New results on the prime gap conjecture

The twin prime and prime gap conjectures

Paul Erdos, one of the 20th century’s most famed mathematicians (1913-1996) was well-known for offering prizes for the solution of various mathematical problems. Although most of these prizes were a token USD$25 or the like, he did offer a UDS$10,000 prize for the solution to a certain conjecture of his on prime gaps. Until very recently, it had not been solved.

The twin prime conjecture is that the gap between successive primes, namely g_n = p_n+1 – p_n, is two for infinitely many n. This has not been proven, and is one of the greatest open questions of mathematics. A stronger conjecture is that g_n = 2k infinitely often, for any positive integer k (the weaker twin prime conjecture is the case k = 1).

So how large (and how small) can these gaps be? Both lower and upper bound results have been proved, but “best” bounds are not known in either case.

Although some might think that it is an embarrassment to modern mathematics that such easily stated conjectures remain unresolved, numerous examples could be mentioned in almost any field of mathematics. For example, it is still not known whether the decimal or binary expansions of pi (or any other well-known constant of mathematics) are “normal,” in the sense of any particular digit or string of digits appearing with exactly the frequency, in the limit, that one would expect of a “random” sequence.

Upper bounds

Progress has definitely been achieved lately on upper bounds for prime gaps. In 2005, Daniel Goldston, Janos Pintz and Cem Yildirim proved that

liminf_n (g_n / log p_n) = 0,

or, in other words, that given any epsilon greater than zero, there are infinitely many n for which that quotient g_n / log p_n is less than epsilon (here log means natural logarithm). This was subsequently improved to the statement that

liminf_n g_n / (sqrt (log p_n) (log log p_n)²) < ∞. Then in May 2013, Yitang Zhang proved that liminf g_n < 70,000,000, or, in other words, that there are infinitely many gaps that do not exceed 70 million. At this point, the problem was taken up by the Polymath project, an online collaboration of volunteer mathematicians including, for instance, Terence Tao, one of the inaugural recipients of the Breakthrough Prize in Mathematics. By early 2014, this team had reduced the 70,000,000 figure to a mere 246.

This latest result is truly a remarkable advance, particularly given the fact that many mathematicians worldwide had worked on this problem, mostly in vain, for decades. Also, the involvement of the Polymath project, involving scores of professional mathematician working online, may be a harbinger of future mathematical research work. The rise of Internet-based collaborations such as Polymath, the increasingly pervasive utilization of large-scale computation in mathematical research, and the rise of “compelling plausibility arguments” in lieu of traditional formal proofs, were explored in a July 2014 workshop on experimental mathematics held at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University.

Lower bounds

Lower bounds have also been established for the sequence of prime gaps. In 1938, Scottish mathematician Robert A. Rankin proved that for C = e^γ = 1.78107241799… (here γ is Euler’s constant), the inequality

g_n > C log n log log n log log log log n / (log log log n)²

holds for infinitely many n. This was later shown to hold for C = 2 e^γ = 3.56214483598… The expression on the right-hand side, without the C, grows not much faster than log n, and is not even positive until n is quite large. For n = 10³, 10⁶, 10⁹, 10¹² and 10¹⁵, this expression has the values -12.82944…, -1.37136…, 5.283443…, 11.59908…, 17.96242…, respectively.

With regard’s to Rankin’s inequality above, repeated logarithms commonly appear in number theory, particularly in the mathematics of prime numbers. According to Terence Tao, a joke among number theorists is “What does a drowning number theorist say?” Answer: “Log log log log …”

Paul Erdos conjectured that the constant C in the above could be taken to be any arbitrarily large value. In other words, he conjectured that for any C > 0,

g_n > C log n log log n log log log log n / (log log log n)²

holds infinitely often. Erdos (somewhat rashly, as he later recounted) offered a USD$10,000 prize for either a proof or a disproof.

Two proofs of Erdos’ conjecture

Now, two separate teams of mathematicians have submitted papers with proofs of this conjecture. The first paper is by Kevin Ford, Ben Green, Sergei Konyagin and Terence Tao. The proof presented by Tao and his colleagues “combines existing arguments with a random construction covering a set of primes by arithmetic progression.”

Separately, James Maynard of Oxford University has submitted a paper that proves this result “by incorporating sieve ideas based on the recent results on small gaps between primes … into the Erdos-Rankin method.”

With regards to the USD$10,000 prize for the solution of Erdos’ conjecture, Erdos obviously cannot pay because he died in 1996. But U.C. San Diego mathematician Ronald Graham, who collaborated with Erdos on numerous occasions, has offered to provide the monetary award. Tao is also considering offering additional prize money for anyone who can significantly improve these results.

Additional details, with some background on both Tao and Maynard, are given in a nicely written feature article in Quanta Magazine by Erica Klarreich.