## Prime Number CousinsApril 14, 2016 Prime numbers are those natural number greater than one that have no positive divisors except themselves and one. All the other natural numbers are composite numbers, which are numbers constructed by multiplying prime numbers together. While it seems to be in strange company with all those odd numbers, the number two is prime, since that's the only way we can construct the even numbers. 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. As an example, the taxicab number,1729, is 7 x 13 x 19, and it can't be expressed by any other combination of primes.
One illustration I remember from Irving Adler's Giant Golden Book of Mathematics[1] was the sieve of Eratosthenes This illustration by Lowell Hess, who illustrated quite a few children's books in his career, depicted an actual mechanism for filtering numbered cubes to find the prime numbers. The following figure illustrates such a mechanism.
N is prime is 1/(ln(N)), where ln(N) is the natural logarithm of N.
Scientists, and mathematicians who work in the sciences, often find themselves at scientific conferences, suffering through one boring talk or another while waiting for the speaker that they're really interested in hearing. Mathematician, Stanislaw Ulam found himself at one such meeting in 1963, at which he doodled a spiral of numbers and found that prime numbers were more likely found on some diagonal lines in this spiral, now called the Ulam spiral (see figure).
n is more likely to be prime if it's a solution of the equation,
where b and c are constants.
Another unusual property of prime numbers has been discovered by two mathematicians from Stanford University (Stanford, California). Robert Lemke Oliver and Kannan Soundararajan have discovered that a prime number ending in 9 is about 65% more likely to be followed by a prime number ending in 1 than another prime number ending in 9.[2-3] They present both their empirical evidence and a theoretical reason for this property in a recent arXiv posting.[2]
Evidence has been presented before for such strange correlations. In a 2011 paper, a team of mathematicians from Boston College and Ohio State University followed up on a conjecture, called the prime k-tuples conjecture, by the famous number theory pair, Hardy and Littlewood. They used a computer to amass evidence that prime pairs of the form (p, p+n) occur often. They found that prime pairs of the form (p, p+6) occur twice as often as twin primes, but this is still quite rare.[4-5] The prime k-tuples conjecture, which is unproven but supported by a lot of empirical evidence, gives estimates of occurrence of primes at all given spacings.[3]
(p, p+10) in base-10 would have a good chance of having a prime (p+2), (p+4), (p+6), or (p+8), appear between them. The actual reason is a little deeper than this, since the biases they uncovered are greater than what this simple model would predict.[3]
It's uncertain whether this new property of prime numbers is important to deeper questions about the primes, but it's interesting that such a simple fact lay undiscovered all this time; and, it was a discovery generated by experimental mathematics; i.e., computer mathematics.
## References:- Irving Adler (Author), Lowell Hess (Illustrator), "The Giant Golden Book of Mathematics," Golden Press, first edition 1958, 92 pp. (via Amazon).
- Robert J. Lemke Oliver and Kannan Soundararajan, "Unexpected biases in the distribution of consecutive primes," arXiv, March 11, 2016.
- Erica Klarreich, "Mathematicians Discover Prime Conspiracy," Quanta Magazine, March 13, 2016.
- Avner Ash, Laura Beltis, Robert Gross, and Warren Sinnott, "Frequencies of Successive Pairs of Prime Residues," Experimental Mathematics, vol. 20, no. 4(November 28, 2011), pp. 400-411. A PDF file is available here.
- G.H. Hardy and J.E. Littlewood, "Some problems of Partitio Numerorum III: On the expression of a number as a sum of primes," Acta Mathematica, vol. 44, no. 1 (December, 1923), pp. 1-70.
Linked Keywords: Prime number; natural number; positive; divisor; composite number; multiplication; multiply; even number; theorem; fundamental theorem of arithmetic; product; taxicab number; 1729; illustration; Irving Adler; sieve of Eratosthenes; children's literature; children's book; career; mechanism; cube; physics; physical; Inkscape; number theory; mathematician; enumeration; prime number theorem; probability; natural logarithm; scientist; science; academic conference; scientific conference; boredom; boring; Stanislaw Ulam; doodle; spiral; diagonal line; Ulam spiral; Wikimedia Commons; equation; constant; Stanford University (Stanford, California); Robert Lemke Oliver; Kannan Soundararajan; empirical evidence; theory; theoretical; arXiv; correlation; Boston College; Ohio State University; conjecture; prime k-tuples conjecture; G. H. Hardy; John Edensor Littlewood; computer; twin prime; mentor; intellectual giftedness; marvel; Srinivasa Ramanujan; Mathematician's Apology; randomness; random number; fair coin; James T. Kirk; Spock; computer simulation; source code; coin flip.c; ternary numeral system; base-3; decimal; base-10; radix; base; experimental mathematics; Irving Adler (Author), Lowell Hess (Illustrator), "The Giant Golden Book of Mathematics," Golden Press, first edition 1958, 92 pp. |
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