8,607 research outputs found
Perfect powers in elliptic divisibility sequences
It is shown that there are finitely many perfect powers in an elliptic
divisibility sequence whose first term is divisible by 2 or 3. For Mordell
curves the same conclusion is shown to hold if the first term is greater than
1. Examples of Mordell curves and families of congruent number curves are given
with corresponding elliptic divisibility sequences having no perfect power
terms. The proofs combine primitive divisor results with modular methods for
Diophantine equations.Comment: 16 page
Perfect powers in products of terms of elliptic divisibility sequences
Diophantine problems involving recurrence sequences have a long history and
is an actively studied topic within number theory. In this paper, we connect to
the field by considering the equation \begin{align*} B_mB_{m+d}\dots
B_{m+(k-1)d}=y^\ell \end{align*} in positive integers with
and , where is a fixed integer and
is an elliptic divisibility sequence, an important class
of non-linear recurrences. We prove that the above equation admits only
finitely many solutions. In fact, we present an algorithm to find all possible
solutions, provided that the set of -th powers in is given. (Note
that this set is known to be finite.) We illustrate our method by an example.Comment: To appear in Bulletin of Australian Math Societ
Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4
We say a number is flat if it can be written as a non-trivial power of 2 times an odd squarefree number. The power is the “exponent” and the number of odd primes the “length”. Let N be flat and 4-perfect with exponent a and length m. If a ≢ 1 mod 12, then a is even. If a is even and 3 ∤ N then m is also even. If a ≡ 1 mod 12 then 3 | N and m is even. If N is flat and 3-perfect and 3 ∤ N, then if a a ≡ 1 mod 12, a is even. If
a ≡ 1 mod 12 then m is odd. If N is flat and 3 or 4-perfect then it is divisible by at least one Mersenne prime, but not all odd prime divisors are Mersenne. We also give some conditions for the divisibility by 3 of an arbitrary even 4-perfect number
On the degree of non-Markovianity of quantum evolution
We propose a new characterization of non-Markovian quantum evolution based on
the concept of non-Markovianity degree. It provides an analog of a Schmidt
number in the entanglement theory and reveals the formal analogy between
quantum evolution and the entanglement theory: Markovian evolution corresponds
to a separable state and non-Markovian one is further characterized by its
degree. It enables one to introduce a non-Markovinity witness -- an analog of
an entanglement witness -- and a family of measures -- an analog of Schmidt
coefficients -- and finally to characterize maximally non-Markovian evolution
being an analog of maximally entangled state. Our approach allows to classify
the non-Markovianity measures introduced so far in a unified rigorous
mathematical framework.Comment: 5 page
- …