So said a colleague of Princeton's Professor Andrew Wiles after Dr. Wiles culminated seven years of semi-secretive thinking about Fermat's Last Theorem with a proof which was lauded as one of the great achievements of mathematics in this half-century. In the seventeenth century, Fermat had teased future readers by scribbling in the margin of his book the impossibility of positive whole number solutions to a^n + b^n = c^n, other than n=2--Pythagoras's Theorem--and the fact he did not have room enough for the proof.
Professional mathematicians love their work, and they love to discuss their work with their colleagues. Mathematicians do not mind talking about the same puzzle for years. Many a time I have discussed with advanced students an extremely difficult problem whose solution was either ambiguous or beyond my capability or sometimes beyond that of any one I know. You could separate the real math lovers from the ones just along for the credit and the impressive transcript by their reaction to the lack of resolution of the discussion.
A common theme in our teaching is the frequent "why" expressed, sometimes in despair, by our students. Why are we factoring, why are we proving trig identities, why are we simplifying algebraic radicals (I don't know the answer to that one), etc. I try to tell them not to worry about utility: just enjoy it! Ultimately a practical use is found for nearly everything mathematicians do. The algorithm for searching for prime numbers, for instance, is a great aid for testing computer systems. Click here for more information about large primes.
A current favorite problem regards the relationship between the sparsity of primes and Riemann's related hypothesis. We know that primes get sparser as numbers get larger; Gauss discovered at age 15 that the increase is logarithmic. Gauss's student, Riemann, did his instructor one better by inventing a function of complex numbers (including imaginaries) called the zeta function that accurately predicts the distribution of primes, based on when the zeta function is equal to zero. (For the truly curious, click here for a description of the Riemann function.) Thus the distribution of primes has been likened to calculating the "zeros" of the zeta function. Recently Princeton's Freeman Dyson (sorry, Harvard and Yale) noticed that the calculations of these zeros was amazingly similar to the calculations on energy levels of large atoms. But it is only a clue. The current thinking is to look at families of functions that would include the Riemann zeta function as just one. It took more than 350 years to prove Fermat's Last Theorem. Perhaps our President could decree that "by the end of the decade . . . "
On a more applied level, statisticians are salivating at the prospect of the migration of euro coins around and among the twelve populous European Community countries: Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, and Spain. The problem resembles the simple taxi problem: taxis are located by varying proportions downtown, uptown, or midtown. We also know where riders will want to go, proportionally, from each of these areas to any of the other three areas. Using Markov Chains, made more accessible in the last ten years thanks to our graphing calculators, we can determine where all our cabs are going to eventually end up.
The probability theorists at the University of Amsterdam are hypothesizing that a fixed proportion of euros will leave France for Germany each month, for example, and that every other pair of countries will also have fixed percentages. Euros will also leave and return to each country. The result is a 12x12 matrix, which when raised to the nth power (the Markov chain), will hopefully give a solution to where the euros will settle. In July of 2002, when preliminary reports were coming in regarding euro migration, it was found that coins of larger denominations, for example 1 and 2 euros, were migrating more often than the smaller change, which may be resting in pockets, couches, or dressers.
There are many other issues to be concerned with: Seasonal changes, the coming potential war with Iraq, countries entering and leaving the European Union, etc. The Amsterdam group believes that within five to seven years, coin stability will be accomplished. A group of statisticians at the University of Freiberg, however, believes it will take several decades. In any case, the migration of the euro will be invaluable in studying more important types of migration, like epidemics.
Source: The New York Times