BY DANA MACKENZIE
A famous conjecture in number theory
has stood unproven for more than 150
years, but for the second time this year
mathematicians have gotten dramatically closer to proving it. With a strategy
others had abandoned, a young mathematician has narrowed the gap between
primes, in hopes of ultimately proving
the twin prime conjecture.
His work has also shown that prime
numbers bunch together in clusters as
well as in pairs.
The twin prime conjecture asserts
that infinitely many pairs of prime numbers are separated by only two, as are 3
and 5 or 1,997 and 1,999. Prime numbers
are divisible only by themselves and 1.
Number theorists know that as numbers
get larger, primes gradually get sparser.
Nonetheless, if the twin prime conjecture is true, pairs of primes separated by
two continue to pop up forever.
Number theorists have been revved up
since May, when Yitang “Tom” Zhang,
a University of New Hampshire mathematician, announced a partial solution to the twin prime problem, which
will appear in Annals of Mathematics.
Researchers have since been refining
his methods, getting closer to solving the
problem (SN: 12/28/13, p. 32).
In October, James Maynard, a postdoctoral researcher at the University
of Montreal, announced at a workshop
in Germany that he had improved
Zhang’s estimate for prime pairs. In
doing so, he had also resuscitated a previously discarded technique to prove
that primes also occur in clusters.
Maynard posted his results November
19 at arXiv.org.
Zhang had shown that pairs of
primes with gaps no larger than 70 million keep occurring forever. He was the
first to demonstrate a finite cap on the
Math & technology
may go all the way
Major advance made toward
proving legendary conjectures
minimum gaps between primes.
Even though 70 million is a long way
from 2, it’s an even longer way from
infinity. “He didn’t waste time trying
to get a smaller number,” says John
Friedlander of the University of
Toronto. Zhang knew that just getting
any finite number would be a sensational result.
But other mathematicians enthusiastically pounced on the problem.
In June, hardly a day passed without a new world record for the smallest prime gap that repeats infinitely.
By the end of July, the record stood
at 4,680. The researchers made these
gains using tweaks and refinements of
Zhang’s argument, not by breaking new
Meanwhile, as he was finishing up his
doctorate at the University
of Oxford, Maynard pondered a related question: Do
primes occur infinitely only
in pairs like cherries or also
in bunches like grapes?
In addition to twin
primes with their gaps of
2, triplets may occur in
baskets of width 6. Such a
basket catches 7, 11 and 13
or 2,707, 2,711 and 2,713,
and the harvest presumably
continues forever. This statement is the
prime triplets conjecture, and there is
an analogous conjecture for prime quadruplets and larger.
Zhang’s work says nothing about
prime triplets or other multiples. It
uses a prime detection tool called the
one-dimensional Selberg sieve that, for
reasons not completely understood, can
detect only pairs of primes. The sieve
is a theoretical function (not an actual
device or program) that weights numbers roughly according to their probability of being prime.
The original multidimensional
Selberg sieve, discovered in the 1940s
by the Norwegian mathematician Atle
Selberg, does not have the pair limita-
tion. Zhang had borrowed the simpler
one-dimensional version from 2005
work on prime gaps by mathematicians
Daniel Goldston, János Pintz and
That paper was itself a modification
of an argument from a 2003 paper by
Goldston and Yıldırım, which had used
the multidimensional sieve to address
the twin prime conjecture but had to be
retracted because of a fatal error.
Because of this history, many number
theorists considered the multidimensional sieve inherently flawed. “I think
the issue was perhaps more psychological than technical,” says Terence Tao
But Maynard, who had nothing to
lose because he was essentially done
with his doctoral dissertation, decided
to play around with the discredited
sieve. Nearly on the eve of his dissertation defense, he figured out how to
make it work.
Using the sieve, Maynard
found a list of 105 numbers
(0, 10, 12, 24, … 594, 598,
600) that serves as a template for prime pairs. This
means that infinitely many
numbers, when added to
the numbers in the template, produce at least two
primes. (For example, 3
works because 3+0 and
Even more important, Maynard
showed that longer templates exist for
prime triplets, quadruplets and higher-order prime multiples, thereby establishing world records that had not even
existed for prime bunches of varying
size. (Tao independently used the same
ideas to reach a similar but slightly
“Maynard’s proof is much shorter
than Zhang’s and much more elementary, and it produces stronger results,”
Friedlander says. “But the proofs are
quite different, so at some point in time
Zhang’s ideas and Maynard’s could be
incorporated together to get results
stronger than either one got alone. It’s
a wonderful situation.” s
gap between primes
that appears infinitely,
according to conjecture
Smallest prime gap
now proven to