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You can’t always solve a mathematical problem by reducing it to something you’ve already solved. Sometimes, you need to invent an entirely new field of mathematics. Last month, Shinichi Mochizuki of Kyoto University in Japan announced that a new field he’s been developing for several years—which he calls Inter-universal Teichmüller theory—has proved a famous conjecture in number theory known as the “abc conjecture.” But the abc conjecture is only the beginning: If Mochizuki’s theory proves correct, it will settle a raft of open problems in number theory and other branches of math.
The conjecture grows out of a seemingly trivial equation: a + b = c. Unlike the equation a2 + b2 = c2, which requires some algebraic finesse to produce solutions (not to mention its famously unsolvable cousin a n + bn = cn with exponent n greater than two), the equation a + b = c essentially solves itself: Just pick two numbers a and b, add them together, and voilà.
But when you bring in prime numbers, things get interesting. In the mid 1980s, mathematicians David Masser of the University of Basel in Switzerland and Joseph Oesterlé of Pierre and Marie Curie University in Paris observed that when a and b are divisible by small primes raised to large powers—numbers such as a = 210 and b = 34—their sum c tends to factor into large primes to small powers. (In this example, 1024 + 81 = 1105 = 5 x 13 x 17.) The abc conjecture describes this connection in precise mathematical language, highlighting how the underlying “tension” between the operations of addition and multiplication produce such lopsided equations: many small primes on one side, a few relatively large primes on the other.