Number theory offers a host of problems that are remarkably easy to state but fiendishly difficult to solve. Many of these questions and conjectures feature prime numbers–integers evenly divisible only by themselves and 1.
For instance, primes often occur as pairs of consecutive odd integers: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. So-called twin primes are scattered throughout the list of all prime numbers. There are 16 twin prime pairs among the first 50 primes. The largest known twin prime is the 32,220-digit pair 318032361 x 2107001 +/–1, found recently by David Underbakke and Phil Carmody.
Although most mathematicians believe that there are infinitely many twin primes, no one has yet proved this conjecture to be true. Indeed, the twin prime conjecture is considered one of the major unsolved problems in number theory. It was even mentioned in the 1996 movie A Mirror Has Two Faces, which starred Barbra Streisand.
In 1919, Norwegian mathematician Viggo Brun (1885–1978) proved that if you add together the reciprocals of successive twin primes, the sum converges to a specific numerical value now called Brun’s constant. The fact that this sum converges demonstrates that twin primes are relatively scarce–even though there may be infinitely many of them! In contrast, the sum of the reciprocals of all primes diverges.
In recent years, twin primes and Brun’s constant have been the focus of several massive computing efforts. One of these projects was initiated in 1993 by Thomas R. Nicely of Lynchburg, Va. He has now extended his enumeration of twin primes and the sum of their reciprocals to 3.155 x 1015, having found 3,471,427,262,962 twin primes. His estimate of Brun’s constant stands at 1.90216 05823 10.
Early in the course of his computations, Nicely also happened upon a flaw in a Pentium microprocessor that caused certain types of arithmetic errors. His discovery forced the chip’s manufacturer to embark on a costly program to replace the faulty processors.
Another twin primes computing effort, led by computer scientist Patrick Fry and his colleagues at the Rensselaer Polytechnic Institute in Troy, N.Y., harnessed unused time on many computers to enumerate twin primes up to 1 x 1016. In a span of just over 2 years, the team surpassed Nicely’s numbers and eventually reached a total of 8,494,836,459,690 twin primes.
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In 1996, Jörg Richstein of the Institute of Informatics at the University of Giessen in Germany and his collaborators also mounted efforts to find twin primes. They identified all twin primes up to 1014, confirming the values obtained by Nicely.
These collections of data are of great interest to investigators examining the distribution of twin primes among all primes and the gaps between consecutive twins. The data show that, like primes, twin primes tend to become more scarce as their numerical value increases.
In a new analysis, physicists P.F. Kelly and Terry Pilling of North Dakota State University in Fargo focus on the distribution of twin primes within the set of primes, rather than their distribution within the set of all whole numbers. They have determined how many “singleton” primes occur between consecutive pairs of twin primes. For example, there are no singleton primes between the twin pairs (5, 7) and (11, 13), and there is one prime (23) between the pairs (17, 19) and (29, 31). Two primes (47 and 53) occur between the pairs (41, 43) and (59, 61). The number of singleton primes between consecutive pairs is termed the prime separation.
Kelly and Pilling determined all prime separations between pairs of twins in various ranges selected from integers between 79,561 and 4,020,634,603. They then determined the relative frequency of occurrence of each separation in a given range. They observed that, for all sufficiently large ranges, the relative frequencies appear to obey a surprisingly simple logarithmic relationship.
“This remarkable behavior is perfectly characteristic of a completely random system,” Kelly and Pilling comment in a paper describing their results. “We infer that as one approaches each prime number in the sequence of primes following a twin, the likelihood of it being the first member of the next twin prime is constant!”
Kelly and Pilling liken the numerical behavior of twin primes to that a radioactive substance, where the likelihood of one of its atoms decaying in any short time interval is fixed. “The measured slope of the line fit to our data provides a decay constant which is particular to the twin primes,” they say.
Interestingly, this “constant” probability isn’t universal, the researchers note. It varies in a rather simple way with the length of the sequence of primes in the selected range of whole numbers.
Marek Wolf of the Institute of Theoretical Physics at the University of Wroclaw in Poland, has been studying gaps between consecutive twin primes, typically measured as the arithmetical distance between the last primes constituting consecutive twins. For example, the distance between the twins (29, 31) and (17, 19) is 31 – 19, or 12. The distances are multiples of 6 because all twins are of the form 6k +/–1.
Performing a computer search to count the number of primes between consecutive twins in a range up to 1.76 x 1013, Marek extracted data that confirmed the relationship found by Kelly and Pilling. His analysis also provided additional insights into how this heuristic formula arises.
Twin primes continue to fascinate!