Joint Colloquium: Norman Wildberger -- Primes, Complexity and Computation: How Big Number theory resolves the Goldbach Conjecture

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 – the investigation of large numbers exceeding the computational
capacity of our computers, starting from Ackermann’s and Goodstein’s
hyperoperations, to the presenter’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.  We’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.