Pattern hunters spy order among prime numbers.Mathematicians have taken a step forward in understanding patterns within the primes, numbers divisible DIVISIBLE. The susceptibility of being divided.
2. A contract cannot, in general, be divided in such a manner that an action may be brought, or a right accrue, on a part of it. 2 Penna. R. 454. only by 1 and themselves. According to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. the new work, the population of prime numbers There are infinitely many prime numbers. The first 500 are listed below, followed by lists of the first prime numbers of various types in alphabetical order. The first 500 prime numbers
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71 contains an infinite collection of arithmetic progressions--number sequences in which each term differs from the preceding one by the same fixed amount.
For example, in the sequence 3, 5, 7, each prime number is 2 more than the preceding one. Another example of such a sequence is 5, 11, 17, 23, 29, in which successive primes differ by 6.
For centuries, mathematicians have wondered how many arithmetic progressions such as these exist among the set of prime numbers and how long the progressions can get. In 1939, the Dutch mathematician Johannes van der Corput Johannes Gualtherus van der Corput (Rotterdam, September 4, 1890 - Amsterdam, September 16, 1975) was a Dutch mathematician, working in the field of analytic number theory. proved that there are infinitely many progressions with three terms. Whether longer progressions are infinitely plentiful or limited in number and size had remained a matter of conjecture.
The longest known progressions have just 22 terms and lie in remote stretches of the number line. For instance, one 22-term progression starts at 11,410,337,850,553 and the difference between successive terms is 4,609,098,694,200.
Now, a pair of mathematicians offers a proof that in one fell swoop demonstrates that there are infinitely many prime progressions of every finite length. Ben Green of the University of British Columbia Locations
The Vancouver campus is located at Point Grey, a twenty-minute drive from downtown Vancouver. It is near several beaches and has views of the North Shore mountains. The 7. in Vancouver and Terence Tao Terence Chi-Shen Tao (陶哲軒) (born 17 July 1975, Adelaide, South Australia) is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. of the University of California, Los Angeles UCLA comprises the College of Letters and Science (the primary undergraduate college), seven professional schools, and five professional Health Science schools. Since 2001, UCLA has enrolled over 33,000 total students, and that number is steadily rising. report their findings in a preprint pre·print
Something printed and often distributed in partial or preliminary form in advance of official publication: a preprint of a scientific article.
tr.v. that they posted on the Internet on April 8.
It may be months before mathematicians have finished checking the proof. Nevertheless, Green and Tao's report has sparked excitement in the math community.
Proving anything about progressions with more than three terms had seemed beyond reach, says Andrew Granville Andrew Granville is a British mathematician, working in the field of number theory.
He has been a faculty member at the Université de Montréal since 2002. Before moving to Montreal he was a mathematics professor at University of Georgia (UGA) from 1991 until 2002. , a mathematician at the University of Montreal. "If they've succeeded in breaking that barrier, it's an extraordinary achievement."
In their proof, Green and Tao considered how the primes relate to a larger set of numbers that the pair calls almost-primes--numbers that are a product of at most 10 primes. Although prime numbers are scarce among the whole numbers, they are more plentiful in the narrower setting of the almost-primes.
Green and Tao showed that the sequence of almost-primes is pseudorandom pseu·do·ran·dom
Of, relating to, or being random numbers generated by a definite, nonrandom computational process. ; roughly speaking, the almost-primes are "nicely spread out all over the place." Green says. The mathematicians then deduced that the prime numbers are arranged within the spread of almost-primes with enough regularity to ensure that the overall sequence of primes does indeed contain arithmetic progressions of every length.
Unfortunately, the new insight about prime numbers won't give mathematicians much of a handle on where to look for long progressions. All it does is guarantee that for any length k, there's a prime progression of that length that starts at a number smaller than 2 raised to the power 2 raised to the power 2 raised to the power 2, repeated k times. But for even small values of k, that upper limit quickly becomes astronomical.
"The bounds our argument gives are ridiculously bad," Green acknowledges. They are essentially useless, even for mathematicians with access to the world's most powerful computers, he says.
Green and Tao are now trying to pin down the location of prime arithmetic progressions more precisely. "It's going to force us to understand the primes better," Green says.