Sums of Digits and the Distribution of Generalized Thue
... have been some new results about the distribution of the digit sum sequence (sq (k))∞ k=1 . Morgenbesser, Shallit, and Stoll [1] considered the classical Thue-Morse sequence (s2 (k) mod 2)∞ k=1 . They proved that the least number k satisfying s2 (d · k) ≡ 1 (mod 2) is at most d + 4, for every fixed ...
... have been some new results about the distribution of the digit sum sequence (sq (k))∞ k=1 . Morgenbesser, Shallit, and Stoll [1] considered the classical Thue-Morse sequence (s2 (k) mod 2)∞ k=1 . They proved that the least number k satisfying s2 (d · k) ≡ 1 (mod 2) is at most d + 4, for every fixed ...
Absolute Values for Rational Numbers and More Definition: A
... holds, we say the absolute value is non-archimedean. If not, the absolute value is called archimedean. Typically one allows the absolute value to take non-negative real values. There really isn’t any loss of generality here, though (see the comment before Lemma 8). Our definition also has the advant ...
... holds, we say the absolute value is non-archimedean. If not, the absolute value is called archimedean. Typically one allows the absolute value to take non-negative real values. There really isn’t any loss of generality here, though (see the comment before Lemma 8). Our definition also has the advant ...
Chapter 6
... compatible,” then we can paste them together to actually construct a global solution in Z. In fact it suffices to consider the cases where m = pe is a prime power. What it means for the solutions to be sufficiently compatible means is the following. Consider x2 + y 2 = 244. A solution to this in Z would ...
... compatible,” then we can paste them together to actually construct a global solution in Z. In fact it suffices to consider the cases where m = pe is a prime power. What it means for the solutions to be sufficiently compatible means is the following. Consider x2 + y 2 = 244. A solution to this in Z would ...
number systems
... In this section, we are going to study rational and irrational numbers from a different point of view. We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals. We will also explain how to visualise the representatio ...
... In this section, we are going to study rational and irrational numbers from a different point of view. We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals. We will also explain how to visualise the representatio ...
5.1.1 Integers - OpenTextBookStore
... written using a negative sign in front, such as −1, −5, and −30. These are read "negative one," "negative five," and "negative thirty." (The negative sign should not be read as "minus"; minus means subtraction.) The numbers greater than 0 are called positive numbers and can be written with or withou ...
... written using a negative sign in front, such as −1, −5, and −30. These are read "negative one," "negative five," and "negative thirty." (The negative sign should not be read as "minus"; minus means subtraction.) The numbers greater than 0 are called positive numbers and can be written with or withou ...
INTEGERS
... positive. When you add two negative integers or losses, you have a bigger loss so the result remains negative. When you add a positive integer and a negative integer, or a gain and a loss, your result is the DIFFERENCE of these values and the sign will be that of which was larger, the gain or loss. ...
... positive. When you add two negative integers or losses, you have a bigger loss so the result remains negative. When you add a positive integer and a negative integer, or a gain and a loss, your result is the DIFFERENCE of these values and the sign will be that of which was larger, the gain or loss. ...
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.