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# Mathematical Digest

### Short Summaries of Articles about Mathematics

in the Popular Press

"Prime Finding: Mathematicians mind the gap," by Erica Klarreich. *Science News,* Week of March 29, 2003; Vol. 163, No. 13.

"Prime-number analysis sees advances." *USA Today,* 25 March 2003, page 7D.

"SJSU math professor breaks barrier: He and Turkish colleague solve number theory," by Glennda Chui. *San Jose Mercury News,* 26 March 2003.

These articles describe a major advance in prime number theory obtained by two mathematicians, Daniel A. Goldston and Cem Y. Yildirim. One of the most famous questions in number theory is the so-called twin primes conjecture, which states that there are infinitely many prime numbers that differ by two. Examples of twin primes are 17 and 19, and 41 and 43. But are there infinitely many? No one knows the answer, though mathematicians suspect it's yes. In thinking about this conjecture, mathematicians have explored more general questions about the average spacing between prime numbers. Klarreich's article sums up nicely the Goldton-Yildirim result: "Given any fraction, no matter how small, there are infinitely many prime pairs closer together than that fraction of the average." This result definitively improves on all previous results about the average spacing of primes and may provide new information about the zeta function, which is at the heart of the celebrated Riemann Hypothesis.

*--- Allyn Jackson*