*abc*conjecture which, if true, may solve Fermat's Last Theorem from a new angle.

From

*Nature*:

TheFermat's Last Theorem, famously scribbled in the margin of Fermat's Arithmetica, states that no three positive integersabcconjecture, proposed independently by David Masser and Joseph Oesterle in 1985, might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. “Theabcconjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem,” says Dorian Goldfeld, a mathematician at Columbia University in New York. “If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century.”

*a*,

*b*, and

*c*can satisfy the equation

*a*

^{n}+

*b*

^{n}=

*c*

^{n}for any integer value of

*n*greater than two. Fermat claimed his proof was too large to fit in the margin and problem remained one of the greatest mysteries of mathematics for 358 years.

Previously, Andrew Wiles spent seven years of his life toiling over Fermat's Last Theorem before offering his proof, and he was knighted for the successful effort. He proved Fermat's Last Theorem in 1994.

Mochizuki's work, while building on the work that came before him, as mathematics does, differs in that he has developed techniques that "very few other mathematicians fully understand and that invoke new mathematical ‘objects’ — abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices." This is the exciting part in relation to the Imagination Age: imagination is required to understand the relationship between objects, concepts and their mathematical realities.