## The Twin Prime ConjectureJune 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.
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 1, where ^{n} x 2 x 2 x 13n 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]
"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.
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. |
RSS Feed
## Google Search
Latest Books by Dev Gualtieri
- Rough Microparticles - July 17, 2017
- Robot Musicians - July 10, 2017
- Walter Noll (1925-2017) - July 6, 2017
- cosmogony - July 3, 2017
- Crystal Prototypes - June 29, 2017
- Voice Synthesis - June 26, 2017
- Refining Germanium - June 22, 2017
- Granular Capillarity - June 19, 2017
- Kirchhoff–Plateau Problem - June 15, 2017
- Self-Assembly - June 12, 2017
- Physics, Math, and Sociology - June 8, 2017
- Graphene from Ethylene - June 5, 2017
- Crystal Alignment Forces - June 1, 2017
- Martian Brickwork - May 29, 2017
- Carbon Nanotube Textile - May 25, 2017
- The Scent of Books - May 22, 2017
- Patterns from Randomness - May 18, 2017
- Terpene - May 15, 2017
- The Physics of Inequality - May 11, 2017
- Asteroid 2015 BZ509 - May 8, 2017
- Fuzzy Fibers - May 4, 2017
- The Sofa Problem - May 1, 2017
- The Wisdom of Composite Crowds - April 27, 2017
- J. Robert Oppenheimer and Black Holes - April 24, 2017
- Modeling Leaf Mass - April 20, 2017
- Easter, Chicks and Eggs - April 13, 2017
- You, Robot - April 10, 2017
- Collisions - April 6, 2017
- Eugene Garfield (1925-2017) - April 3, 2017
- Old Fossils - March 30, 2017
- Levitation - March 27, 2017
- Soybean Graphene - March 23, 2017
- Income Inequality and Geometrical Frustration - March 20, 2017
- Wireless Power - March 16, 2017
- Trilobite Sex - March 13, 2017
- Freezing, Outside-In - March 9, 2017
- Ammonia Synthesis - March 6, 2017
- High Altitude Radiation - March 2, 2017
- C.N. Yang - February 27, 2017
- VOC Detection with Nanocrystals - February 23, 2017
- Molecular Fountains - February 20, 2017
- Jet Lag - February 16, 2017
- Highly Flexible Conductors - February 13, 2017
- Graphene Friction - February 9, 2017
- Dynamic Range - February 6, 2017
- Robert Boyle's To-Do List for Science - February 2, 2017
- Nanowire Ink - January 30, 2017
- Random Triangles - January 26, 2017
- Torricelli's law - January 23, 2017
- Magnetic Memory - January 19, 2017
- Graphene Putty - January 16, 2017
- Seahorse Genome - January 12, 2017
- Infinite c - January 9, 2017
- 150 Years of Transatlantic Telegraphy - January 5, 2017
- Cold Work on the Nanoscale - January 2, 2017
### Deep ArchiveDeep Archive 2006-2008
Blog Article Directory on a Single Page |

Copyright © 2017 Tikalon LLC, All Rights Reserved.

Last Update: 07-17-2017