S-parts of terms of integer linear recurrence sequences Yann
... The recurrence sequence (un )n≥0 is said to be degenerate if there are integers i, j with 1 ≤ i < j ≤ t such that αi /αj is a root of unity. We keep the above notation throughout the present paper. The case t = 1, that is, of sequences (f (n)an )n≥0 where f (X) is an integer polynomial and a a non-z ...
... The recurrence sequence (un )n≥0 is said to be degenerate if there are integers i, j with 1 ≤ i < j ≤ t such that αi /αj is a root of unity. We keep the above notation throughout the present paper. The case t = 1, that is, of sequences (f (n)an )n≥0 where f (X) is an integer polynomial and a a non-z ...
LOCAL FIELDS AND p-ADIC GROUPS In these notes, we follow [N
... We always have the trivial absolute value on any field K, defined by |x|v = 1 for every nonzero x ∈ K. In these notes, we will eliminate the trivial absolute value from discussion, and an absolute value will always assumed to be nontrivial. Example 1. Consider the field Q of rational numbers. There ...
... We always have the trivial absolute value on any field K, defined by |x|v = 1 for every nonzero x ∈ K. In these notes, we will eliminate the trivial absolute value from discussion, and an absolute value will always assumed to be nontrivial. Example 1. Consider the field Q of rational numbers. There ...
Periodicity and Correlation Properties of d
... hardware or software), (b) have good distribution properties which make them appear (statistically) to be ‘‘random’’, (c) have low crosscorrelation values so that each sequence may be separated from the others in the family, and (d) arise from some underlying algebraic structure so they can be analy ...
... hardware or software), (b) have good distribution properties which make them appear (statistically) to be ‘‘random’’, (c) have low crosscorrelation values so that each sequence may be separated from the others in the family, and (d) arise from some underlying algebraic structure so they can be analy ...
[Write on board:
... (1) Show you a different proof that sqrt(2) is irrational which has the virtue that, if you turn it around, gives you a procedure for generating rational numbers that get closer and closer to sqrt(2). (2) Discuss the holes in the rationals, and state an axiom capturing the idea that “there are no ho ...
... (1) Show you a different proof that sqrt(2) is irrational which has the virtue that, if you turn it around, gives you a procedure for generating rational numbers that get closer and closer to sqrt(2). (2) Discuss the holes in the rationals, and state an axiom capturing the idea that “there are no ho ...
Unit 1 Numbers Student Edition
... • Any number that can be written as a ratio of two integers. (Lesson 3.1) • The greatest factor shared by two or more numbers. (Lesson 2.1) • A diagram used to show the relationship between two sets or groups. (Lesson 3.1) • The set of all whole numbers and their opposites. (Lesson 1.1) • The distan ...
... • Any number that can be written as a ratio of two integers. (Lesson 3.1) • The greatest factor shared by two or more numbers. (Lesson 2.1) • A diagram used to show the relationship between two sets or groups. (Lesson 3.1) • The set of all whole numbers and their opposites. (Lesson 1.1) • The distan ...
Sample Segment
... Problem(Erdos-Szekeres). Let n ≥ 3 be an integer. Prove that there is an integer n0 > 0 such that in each set of at least n0 points in the plane, no three collinear, we can find n that are the vertices of a convex n-gon. Solution. We will use two lemmas: Lemma 1. Among any set of 5 points in the pl ...
... Problem(Erdos-Szekeres). Let n ≥ 3 be an integer. Prove that there is an integer n0 > 0 such that in each set of at least n0 points in the plane, no three collinear, we can find n that are the vertices of a convex n-gon. Solution. We will use two lemmas: Lemma 1. Among any set of 5 points in the pl ...
On integers of the forms k ± 2n and k2 n ± 1
... find no counterexample up to k < 3061 (see Jaeschke [24], Baillie, Cormack and Williams [2]). Erdős and Odlyzko [19] proved that the set of odd numbers k for which there exists a positive integer n with k2n + 1 being prime has positive lower asymptotic density in the set of all positive odd integer ...
... find no counterexample up to k < 3061 (see Jaeschke [24], Baillie, Cormack and Williams [2]). Erdős and Odlyzko [19] proved that the set of odd numbers k for which there exists a positive integer n with k2n + 1 being prime has positive lower asymptotic density in the set of all positive odd integer ...
Rational Numbers - Leon County Schools
... A rational number is a number that can be written as a ratio of two integers a and b, where b is not zero. For example, _47 is a rational ...
... A rational number is a number that can be written as a ratio of two integers a and b, where b is not zero. For example, _47 is a rational ...
Rational Numbers
... where the numerator is either 0 or a positive integer and the denominator, a positive integer. You compared two fractions, found their equivalent forms and studied all the four basic operations of addition, subtraction, multiplication and division on them. In this Chapter, we shall extend the number ...
... where the numerator is either 0 or a positive integer and the denominator, a positive integer. You compared two fractions, found their equivalent forms and studied all the four basic operations of addition, subtraction, multiplication and division on them. In this Chapter, we shall extend the number ...
P-adic number
In mathematics the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of ""closeness"" or absolute value. In particular, p-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p – the higher the power the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.p-adic numbers were first described by Kurt Hensel in 1897, though with hindsight some of Kummer's earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility.The p in p-adic is a variable and may be replaced with a prime (yielding, for instance, ""the 2-adic numbers"") or another placeholder variable (for expressions such as ""the ℓ-adic numbers""). The ""adic"" of ""p-adic"" comes from the ending found in words such as dyadic or triadic, and the p means a prime number.