Handout for Pi Day at Science Central by Professor Adam Coffman
... David H. Bailey’s page: http://crd.lbl.gov/˜dhbailey/pi/ David Blatner’s page: http://www.joyofpi.com/ St. Andrew’s History of Mathematics Archive: http://www-groups.dcs.st-and.ac.uk/˜history/ Did the Indiana state legislature really pass a law in 1897 declaring π to be equal to 3.2? No, it almost d ...
... David H. Bailey’s page: http://crd.lbl.gov/˜dhbailey/pi/ David Blatner’s page: http://www.joyofpi.com/ St. Andrew’s History of Mathematics Archive: http://www-groups.dcs.st-and.ac.uk/˜history/ Did the Indiana state legislature really pass a law in 1897 declaring π to be equal to 3.2? No, it almost d ...
A Readable Introduction to Real Mathematics
... is even. Therefore, the prime factorization of c2 includes 2, so the prime factorization of c also includes 2. Since a and b are odd natural numbers, there are nonnegative integers k1 and k2 such that a = 2k1 + 1 and b = 2k2 + 1. Then a2 = 4k12 + 4k1 + 1 and b2 = 4k22 + 4k2 + 1, so a2 + b2 ≡ 4(k12 + ...
... is even. Therefore, the prime factorization of c2 includes 2, so the prime factorization of c also includes 2. Since a and b are odd natural numbers, there are nonnegative integers k1 and k2 such that a = 2k1 + 1 and b = 2k2 + 1. Then a2 = 4k12 + 4k1 + 1 and b2 = 4k22 + 4k2 + 1, so a2 + b2 ≡ 4(k12 + ...
Proof Technique
... Methods of proving theorems: Existence Proofs Many theorems state that an object with certain properties exists, i.e., xP(x) where P is a predicate. A proof of such a theorem is called an existence proof. There are two kinds: Constructive Existence Proof: The proof is established be giving exa ...
... Methods of proving theorems: Existence Proofs Many theorems state that an object with certain properties exists, i.e., xP(x) where P is a predicate. A proof of such a theorem is called an existence proof. There are two kinds: Constructive Existence Proof: The proof is established be giving exa ...
Full text
... DIVISIBILITY TESTS IN N James E. Voss 129 Woodland Avenue #7, San Rafael, CA 94901 (Submitted April 1996-Final Revision September 1996) ...
... DIVISIBILITY TESTS IN N James E. Voss 129 Woodland Avenue #7, San Rafael, CA 94901 (Submitted April 1996-Final Revision September 1996) ...
MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS (III)
... (c) (Lagrange’s theorem) Every n ∈ N is a sum of four squares of integers. Those integers Tx = x(x + 1)/2 with x ∈ Z are called triangular numbers. Note that Tx = T−x−1 and 8Tx + 1 = (2x + 1)2 . In 1638 P. Fermat asserted that each n ∈ N can be written as a sum of three triangular numbers (equivalen ...
... (c) (Lagrange’s theorem) Every n ∈ N is a sum of four squares of integers. Those integers Tx = x(x + 1)/2 with x ∈ Z are called triangular numbers. Note that Tx = T−x−1 and 8Tx + 1 = (2x + 1)2 . In 1638 P. Fermat asserted that each n ∈ N can be written as a sum of three triangular numbers (equivalen ...
On the Sum of a Prime and a Square
... primes. Chen’s result is, at present, the closest one seems to be able to get towards a proof of the Goldbach conjecture, which famously asserts that every even integer greater than two can be written as the sum of two primes. It would be interesting to see if one could modify the explicit proof of ...
... primes. Chen’s result is, at present, the closest one seems to be able to get towards a proof of the Goldbach conjecture, which famously asserts that every even integer greater than two can be written as the sum of two primes. It would be interesting to see if one could modify the explicit proof of ...
Primality tests and Fermat factorization
... Even without such a result it turns out that in practice the Miller-Rabin test almost always detects compositeness very quickly. In fact, even if we only apply the Miller-Rabin test with the four bases 2, 3, 5, and 7 then there is only one composite number less than 2.5 × 1010 which will not be dete ...
