Type: Seminar

Modified: Wed 5 Oct 2016 1623; Sat 15 Oct 2016 2036

Distribution: World

Expiry: 5 Oct 2017

CalTitle1: Primes, Complexity and Computation: How Big Number theory resolves the Goldbach Conjecture

Auth: leo@121.209.16.59 (ltzo2369) in SMS-WASM

The Goldbach Conjecture states that every even number greater than 2 can be written as the sum of two primes, and it is one of the most famous unsolved problems in number theory. In this lecture, we look at the problem from the novel point of view of Big Number theory â€“ the investigation of large numbers exceeding the computational capacity of our computers, starting from Ackermannâ€™s and Goodsteinâ€™s hyperoperations, to the presenterâ€™s successor-limit hierarchy which parallels ordinal set theory. This will involve a journey to a distant, seldom visited corner of number theory that impinges very directly on the Goldbach conjecture, and also on quite a few other open problems. Along the way we will meet some seriously big numbers, and pass by vast tracts of dark numbers. We will also bump into philosophical questions about the true nature of natural numbers---and the arithmetic that is possible with them. Weâ€™ll begin with a review of prime numbers and their distribution, notably the Prime Number Theorem of Hadamard and de la Vallee Poussin. Then we look at how complexity interacts with primality and factorization, and present simple but basic results on the compression of complexity. These ideas allow us to slice through the Gordian knot and resolve the Goldbach Conjecture: using common sense, an Aristotelian view on the foundations of mathematics as espoused by James Franklin and his school, and back of the envelope calculations.