Chapter 08: Divisibility and Prime Numbers
... Given any natural number n > 1, there is a prime factorization of n. This is a consequence of the following general argument. It is based on a fundamental property of natural numbers (Section 4.7): if there is a natural number with a certain property, then there is a least one. Suppose there is a ‘b ...
... Given any natural number n > 1, there is a prime factorization of n. This is a consequence of the following general argument. It is based on a fundamental property of natural numbers (Section 4.7): if there is a natural number with a certain property, then there is a least one. Suppose there is a ‘b ...
The factor-difference set of integers
... In other words, each integer 4p2k will have to contain in its factor-difference set the n integers {4ki + 2 : i = 1, . . . , n}. (An affirmative answer to Erdős’ question was found by Piepmeyer [3].) In Section 3 we deal with the intersection question. In Section 4 we investigate questions related ...
... In other words, each integer 4p2k will have to contain in its factor-difference set the n integers {4ki + 2 : i = 1, . . . , n}. (An affirmative answer to Erdős’ question was found by Piepmeyer [3].) In Section 3 we deal with the intersection question. In Section 4 we investigate questions related ...
Olympiad Hamilton Paper
... corner. So consider an alternative numbering system: in each shaded cell write the number of cells with which it shares an edge or corner; leave each unshaded cell blank. For example, for the shading pattern given in the question we obtain: ...
... corner. So consider an alternative numbering system: in each shaded cell write the number of cells with which it shares an edge or corner; leave each unshaded cell blank. For example, for the shading pattern given in the question we obtain: ...
Lesson 4.2 Irrational Numbers Exercises
... v) All rational numbers can be written as the quotient of two integers. Since all natural numbers can be written as a quotient with denominator 1, some rational numbers are natural numbers. The statement is true. b) iii) The number 0 is a whole number, but it is not a natural number. iv) π is irrati ...
... v) All rational numbers can be written as the quotient of two integers. Since all natural numbers can be written as a quotient with denominator 1, some rational numbers are natural numbers. The statement is true. b) iii) The number 0 is a whole number, but it is not a natural number. iv) π is irrati ...
Powers of rationals modulo 1 and rational base number systems
... proving whether they form a dense set or not, is a frustrating question: “This very old problem of Pisot and Vijayaraghavan is still unanswered.” writes Michel Mendès France in [16] and he goes on: “Pisot, Vijayaraghavan and André Weil did however show that there are infinitely many limit points.” ...
... proving whether they form a dense set or not, is a frustrating question: “This very old problem of Pisot and Vijayaraghavan is still unanswered.” writes Michel Mendès France in [16] and he goes on: “Pisot, Vijayaraghavan and André Weil did however show that there are infinitely many limit points.” ...
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.