the only way we can construct the
even numbers. It's the only even prime number.
The fundamental theorem of arithmetic has been proven, but there are many unproven
conjectures involving the prime numbers.
Goldbach's conjecture states that any even integer greater than two can be expressed as the sum of two primes.
Computers have enabled
experimental mathematics, and they've given substantial credence to this conjecture, since no
counterexample has been found up to 4 x 10
18.
Anecdotal evidence, however, is not proof, so mathematicians are still examining this conjecture.
Examination of any list of prime numbers illustrates that they become rare as we go to larger numbers. This behavior has been quantified by the
prime number theorem, which states that the
probability that a number
N is prime will closely follow the function
1/ln(N).
Wikipedia lists 29 prime number conjectures, one of which, the
first Hardy–Littlewood conjecture, gives us the mysterious twin prime constant,
.6601618158468695739278121100145557784326233602847334133...,
also known as sequence
A005597 in the
On-Line Encyclopedia of Integer Sequences (OEIS). This constant relates to numbers known as the
twin primes.
Twin primes are prime numbers that differ from each other by the smallest possible interval; namely, two. Thus, (3,5) and (5,7) are twin primes, as are (617,619) and many others.
OEIS sequence A077800 is the list of twin primes. According to Wikipedia, there are 808,675,888,577,436 twin prime pairs below 10
18.
The
density of prime numbers decreases as numbers get larger, so the density of twin primes decreases as well. An open problem is whether twin primes exist when we reach arbitrarily large numbers. The
twin prime conjecture is that there's an infinite number of twin primes.
Euclid, the famous
Greek geometer, is the supposed author of this conjecture, which makes it one of the oldest conjectures in
number theory.[2]
Euclid, from a 15th century Latin manuscript entitled, "Artes Liberales" (The Liberal Arts).
(Via Wikimedia Commons.)
Recent work by mathematician,
Zhang Yitang, of the
University of New Hampshire (Durham) has produced a result that's the closest yet to a proof of this conjecture, but still far from the mark.[2-4] As an example that mathematics sometimes works in strange ways, what's been proven is that there are infinitely many primes which differ in distance by at most 70 million. Seventy million, of course, is quite a ways from two, but Zhang's result shows that this conjecture can be addressed. Zhang's proof will appear in the
Annals of Mathematics.
This proof advances the status of the twin prime conjecture beyond a result obtained a few years ago that there is an infinite number of prime pairs, but only if another unproven conjecture is true.[2-5] That proof showed that there will always be a pair of primes which are closer together than the average gap between primes anywhere on the
number line.[3] Zhang was able to prove his result using standard mathematical techniques, so it was something that others could have done before, but they hadn't. Zhang thinks that his techniques can be used to push his 70 million limit downwards.[2]
As is common in
creative endeavors, such as
science and mathematics, individuals will carry a problem around with them for years and make no progress towards a solution; then, suddenly, the answer comes in a
flash of inspiration. Zhang thought about the problem for three years, but it was only on
holiday at a friend's house, during a short wait before leaving for a
concert, that the solution came to him.[3] Says
Daniel Goldston, a
number theorist at
San Jose State University,
"It's one of those problems you weren't sure people would ever be able to solve.[3]
References:
- Irving Adler (Author), Lowell Hess (Illustrator), "The Giant Golden Book of Mathematics [Hardcover]," Golden Press, first edition 1958, 92 pages.
- Maggie McKee, "First proof that infinitely many prime numbers come in pairs," Nature News, May 14, 2013.
- Erica Klarreich, "Unheralded Mathematician Bridges the Prime Gap," Simons Foundation Press Release, May 19, 2013. Same article on Wired.
- Kenneth Chang, "Solving a Riddle of Primes," The New York Times, May 20, 2013.
- Daniel A. Goldston, János Pintz and Cem Y. Yíldírím, "Primes in tuples I," Annals of Mathematics, vol. 170, no. 2 (September, 2009), pp. 819-862.
Permanent Link to this article
Linked Keywords: Book; child; mathematician; Irving Adler; sieve of Eratosthenes; cube; prime number; Inkscape; natural number; sign; positive; divisor; composite number; multiplication; multiplying; theorem; the fundamental theorem of arithmetic; infinite number; integer factorization; prime factor; even number; conjecture; Goldbach's conjecture; computer; experimental mathematics; counterexample; anecdotal evidence; prime number theorem; probability; Wikipedia; first Hardy–Littlewood conjecture; A005597; On-Line Encyclopedia of Integer Sequences; twin prime; OEIS sequence A077800; probability density function; density; twin prime conjecture; Euclid; Greek; Euclidean geometry; geometer; number theory; 15th century; Latin; manuscript; Wikimedia Commons; Zhang Yitang; University of New Hampshire (Durham); Annals of Mathematics; number line; creative professional; creative endeavor; science; Eureka effect; flash of inspiration; vacation; holiday; concert; Daniel Goldston; number theorist; San Jose State University; Irving Adler (Author), Lowell Hess (Illustrator), "The Giant Golden Book of Mathematics [Hardcover]," Golden Press, first edition 1958, 92 pages.