Pedro Berrizbeitia (Ulam Visitor from Universidad Simon Bolivar)
X
Let an odd prime and a non quadratic residue mod . Let be the ring of integers of . If is coprime to , then has order in the multiplicative group , hence must be congruent mod to (why?). Using properties of the classical biquadratic residue symbol including the Biquadratic Reciprocity Law we present two different methods to compute the sign, which leads to what we call the "Norm Relation Theorem". As an application we prove the following rather curious property of Mersenne primes that we believe had not been noticed: Let be prime. Let be prime (i.e. a Mersenne prime). It is known that has integer solutions . Moreover, is unique with . We prove that !!
Examples:
; and $\frac{L}{2}= 1 \equiv 5 \pmod 4$
; $L=-10 \equiv 2\pmod 3$ and
The Biquadratic Symbol in . A new property of the Mersenne Primes
Mar. 29, 2016 1:10pm (MATH 3…
Kempner
Daniel Goncalves (Universidade Federal de Santa Caterina)
X
Symbolic dynamics is a fundamental area of dynamical systems, and is typically concerned with a finite alphabet , spaces of infinite sequences $A^\N$ or $A^\Z$, and subspaces of $A^\N$ or $A^\Z$ which are closed under the shift map (shift spaces). Over the years many approaches have been proposed to generalize shift spaces to the infinite alphabet case, one of the hurdles to overcome is the fact that the countable product of an infinite discrete space is not locally compact.
In this talk I will describe a recent approach to infinite shift spaces introduced by Ott, Tomforde and Willis. I intend to show that, while many of the finite alphabet symbolic dynamics results still hold for these infinite alphabet shifts (as for example the connection of conjugacy of edge shifts of graphs with isomorphism of the associated C*-algebras), some surprising results appear (for example there are (M+1)-step shifts not conjugate to an M-step shift).