5,182 research outputs found
Using Dynamical Systems to Construct Infinitely Many Primes
Euclid's proof can be reworked to construct infinitely many primes, in many
different ways, using ideas from arithmetic dynamics.
After acceptance Soundararajan noted the beautiful and fast converging
formula: Comment: To appear in the American Mathematical Monthl
Squares in arithmetic progressions and infinitely many primes
We give a new proof that there are infinitely many primes, relying on van der
Waerden's theorem for coloring the integers, and Fermat's theorem that there
cannot be four squares in an arithmetic progression. We go on to discuss where
else these ideas have come together in the past.Comment: To appear in the American Mathematical Monthl
Primitive prime factors in second order linear recurrence sequences
For a class of Lucas sequences , we show that if is a positive
integer then has a primitive prime factor which divides to an odd
power, except perhaps when . This has several desirable
consequences.Comment: To Andrzej Schinzel on his 75th birthda
What is the best approach to counting primes?
As long as people have studied mathematics, they have wanted to know how many
primes there are. Getting precise answers is a notoriously difficult problem,
and the first suitable technique, due to Riemann, inspired an enormous amount
of great mathematics, the techniques and insights permeating many different
fields. In this article we will review some of the best techniques for counting
primes, centering our discussion around Riemann's seminal paper. We will go on
to discuss its limitations, and then recent efforts to replace Riemann's theory
with one that is significantly simpler.Comment: To appear in a volume dedicated to the MAA Centennial in 201
Beyond the LSD method for the partial sums of multiplicative functions
The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the
partial sums of a multiplicative function whose prime values are
on average. In the literature, the average is usually taken to be with
a very strong error term, leading to an asymptotic formula for the partial sums
with a very strong error term. In practice, the average at the prime values may
only be known with a fairly weak error term, and so we explore here how good an
estimate this will imply for the partial sums of , developing new techniques
to do so.Comment: Addressed referee's comments; added some references; corrected and
simplified the proof of Theorem 9. 26 page
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