Prime Numbers

14 Chapter 1 PRIMES!
Erdős–Turán conjecture: A subset S of the natural numbers that contains
a positive proportion of the natural numbers (that is, the limsup of the
proportion of S ∩ [1,x]in{1, 2,...,⌊x⌋} is positive) must contain arbitrarily
long arithmetic progressions. This result appears not to apply, since the primes
do not form a positive proportion of the natural numbers. However, Green
and Tao actually prove a version of Szemerédi’s theorem where the universe
of natural numbers is allowed to be somewhat generalized. They then proceed
to give an appropriate superset of the primes for which the Szemerédi analogue
is valid and for which the primes form a positive proportion. Altogether, the
Green–Tao development is quite amazing.
1.2 Celebrated conjectures and curiosities
We have indicated that the definition of the primes is so very simple, yet
questions concerning primes can be so very hard. In this section we exhibit
various celebrated problems of history. The more one studies these questions,
the more one appreciates the profundity of the games that primes play.
1.2.1 Twin primes
Consider the case of twin primes, meaning two primes that differ by 2. It is
easy to find such pairs, take 11, 13 or 197, 199, for example. It is not so easy,
but still possible, to find relatively large pairs, modern largest findings being
the pair
835335 · 2 39014 ± 1,
found in 1998 by R. Ballinger and Y. Gallot, the pair
361700055 · 2 39020 ± 1,
found in 1999 by H. Lifchitz, and (see [Caldwell 1999]) the twin-prime pairs
discovered in 2000:
2409110779845 · 2 60000 ± 1,
byH.Wassing,A.Járai, and K.-H. Indlekofer, and
665551035 · 2 80025 ± 1,
by P. Carmody. The current record is the pair
154798125 · 2 169690 ± 1,
reported in 2004 by D. Papp.
Are there infinitely many pairs of twin primes? Can we predict,
asymptotically, how many such pairs there are up to a given bound? Let
us try to think heuristically, like the young Gauss might have. He had guessed
that the probability that a random number near x is prime is about 1/ ln x,
and thus came up with the conjecture that π(x) ≈ x
dt/ ln t (see Section
2
1.1.5). What if we choose two numbers near x. If they are “independent prime