The question of representing a positive integer as a sum of
a certain number of squares was historically a central problem
in the development of modern number theory. So our starting
point will be the classical problems of representability of
a positive integer as the sum of two, three, or four squares
(theorems of Fermat, Gauss, and Lagrange). While these problems
are solved completely, they are only a small step away from
unsolved, notoriously difficult problems involving sums of
cubes and higher powers (Waring's problem). Using a strong
form of Dirichlet's theorem for arithmetic progressions, we
will give Watson's short proof of Linnik's result that every
sufficiently large positive integer is the sum of at most
seven cubes. Then we will finally meet the powerful Hardy-Littlewood
method, through Hua's elegant proof that at most 2k+1
summands suffice to represent every sufficiently large positive
integer as the sum of kth powers. We will
conclude with a survey of the state of knowledge in Waring's
problem, Goldbach's Conjecture (that every even number larger
than 2 is the sum of two primes), and the theory of partitions.
Afternoon
(break-out) sessions will include problems meant to fill in
proofs from the lectures, as well as theorems and problems
from the theory of partitions. For example, we might attempt
to prove, in digestible steps, the Hardy-Ramanujan-Rademacher
formula for the number p(n) of partitions of n.
Reading list:
The first volume of Mel Nathansons's two Springer-Verlag
books, "Additive Number Theory", subtitled "The Classical
Bases", contains much of the material we wish to cover, and
are easily obtained.
Books
abound on partitions: Apostol's two Springer-Verlag books
on Analytic Number Theory are both in print and contain both
elementary and advanced theorems.
I
recommend reading the original paper of Hardy and Ramanujan
which was the genesis of the circle method: "Asymptotic formulae
in combinatory analysis" Proc. London Math. Soc. (2) 17 (1918),
75-115.
Parts of this and other articles will be required reading
during the week.