... Even without such a result it turns out that in practice the Miller-Rabin test almost always detects compositeness very quickly. In fact, even if we only apply the Miller-Rabin test with the four bases 2, 3, 5, and 7 then there is only one composite number less than 2.5 × 1010 which will not be dete ...
Recap: complex numbers
... b. (x – 4)(x – (3 – 2i)), then either distribute or apply the formula from 2a, to get ...
... b. (x – 4)(x – (3 – 2i)), then either distribute or apply the formula from 2a, to get ...
The congruent number problem - Institut für Mathematik
... The beauty of number theory lies in the fact that many of its well-known conjectures can be easily stated and comprehended without having a solid background in mathematics, whereas their proof needs tools from many branches of higher and modern mathematics. One of the most widespread and best known ...
... The beauty of number theory lies in the fact that many of its well-known conjectures can be easily stated and comprehended without having a solid background in mathematics, whereas their proof needs tools from many branches of higher and modern mathematics. One of the most widespread and best known ...
![arXiv:math/0608068v1 [math.NT] 2 Aug 2006](http://s1.studyres.com/store/data/016444901_1-15a2bfb469428d063b2c2745e8d56343-300x300.png)
arXiv:math/0608068v1 [math.NT] 2 Aug 2006
... This equation has no non-trivial solutions, if c = 0 for instance, according to the Gauss’ characterization for the numbers that can be written as sums of two perfect squares. Our study of the existence of such triangles started with an American Mathematics Competition problem in the beginning of 20 ...
... This equation has no non-trivial solutions, if c = 0 for instance, according to the Gauss’ characterization for the numbers that can be written as sums of two perfect squares. Our study of the existence of such triangles started with an American Mathematics Competition problem in the beginning of 20 ...
(pdf)
... |α + β| ≤ max{|α|, |β|}. Definition 2.4. A class of equivalent valuations on k is called a place or prime. We will call the nonarchimedean primes finite, and the archimedean primes infinite. We make the following remark for future use. Remark 2.5. If |α| < |β|, then |α + β| = |β|. This is because |α ...
... |α + β| ≤ max{|α|, |β|}. Definition 2.4. A class of equivalent valuations on k is called a place or prime. We will call the nonarchimedean primes finite, and the archimedean primes infinite. We make the following remark for future use. Remark 2.5. If |α| < |β|, then |α + β| = |β|. This is because |α ...
An Example of Induction: Fibonacci Numbers
... An Example of Induction: Fibonacci Numbers Art Duval University of Texas at El Paso February 12, 2007 This short document is an example of an induction proof. Our goal is to rigorously prove something we observed experimentally in class, that every fifth Fibonacci number is a multiple of 5. As usual ...
... An Example of Induction: Fibonacci Numbers Art Duval University of Texas at El Paso February 12, 2007 This short document is an example of an induction proof. Our goal is to rigorously prove something we observed experimentally in class, that every fifth Fibonacci number is a multiple of 5. As usual ...
CS 103X: Discrete Structures Homework Assignment 3 — Solutions
... the well-ordering principle, if the premises of induction hold for a set A then A = N+ , which proves the induction principle. (b) Recall that in the last homework, we proved that strong induction follows from the induction principle, so proving well-ordering from strong induction will suffice. Agai ...
... the well-ordering principle, if the premises of induction hold for a set A then A = N+ , which proves the induction principle. (b) Recall that in the last homework, we proved that strong induction follows from the induction principle, so proving well-ordering from strong induction will suffice. Agai ...
PDF - UNT Digital Library
... proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. ...
... proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. ...
Full text
... For a positive integer n, let f(ri) be the number of multiplicative partitions of n. That is, f(n) represents the number of different factorizations of n, where two factorizations are considered the same if they differ only in the order of the factors. For example, /"(12) = 4, since 12 = 6*2 = 4 • 3 ...
... For a positive integer n, let f(ri) be the number of multiplicative partitions of n. That is, f(n) represents the number of different factorizations of n, where two factorizations are considered the same if they differ only in the order of the factors. For example, /"(12) = 4, since 12 = 6*2 = 4 • 3 ...
