![]() cryptography) involving primes, have also proven to be of little utility for these sorts of problems again, see Remark 2. The various primality tests in the literature, while useful for practical applications (e.g. One of course can create non-useful functions of this form, such as the ordered parameterisation that maps each natural number to the prime, or one could invoke Matiyasevich’s theorem to produce a polynomial of many variables whose only positive values are prime, but these sorts of functions have no usable structure to exploit (for instance, they give no insight into any of the Landau problems listed above see also Remark 2 below). One of the main reasons that the prime numbers are so difficult to deal with rigorously is that they have very little usable algebraic or geometric structure that we know how to exploit for instance, we do not have any useful prime generating functions. We also now have some understanding of the barriers we are facing to fully resolving each of these problems, such as the parity problem this will also be discussed in the course. Īll four of Landau’s problems remain open, but we have convincing heuristic evidence that they are all true, and in each of the four cases we have some highly non-trivial partial results, some of which will be covered in this course. There are infinitely many primes of the form.Legendre’s conjecture: for every natural number, there is a prime between and.Twin prime conjecture: there are infinitely many pairs which are simultaneously prime.Even Goldbach conjecture: every even number greater than two is expressible as the sum of two primes. ![]() ![]() The type of results about primes that one aspires to prove here is well captured by Landau’s classical list of problems: As with my previous courses, I will place lecture notes online on my blog in advance of the physical lectures. I will list the topics I intend to cover in this course below the fold. As the name suggests, this is a course covering many of the analytic number theory techniques used to study the distribution of the prime numbers. In the winter quarter (starting January 5) I will be teaching a graduate topics course entitled “ An introduction to analytic prime number theory“.
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