Prime conjecture verified to new heights

Like the elements in chemistry, prime numbers serve as building blocks in the mathematics of whole numbers. Evenly divisible only by themselves and one, primes are a rich source of speculative ideas that mathematicians often find simple to state but difficult to prove.

The Goldbach conjecture, devised by historian and mathematician Christian Goldbach in 1742, proposes that every even number is the sum of two primes; for example, 8 = 3 + 5. No one has yet proved the conjecture, but a researcher in Germany has now verified that all even numbers up to 4 x 10¹⁴ satisfy this relationship.

Jürg Richstein of the Institute of Informatics at the University of Giessen reports his results in a paper to be published in Mathematics of Computation. He used a variant of an older method, making it possible to perform the computations with a network of relatively modest computers.

“Such computations prove the truth of the Goldbach conjecture for a finite set of even numbers,” says Herman te Riele of the National Research Institute for Mathematics and Computer Science in Amsterdam. “Such computations [also] could possibly give a hint toward the proof or disproof of the Goldbach conjecture for the infinite set of all the even numbers.”

Progress in proving the Goldbach conjecture has been slow. In the best effort to date, a mathematician proved in the 1960s that beyond some large number, every even integer may be written as the sum of a prime number and a number that is either a prime or a product of two primes.

In recent decades, mathematicians and computer scientists have turned to computers to test the conjecture against larger and larger even numbers. In 1998, te Riele and his coworkers used a Cray supercomputer as well as improved computational techniques to push the upper limit to 10¹⁴. They also checked the conjecture for a sample of larger even numbers, up to 10³⁰⁰.

In extending that effort, Richstein’s innovative approach also enabled him to investigate the number of different ways in which an even number can be expressed as the sum of two primes. In general, as the even integers get larger, the number of such prime-pairs increases.

For example, there are two such pairs that add up to 20, yet five pairs that add up to 48. This observation suggests that the likelihood of finding the exceptional even number that is not the sum of any two primes diminishes as one searches among ever larger even numbers.

In recent work, Richstein found the number of such sums for all even integers up to 5 x 10⁸. For example, 291,400 distinct pairs of primes sum to 100 million. This evidence supports the Goldbach conjecture, Richstein notes.

Mathematical proofs of conjectures, however, require more than overwhelming numerical evidence. A prize of $1 million recently offered by a British publisher, promoting a novel in which a mathematician seeks to prove the Goldbach conjecture, could well stimulate the additional research needed to crack the problem.