Full text
... expressed as a sum of elements of Xx,..., Xk. Let m be the least such number. Then we have Xl+-- + Xi1. We claim that mXi+l, then 0
... expressed as a sum of elements of Xx,..., Xk. Let m be the least such number. Then we have Xl+-- + Xi
COMP4690 Tutorial
... The powers of an integer a, modulo n a, a2, a3, … (mod n) If a and n are relatively prime, based on Euler’s theorem, we have aø(n) ≡ 1 mod n a, a2, a3, … will have a repeated pattern E.g., ø(5)=4, 3ø(5)=81≡1 mod 5 ...
... The powers of an integer a, modulo n a, a2, a3, … (mod n) If a and n are relatively prime, based on Euler’s theorem, we have aø(n) ≡ 1 mod n a, a2, a3, … will have a repeated pattern E.g., ø(5)=4, 3ø(5)=81≡1 mod 5 ...
14(4)
... for a natural algebraic and geometric setting for their analysis. In this way many known results are unified and simplified and new results are obtained. Some of the results extend to Fibonacci representations of higher order, but we do not present these because we have been unable to extend the the ...
... for a natural algebraic and geometric setting for their analysis. In this way many known results are unified and simplified and new results are obtained. Some of the results extend to Fibonacci representations of higher order, but we do not present these because we have been unable to extend the the ...
35(2)
... In one of his famous results, Fermat showed that there exists no Pythagorean triangle with integer sides whose area is an integer square. His elegant method of proof is one of the first known examples in the history of the theory of numbers where the method of infinite descent is employed. Mohanty [ ...
... In one of his famous results, Fermat showed that there exists no Pythagorean triangle with integer sides whose area is an integer square. His elegant method of proof is one of the first known examples in the history of the theory of numbers where the method of infinite descent is employed. Mohanty [ ...
OPEN DIOPHANTINE PROBLEMS 1. Diophantine Equations 1.1
... This conjecture may be rephrased as follows. Given an increasing sequence of positive integers n1 < · · · < nk for which the product n1 · · · nk has fewer than k distinct prime factors, there is a prime p in the range n1 ≤ p ≤ nk . The equivalence of this with the original formulation follows from t ...
... This conjecture may be rephrased as follows. Given an increasing sequence of positive integers n1 < · · · < nk for which the product n1 · · · nk has fewer than k distinct prime factors, there is a prime p in the range n1 ≤ p ≤ nk . The equivalence of this with the original formulation follows from t ...
Full text
... E x a m p l e 1: The partition (1 4) is a 2-subcomplete partition of 5. To see this we list 2representations of numbers from 1 to 10; 1 = 1, 2 = 2 - 1 + 0 - 4 , 3 = —1 • 1 + 1 • 4,4 = ...
... E x a m p l e 1: The partition (1 4) is a 2-subcomplete partition of 5. To see this we list 2representations of numbers from 1 to 10; 1 = 1, 2 = 2 - 1 + 0 - 4 , 3 = —1 • 1 + 1 • 4,4 = ...
The k-Binomial Transforms and the Hankel Transform
... The case k = 0 must be dealt with separately because 00 would occur in the formulas otherwise. Our definitions effectively take 00 to be 1. These turn out to be “good” definitions, in the sense that all the results discussed subsequently hold under our definitions for the k = 0 case. When k = 0, the ...
... The case k = 0 must be dealt with separately because 00 would occur in the formulas otherwise. Our definitions effectively take 00 to be 1. These turn out to be “good” definitions, in the sense that all the results discussed subsequently hold under our definitions for the k = 0 case. When k = 0, the ...
Modeling Chebyshev`s Bias in the Gaussian Primes as a Random
... Definition 5. a and b are said to be relatively prime, or coprime if gcd(a, b) = 1. Two prime numbers, p and q, will always be coprime to each other. A composite number, a, will be coprime to prime number, p, if and only if a is not a multiple of p. Definition 6. Euler’s totient function, denoted φ( ...
... Definition 5. a and b are said to be relatively prime, or coprime if gcd(a, b) = 1. Two prime numbers, p and q, will always be coprime to each other. A composite number, a, will be coprime to prime number, p, if and only if a is not a multiple of p. Definition 6. Euler’s totient function, denoted φ( ...
