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 NCB Deposit  # 19

Chris K. Caldwell
Department of Mathematics
University of Tennessee, Martin
Martin, Tennessee

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Prime Numbers

Twin Primes

From ( IX, 20) of the Elements.
Image from the first English "Elements"

Math in the news:  Are there an infinite number of pairs of "twin primes?"

For the student . . . .
Prime number explanations

Prime numbers explanation continued
Background . . . .

Twin primes are pairs of prime numbers that only differ by two. 
Examples:   3   and  5
   5   and  7
  11  and  13
  17  and  19
  29  and  31
  41  and  43
  59  and  61

Until recently, it had been conjectured that there are infinitely many twin primes. If the probability of a random integer  n  and the integer  n+2  being prime were statistically independent events, then it would follow from the prime number theorem that there are about n/(log n)2 twin primes less than or equal to n. These probabilities are not independent.

A famous team of British mathematicians - hmm, another pair so to speak, Hardy and Littlewood, conjectured that the correct estimate should be the following:


But conjecture is not  a proof.   Recently, in March 2003, a new team of mathematicians  -  Dan Goldston of San Jose State University in California and Cem Yalcin Yildirim of Bogazici University in Istanbul, Turkey -  announced they had at least made progress in proving the suspicion that pairs of primes keep going off to infinity. 

Goldston attacked the proof in a novel manner.  In a sense, he avoided the question by working on a more manageable piece of the problem:

Taking into account the fact that the larger numbers become, the sparser the occurrence of primes,  is it possible to always find prime numbers that may not be twins, but that are much closer together than average?

Goldston and Yildrim believe they can prove the answer is an emphatic "Yes!"

Question on prime numbers

The Prime Number theorem is seemingly about as intertwined with the Zeta function as anything can be.

Thus, the distribution of primes is closely related to one of the most renowned unsolved questions in mathematics, the Riemann hypothesis, which concerns an infinite sum of numbers called the Zeta function.

Hilbert listed the Riemann hypothesis as the eighth problem in his famous address before the Paris Congress in 1900.  Smale listed it first in his selection of unsolved problems in 2000.

Currently, the Clay Mathematics Institute is offering $1 million to anyone who can prove the Riemann hypothesis.

Riemann zeta function

Experts believe that the new result may pave the way for a proof of the Zeta function and thus, the Riemann hypothesis.

Graph of primes
More graphs of primes

Study the two curves.  On the left, the number of primes between 0 and 100 would appear to be "locally" irregular.  When the scale is increased on the right graph, the curve would appear to be far smoother and far more predictable.

More graphs of primes More graphs of primes

For several years Chris K. Caldwell of the University of Tennessee, Martin, has created and maintained an indispensable web site on prime number theory.  The NCB was lucky to have had his timely Deposit # 19 a few weeks before the announcement of Goldston and Yildrim's new work.  Unfortunately, Goldston and Yildrim had to withdraw their findings.  But the search for primes continues to fascinate researchers.

The “largest” known prime on May, 2004 was also the forty-first known Mersenne prime.  This prime was found by Josh Findley of the National Oceanic and Atmospheric Administration.  The number can be expressed as
largest prime 2004
  and when written out, is made up of 7,235,733 according to the Great Internet Mersenne Prime Search, or GIMPS.

On February 26, 2005 a still larger prime was announced by GIMPS organizer George Woltman. He reported the forty-second Mersenne prime with the work being verified by Tony Reix.  This prime has 7,816,230 digits, the largest of any type, and may be expressed as

largest prime 2005
The NCB recommends the following sources:
How Many Primes Are There?  See  <  >
Frequently Asked Questions about Prime Numbers.  See  <  > 
What are the ten largest known  "twin primes?"  See  <  >
"Great Internet Mersenne Prime Search"  See <  >  and Focus, The Newsletter of the Mathematical Association of America, vol. 24, February, 2004, p. 4.
Jeffrey J. Wanko, "The Legacy of Marin Mersenne," Mathematics Teacher, vol. 98, No. 8, April, 2005, pp. 525-529.
and from the BBC News on April 4, 2003,  see  <  >
San Jose Mercury article Dan Goldston photo

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