Following footsteps of Gauss, Euler, Riemann, Hurwitz, Smith, Hardy, Littlewood, Hedlund, Khinchin and Chebyshev, we visit some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We experiment then with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These con...

The Iwasawa theory of CM fields has traditionally concerned modules that are abelian pro-p Galois groups with ramification allowed at a maximal set primes over p such the module is torsion. A main conjecture for an describes its codimension one support in terms p-adic L-function attached to ramification. In this paper, we study more general and potentially much smaller quotients exterior powers...

The best algorithms for discrete logarithms in Jacobians of algebraic curves of small genus are based on index calculus methods coupled with large prime variations. For hyperelliptic curves, relations are obtained by looking for reduced divisors with smooth Mumford representation [4]; for non-hyperelliptic curves it is faster to obtain relations using special linear systems of divisors [2, 3]. ...

In [4], Ash and Sinnott conjecture that any Galois representation having niveau 1 which satisfies a certain parity condition is attached in a specific way to a Hecke eigenclass in cohomology, and they make a prediction about exactly where the relevant cohomology class should lie. They give examples of reducible three-dimensional representations which appear to be attached to cohomology eigencla...

The Somos-5 sequence is defined by a0 = a1 = a2 = a3 = a4 = 1 and am = am−1am−4+am−2am−3 am−5 for m ≥ 5. We relate the arithmetic of the Somos-5 sequence to the elliptic curve E : y2 + xy = x3 + x2 − 2x and use properties of Galois representations attached to E to prove the density of primes p dividing some term in the Somos-5 sequence is equal to 5087 10752 .

A prime number sieve is an algorithm that nds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O(p n=(log log n) 2) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l > 1...

An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the rst prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: { it employs only (n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for xed n; { the transition from n to n + 1 takes on...

We prove two “large images” results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p > 13 not dividing d, provided that d is divisible by q (but d 6= q) with q = 2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing dD, where D = disc(...

a power digraph, denoted by $g(n,k)$, is a directed graph with $z_{n}={0,1,..., n-1}$ as the set of vertices and $l={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. in this paper, the structure of $g(2q+1,k)$, where $q$ is a sophie germain prime is investigated. the primality tests for the integers of the form $n=2q+1$ are established in terms of th...

In this paper, we investigate the large deviations of sums weighted random variables that are approximately independent, generalizing and improving some results Montgomery Odlyzko. We motivated by examples arising from number theory, including sequences pit, χ(p), χd(p), λf (p), Klq(a − n, b), where p ranges over primes, t varies in a interval, χ among all characters modulo q, χd quadratic atta...