Additive decompositions of sets with restricted prime factors
... Wirsing [39] showed that almost all sets of integers are asymptotically additively irreducible. But it seems very difficult to prove whether a given set S that “occurs in nature” is asymptotically additively irreducible or not. Shortly we will describe some of the results obtained previously. The m ...
... Wirsing [39] showed that almost all sets of integers are asymptotically additively irreducible. But it seems very difficult to prove whether a given set S that “occurs in nature” is asymptotically additively irreducible or not. Shortly we will describe some of the results obtained previously. The m ...
Notes for Number Theory
... Solution. All we need is a single counterexample. Setting a = 5, b = 3, and c = 7 does the trick. 2. Find a divisor of 359951 besides 1 and itself. Solution. The only answers are 593 and 607. It would be tedious to find them by checking divisors starting with 2, 3, etc. A better way is to use the al ...
... Solution. All we need is a single counterexample. Setting a = 5, b = 3, and c = 7 does the trick. 2. Find a divisor of 359951 besides 1 and itself. Solution. The only answers are 593 and 607. It would be tedious to find them by checking divisors starting with 2, 3, etc. A better way is to use the al ...
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... Here are your solutions to HW # 2, I’m still working on the grading but thought you would enjoy these solutions. Enjoy!!! 2.2.3 There are three inequalities to prove, the middle inequality is obvious since A is nonempty. So suppose A ⊆ B. We must prove inf(B) ≤ inf (A), and sup(A) ≤ sup(B). To prove ...
... Here are your solutions to HW # 2, I’m still working on the grading but thought you would enjoy these solutions. Enjoy!!! 2.2.3 There are three inequalities to prove, the middle inequality is obvious since A is nonempty. So suppose A ⊆ B. We must prove inf(B) ≤ inf (A), and sup(A) ≤ sup(B). To prove ...
The maximum modulus of a trigonometric trinomial
... Dirac measures. Section 11 replaces our computation of the Sidon constant in a general context; it describes the initial motivation for this research. Part of these results appeared previously, with a different proof, in [19, Chapter II.10] and in [17]. Notation. Throughout this article, λ1 , λ2 and ...
... Dirac measures. Section 11 replaces our computation of the Sidon constant in a general context; it describes the initial motivation for this research. Part of these results appeared previously, with a different proof, in [19, Chapter II.10] and in [17]. Notation. Throughout this article, λ1 , λ2 and ...
A Polynomial Time Algorithm for Prime Recognition
... Kayal and Saxena had been students of Agrawal at the time. Agrawal received the Clay Research Award of the Clay Mathematical Institute which is presented ”as its highest recognition of general achievement in mathematical research to one or more mathematicians”. Agrawal and his students were invited ...
... Kayal and Saxena had been students of Agrawal at the time. Agrawal received the Clay Research Award of the Clay Mathematical Institute which is presented ”as its highest recognition of general achievement in mathematical research to one or more mathematicians”. Agrawal and his students were invited ...
Sequences, Series, and the Binomial Theorem
... This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/) license. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it avai ...
... This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/) license. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it avai ...
An Introduction to Complex Analysis and Geometry
... experience. A second example concerns certain definite integrals. The techniques of complex analysis allow for stunningly easy evaluations of many calculus integrals and seem to lie within the realm of science fiction. This book is meant to be readable, but at the same time it is precise and rigorou ...
... experience. A second example concerns certain definite integrals. The techniques of complex analysis allow for stunningly easy evaluations of many calculus integrals and seem to lie within the realm of science fiction. This book is meant to be readable, but at the same time it is precise and rigorou ...
Sample pages 2 PDF
... 1. One of the oldest mathematical problems concerns perfect numbers. A positive integer N is called perfect, if it equals the sum of its proper divisors, i.e., the equality σ (N) = 2N holds1 . It had been noted already by Euclid that if the numbers 2p − 1 and p are both prime, then 2p−1 (2p − 1) is ...
... 1. One of the oldest mathematical problems concerns perfect numbers. A positive integer N is called perfect, if it equals the sum of its proper divisors, i.e., the equality σ (N) = 2N holds1 . It had been noted already by Euclid that if the numbers 2p − 1 and p are both prime, then 2p−1 (2p − 1) is ...
