Shinichi Mochizuki, a mathematician at Kyoto University in Japan, has released a 500-page proof of the “abc” conjecture, a celebrated unsolved problem originally posed in 1985.<\/p>\n
Let sqp(n) denote the square-free part of an integer n, or in other words the product of the prime factors of n. For example, sqp(18) = 2 * 3 = 6 (here * denotes multiplication). The abc conjecture asserts that for integers a, b and c, where a + b = c, the ratio sqp(a*b*c)r<\/sup>\/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a = 3 and b = 125, so that c = 128, and r = 2, then sqp(a*b*c)=30 and sqp(abc)2<\/sup>\/c = 900\/128 = 7.03125.<\/p>\n It turns out that this conjecture encapsulates many other deep problems of number theory, including Fermat’s Last Theorem (which states that an<\/sup> + bn<\/sup> = cn<\/sup> has no integer solutions if n > 2), which was proven 17 years ago by Princeton mathematician Andrew Wiles. Brian Conrad of Stanford University notes that the abc conjecture “encodes a deep connection between the prime factors of a, b and a + b.”<\/p>\n Needless to say, Mochizuki’s long, multi-step proof must be carefully scrutinized by mathematicians in the field before it can be taken seriously. There have been many cases where a claimed proof of a major mathematical result was subsequently found to have serious flaws. Some general principles to follow in judging whether a mathematical result should be taken seriously are given in our previous Math Drudge blog<\/a>.<\/p>\n For additional details on the abc conjecture and Mochizuki’s manuscript, see the informative Nature article<\/a>, from which some of the above note was taken, Barry Cipra’s Science Now article<\/a>, Kenneth Chang’s NY Times article<\/a> and Jeremy Teitelbaum’s article in UConn Today<\/a>. Additional background information may be seen in the Wikipedia article<\/a> on the abc conjecture.<\/p>\n","protected":false},"excerpt":{"rendered":" Shinichi Mochizuki, a mathematician at Kyoto University in Japan, has released a 500-page proof of the “abc” conjecture, a celebrated unsolved problem originally posed in 1985.<\/p>\n Let sqp(n) denote the square-free part of an integer n, or in other words the product of the prime factors of n. For example, sqp(18) = 2 * 3 = 6 (here * denotes multiplication). The abc conjecture asserts that for integers a, b and c, where a + b = c, the ratio sqp(a*b*c)r\/c always has some minimum value greater than zero for any value of r greater than 1. For example, if <\/p>\n