A random A yields A+A of size closer to the high end of |A|2. Furthermore, any set A whose members are sufficiently separated has |A+A| close to this high end, and the high end is actually tight for sets A that are geometric progressions, e.g. the first n powers of two. What about the low end of |A|: what is the structure of sets A where |A+A| is close to |A|? We can see that this low-end estim...

x1 − 2x2 + x3 = 0 x2 − 2x3 + x4 = 0 are given by (x1, x2, x3, x4) = (±1,±1,±1,±1). Now, the above variety is an intersection between 2 quadrics in P. In general – i.e., except for the possibility of the variety being reducible or singular – an intersection between 2 quadrics in P is (isomorphic to) an elliptic curve and there is an algorithm that brings the curve to Weierstraß form by means of ...

It has been a long conjecture that there are arbitrarily long arithmetic progressions of primes. As of now, the longest known progression of primes is of length 26 and was discovered by Benoat Perichon and PrimeGrid in April, 2010 ([1]): 43142746595714191+23681770·223092870n for n = 0, 1, · · · , 25. Many mathematicians have spent years trying to prove (or disprove) this conjecture, and even mo...

We present a new numerical abstract domain. This domain automatically detects and proves bounds on the values of program variables. For that purpose, it relates variable values to a clock counter. More precisely, it bounds these values with the i-th iterate of the function [X 7→ α×X+β] applied onM , where i denotes the clock counter and the floating-point numbers α, β, and M are discovered by t...

Article history: Received 22 March 2012 Accepted 12 November 2012 Available online 27 November 2012

Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized it, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of this polynomial theorem. We focus on the case farthest f...

Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together w...

Freiman’s Theorem describes the structure of a set A under the condition that A+ A has size close to that of A. If P is a generalized arithmetic progression, then |P +P | is close to |P |. Freiman’s Theorem states the partial converse: if |P + P | is close to P then P must be contained in a small generalized arithmetic progression. The theorem may be stated as follows, and we will give the rema...

The following is a quick set of notes of some properties of Dirichlet characters, in particular, how they are used to prove the infinitude of primes in arithmetic progressions. These notes are from from An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash. As this is a modified snippet from the book, references to other parts of the book are displayed as ??. 1. Dirichlet Ch...

has only the solution (n, k, b, y, l) = (48, 3, 6, 140, 2) in positive integers n, k, b, y and l, where k, l ≥ 2, P (b) ≤ k and P (y) > k. Here, P (m) denotes the greatest prime factor of the integer m (where, for completeness, we write P (±1) = 1 and P (0) = ∞). Rather surprisingly, no similar conclusion is available for the frequently studied generalization of this equation to products of con...