Ivars Peterson's MathTrek

January 18, 1999

# Millennium Primes

The arrival of the year 1999 and the impending transition to 2000 has captured the attention of people who like to play with integers.

Consider 1999. It's a prime number, meaning that it is evenly divisible only by itself and 1 (see Prime Talent, July 6, 1998). As it happens, 1997 is also a prime number. The pair 1997 and 1999 is an example of what are known as twin primes. The next pair doesn't come up until 2027 and 2029.

No one has yet established whether there are infinitely many twin primes, p and p + 2. It's one of the major unsolved problems in number theory (see Prime Theorem of the Century, Dec. 23, 1996).

Clearly, 2000 is not a prime number. Mathematician John B. Cosgrove of St. Patrick's College in Dublin, Ireland, however, has just discovered a 2,000-digit prime number. He teaches an undergraduate course on number theory and cryptography and has always included several primality tests and theorems. For the coming term, Cosgrove decided to introduce his students to a primality-testing scheme based on a result called Pocklington's theorem.

Using the mathematics computer program Maple, "I wanted to put together some numerical illustrations--quirky and beautiful examples, not humdrum ones--of [Pocklington's theorem]," Cosgrove says. From a wide range of possible starting numbers, he selected a few to begin his search for primes.

By the end of the first day, his computer program had proved several numbers to be prime, including one that was 952 digits long. Cosgrove pressed on in the hope of finding a so-called titanic prime. The term "titanic prime" applies to any prime with at least 1,000 digits (see http://www.utm.edu/research/primes/glossary/TitanicPrime.html).

"Last year, I was able to present my third-year students right at the start of their course with two titanic primes," Cosgrove says. "It created a certain excitement."

Cosgrove's program continued its search throughout the night. By the next morning, it had come up with another prime, which happened to have 2000 digits.

Cosgrove didn't immediately connect his newly found prime with the millennium, partly because he is one of those who insist that the next millennium starts on Jan. 1, 2001. Nonetheless, he decided his students would probably find this particular example interesting. And maybe someone will find a 2,001-digit prime.

My own feeling about the proper date of the millennium is that it doesn't matter. You can define it to be whatever you want, and you can celebrate it whenever you like. It has no intrinsic meaning. The numbering of years is a cultural artifact based on some rather arbitrary decisions made along the way.

In that spirit, the "official Web page of the prime number of days until the end of the millennium" provides a truly primal countdown to Dec. 31, 1999. Consulting the list, you would find that Jan. 16 is 349 days and Jan. 18 is 347 days before the end of the millennium. Twin primes!

For those interested in dates on cornerstones or in movie copyright notices, there remains the issue of how to express the number 1999 in Roman numerals. The year 2000, written as MM, has a wonderfully palindromic brevity. In contrast, 1999 gives rise to a mess. Is it MCMXCIX, MDCCCCLXXXXVIIII, MCMXCVIIII, or even MIM?

The Romans never had to face a 1999 situation, and they weren't completely consistent in the way they expressed numbers. They did have rules, though some were applied more strictly than others. One rule, however, was never broken. The Romans represented units, tens, hundreds, and thousands as separate items in their numbers. This strict place rule meant that, when the subtraction rule was applied, only I (1) could be used to the left of V (5) or X (10); only X could be used to the left of L (50) or C (100); and only C could be used to the left of D (500) or M (1,000) . Hence, 99 could be represented as XCIX but never as IC. Similarly, 999 couldn't be IM, and 1999 couldn't be MIM.

That still leaves several possibilities, all apparently consistent with Roman usage: MCMXCIX, MCMXCVIIII, and MDCCCCLXXXXVIIII.

I like the most concise form. Best wishes for MCMXCIX!

References:

Ribenboim, P. 1996. The New Book of Prime Number Records. New York: Springer-Verlag.

Ruane, M.E. 1998. MIM? A.D. 1999 brings a classical question for legions in the arena of Roman numerals. Washington Post (Dec. 31).

John Cosgrove's account of the discovery of a 2000-digit prime is available in the archives of the number theory discussion group at http://listserv.nodak.edu/scripts/wa.exe?A2=ind9901&L=nmbrthry&F=&S=&P=54. His Maple worksheets for performing the necessary calculations can be found at http://web.usna.navy.mil/~wdj/crypto.htm.

Information about titanic primes and the largest known primes is available at http://www.utm.edu/research/primes/largest.html.

A list of the prime days among the 1,000 days preceding Dec. 31, 1999, is available at http://www.khoral.com/staff/ele/play/primes.html.