What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§2. THE GOLDBACH CONJECTURE AND TWIN PRIMES

(page 30)

Goldbach’s conjecture that every even number greater than 2 is a sum of two primes, and the closely associated “twin prime conjecture” that there exist infinitely many primes p for which p + 2 is also prime, remain open. However, a good deal more is now known about both questions.

One of the most powerful methods for tackling some problems in number theory is complex analysis, an idea that goes back to Euler and was exploited in particular by Riemann in his study of the zeta-function ζ(s) (see p. 480). From 1920 onwards Godfrey H. Hardy and John E. Littlewood developed the application of analytic number theory, as it came to be called, to questions about the representation of numbers as sums of numbers of special kinds. In 1937 I. M. Vinogradov used their methods to prove that every sufficiently large odd number is a sum of three primes. This improved upon his four-primes result, cited by Courant and Robbins on p. 31, which was proved in 1934. As they state, his theorem applies only to “sufficiently large” numbers—numbers greater than some particular value n₀—and his proof does not specify how large n₀ should be. In 1956 K. G. Borodzkin filled this gap by showing that n₀ = exp(exp(16.038)) suffices, where exp(x) = e^x. Several mathematicians used Vinogradov’s method to prove that “almost all” even numbers are the sum of two primes; that is, the proportion of such numbers up to some limit ntends to 100% as n tends to infinity.

In 1919 Viggo Brun introduced a different approach, the “sieve method,” which generalizes the sieve of Eratosthenes (see p. 25). He used it to prove that every sufficiently large even integer is a sum of two numbers, each being a product of at most nine primes. A series of improvements to this theorem, made by a number of people, followed. For example, in 1937 G. Ricci proved that every sufficiently large even integer is a sum of two numbers, one being a product of at most two primes and the other a product of at most 366 primes. P. Kuhn used combinatorial ideas of A. A. Buchstab to prove that every sufficiently large even integer is a sum of two numbers, each being a product of at most four primes. In 1957 Wang Yuan proved that every sufficiently large even integer is a sum of a prime and a product of at most three primes, on the assumption that the Generalized Riemann Hypothesis holds.

The classical Riemann Hypothesis, another of Hilbert’s 23 problems and still arguably the biggest unsolved question in the whole of mathematics, concerns the Riemann zeta function ζ(s) when the variable s is complex. Specifically, it states that if ζ(s) = 0 and s is not real, then s = 1/2 +iy for some real y. The consequences of proving this statement would be spectacular: they would revolutionize number theory and algebraic geometry. Moreover, any method for solving such a problem would almost certainly extend to other important variants such as the Generalized Riemann Hypothesis, a considerably stronger statement of the same general kind. Because the Riemann Hypothesis and its generalizations are such a significant obstacle to progress, number theorists have developed the habit of sending out exploratory tendrils into the territory that lies beyond by basing some of their work on the explicit assumption that the Riemann Hypothesis, or a generalization, is true. One justification of this approach is the possibility that it might lead to a contradiction, thereby exposing the Riemann Hypothesis as false, but this is mere rationalization. The number theorists are impatient; they cannot wait to see what lies beyond the Big Obstacle.

Sometimes, once such territory has been explored, new possibilities open up which allow the assumption to be dispensed with. In 1948, without assuming the Generalized Riemann Hypothesis, Alfred Rényi proved that every sufficiently large even integer is a sum of a prime and a product of at most c primes for some fixed but unknown c. In 1961 M. B. Barban showed that c = 9 suffices. In 1962 Pan Cheng Dong reduced this to c = 5; shortly afterwards Barban and Pan independently reduced it to c = 4; and in 1965 Buchstab proved the theorem when c = 3. Finally, in 1966, Chen Jing Run improved the sieve method further and proved the theorem with c = 2. That is, every sufficiently large even integer is a sum of a prime and a product of at most two primes—”prime plus almost-prime.” This is the closest result to the full Goldbach conjecture that is currently known.

The twin prime conjecture has been approached in a similar spirit. Brun’s 1919 paper also proved that there are infinitely many numbers p such that both p and p + 2 are a product of at most nine primes. In line with improvements to Brun’s result on the Goldbach conjecture, there were similar improvements to his work on the twin prime conjecture. In 1924 Rademacher reduced Brun’s number nine to seven. Buchstab reduced it further, to six in 1930 and to five in 1938. In a paper of 1957 Wang noted enigmatically that “corresponding results of twin primes problem have also been obtained,” which in the context amounts to a claim that there are infinitely many numbers p such that both p and p + 2 are a product of at most three primes. Assuming the Generalized Riemann Hypothesis, he showed in 1962 that there exist infinitely many primes p such that p + 2 is a product of at most three primes. In 1965, without making this assumption, Buchstab proved that for some fixed c there exist infinitely many primes p such that p + 2 is a product of at most c primes. Chen’s 1973 paper proved that c = 2 suffices, and again this is the closest known result to the twin primes conjecture. It seems unlikely that current methods can push the result much closer: a genuinely novel idea is needed.