Uncovering a prime failure

Mathematicians have returned to the drawing board after what looked like a dramatic step forward in understanding prime numbers–those whole numbers divisible only by themselves and 1. Daniel A. Goldston of San Jose (Calif.) State University and Cem Y. Yildirim of Bogazii University in Istanbul have acknowledged a flaw in work they announced in March, which appeared to say that tight clusters of primes show up among whole numbers no matter how large the numbers are (SN: 3/29/03, p. 195: Prime Finding: Mathematicians mind the gap).

For more than a century, mathematicians have speculated that there are infinitely many pairs of “twin” primes, such as 11 and 13, which differ only by two. Goldston and Yildirim had created much excitement among number theorists when it appeared that they had come much closer to proving the twin-prime conjecture than others had managed to do in previous attempts.

Mathematicians Andrew Granville of the University of Montreal and Kannan Soundararajan of the University of Michigan in Ann Arbor discovered the error in Goldston and Yildirim’s work after realizing, to their surprise, that they could adapt the new result to prove in just a few additional lines that there are infinitely many pairs of primes differing by 12 or less–a finding almost as strong as the elusive twin-primes conjecture.

This result seemed too good to be true. Scrutinizing Goldston and Yildirim’s work line by line, Granville and Soundararajan found that one term in a complicated expression wasn’t as well behaved mathematically as Goldston and Yildirim had thought, making the final result fall through.

Goldston is now trying to assess which of the earlier findings still hold. “I think some interesting math is going to come out of this, whatever the outcome,” he says.


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