# Proof Positive

From Harvard Magazine:

work to understand the architecture of the universe, they sometimes uncover connections in mysterious places. So it is with Smith professor of mathematics Richard L. Taylor, whose work connects two discrete domains of mathematics: curved spaces, from geometry, and modular arithmetic, which has to do with counting. Taylor has spent his career studying this nexus, and recently proved it is possible to use one domain to solve complex problems in the other. “It just astounded me,” he says, “that there should be a connection between these two things, when nobody could see any real reason why there should be.”

This is not the first instance of finding in geometry an elegant explanation for a seem- ingly unrelated phenomenon. Scholars during the Renaissance, seeking a mathematical basis for our conceptions of beauty, fingered the so-called Golden Ratio (approximately 1.6 to 1). Some analyses find the ratio in structures—most famously the Parthenon—built centuries before its first written formulation. More recently, scientists have found that the faces people find most beautiful are those in which the proportions conform most closely to the ratio. The geometry-arithmetic connection explored by Taylor solves another puzzle that has enticed mathematicians across centuries. In 1637, French mathematician Pierre de Fermat scrawled in a book’s margin a theorem involving equations like the one in the Pythagorean theorem (a2 + b2 = c2), but with powers higher than two. Fermat’s theorem said such equations have no solutions that are whole numbers, either positive or negative. Go ahead, try—it is impossible to find three integers, other than zero, that work in the equation a3 + b3 = c3.

The French mathematician also wrote that he had discovered a way to prove this—but he never wrote the proof down, or if he did, it was lost. For more than 350 years, mathematicians tried in vain to prove what became known as Fermat’s Last Theorem. They could find lots of examples that fit the pattern, and no counterexamples, but could not erase all doubt until Princeton University mathematician Andrew Wiles presented a proof in 1993.

His discovery made the front page of the New York Times, but six months later, the Times reported that another mathematician had found a mistake in the new proof.

More here.