Possible proof is potential breakthrough in number theory

Like Fermat’s theorem, the abc conjecture refers to equations of the form aâ+â bâ=â c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers, but 16 and 18âbeing divisible by 4 ² and 3 ², respectivelyâare not.

The ‘square-free’ part of a number n, sqp( n), is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. For instance, sqp(18)â=â2âÃâ3â=â6.

If you’ve got that, then you should get the abc conjecture. It concerns a property of the product of the three integers abcâmdash;or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers aâ+â bâ=â c, the ratio of sqp( abc) ^r / c always has some minimum value greater than zero for any value of r greater than 1. For example, if aâ=â3 and bâ=â125, so that câ=â128, then sqp( abc)â=â30 and sqp( abc) ²/ câ=â900/128. In this case, in which râ=â2, sqp( abc) ^r / c is nearly always greater than 1, and always greater than zero.

If Mochizuki’s proof is correct, it provides solutions for an entire set of problems, including Fermat’s Last Theorem, involving relations between whole numbers. However, because of the length and complexity of the proof, it may be some time before the result is verified.