Almost all primes have a multiple of small Hamming weight

Recent results of Bourgain and Shparlinski imply that for almost all primes $p$ there is a multiple $mp$ that can be written in binary as $mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k,$ with $k=66$ or $k=16$, respectively. We show that $k=6$ (corresponding to Hamming weight $7$) suffices. We also prove there are infinitely many primes $p$ with a multiplicative subgroup $A=\subset \mathbb{F}_p^*$, for some $g \in \{2,3,5\}$, of size $|A|\gg p/(\log p)^3$, where the sum-product set $A\cdot A+ A\cdot A$ does not cover $\mathbb{F}_p$ completely

Elementary treatment of pa±pb+1=x2p^a \pm p^b + 1 = x^2

We give a shorter simpler proof of a result of Szalay on the equation $2^a + 2^b + 1 = x^2$. We give an elementary proof of a result of Luca on the equation of the title for prime $p > 2$. The elementary treatment is made possible by a lemma which is also used to obtain a bound on $n$ in the equation $x^2 + C = y^n$, where $x$ and $y$ are primes or prime powers, and $2 | C$; the bound depends only on the primes dividing $C$

On quadratic residue codes and hyperelliptic curves

A long standing problem has been to develop "good" binary linear codes to be used for error-correction. This paper investigates in some detail an attack on this problem using a connection between quadratic residue codes and hyperelliptic curves. One question which coding theory is used to attack is: Does there exist a c<2 such that, for all sufficiently large $p$ and all subsets S of GF(p), we have |X_S(GF(p))| < cp?Comment: 18 pages, no figure

Linear Fractional p-Adic and Adelic Dynamical Systems

Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL (2, Q), SL (2, Q) and SL (2, Z) groups. In particular, we investigate behavior of these adelic systems when fixed points are rational. It is shown that any of these rational fixed points is p-adic indifferent for all but a finite set of primes. Thus only for finite number of p-adic cases a rational fixed point may be attractive or repelling. It is also shown that real and p-adic norms of any nonzero rational fixed point are connected by adelic product formula.Comment: 17 page