# Mathematical Digest

### Short Summaries of Articles about Mathematics

in the Popular Press

New result on arithmetic progressions of prime numbers:

"Hardy's prime problem solved," by Dana Mackenzie. *New Scientist*, 8 May 2004, page12.

"Primal Progress: Pattern hunters spy order among prime numbers," by EricaKlarreich. *Science NewsOnline,* week of 24 April 2004.

"Neues aus der Welt der Primzahlen," by George Szpiro. *Neue Zuercher Zeitung,* 28 April 2004.

These stories report on recent research shedding new light on an outstandingold problem. The research concerns arithmetic progressions of prime numbers.Arithmetic progressions are sequences of numbers in which each number differsby a fixed amount from its predecessor; one example is the sequence 3, 5, 7, inwhich each number in the sequence is 2 more than its predecessor. Thisparticular sequence is a prime arithmetic sequence because 3, 5, and 7 are allprime numbers. The question examined in the new research is whether there areinfinitely many prime arithmetic progressions, and how long such progressionscan be. The best result previously known was proved in 1939 by a mathematiciannamed van der Corput, who showed that there are infinitely many primearithmetic progressions of length 3. Ben Green and Terence Tao now claim theycan show that there are infinitely many prime arithmetic progressions of everyfinite length. Their work, posted on the web in the preprint "The primes contain arbitrarily longarithmetic progressions", has yet to be fully checked before being acceptedby mathematicians as correct. Still, the result has generated a good deal ofinterest and enthusiasm.

*--- Allyn Jackson*