1 How to Read and Do Proofs [1]
... These methods are used when you want to show that there is only one object with a certain property such that something happens. There are two methods: direct and indirect. The following table summarizes the direct uniqueness method. When to use it What to assume What to conclude How to do it ...
... These methods are used when you want to show that there is only one object with a certain property such that something happens. There are two methods: direct and indirect. The following table summarizes the direct uniqueness method. When to use it What to assume What to conclude How to do it ...
Full text
... and from these considerations we have proved the following: T h e o r e m 2,3: The total number of partitions Into at most N parts In which every even smaller than the largest part appears at least once is equal to F2N+2The family given in (2.2) has also an interesting property at q = —1. At this po ...
... and from these considerations we have proved the following: T h e o r e m 2,3: The total number of partitions Into at most N parts In which every even smaller than the largest part appears at least once is equal to F2N+2The family given in (2.2) has also an interesting property at q = —1. At this po ...
Fibonacci numbers, alternating parity sequences and
... by a permutation of its rows and a permutation of its columns, where O is a p ×q, nonempty, zero matrix with p +q =n. Lemma 2.2. Any S-matrix has total support and is fully indecomposable. Proof. For m > 1, let H be the m × m matrix defined by hij = 1 if and only if: |i − j | = 1, or i = j ∈ {1, m} ( ...
... by a permutation of its rows and a permutation of its columns, where O is a p ×q, nonempty, zero matrix with p +q =n. Lemma 2.2. Any S-matrix has total support and is fully indecomposable. Proof. For m > 1, let H be the m × m matrix defined by hij = 1 if and only if: |i − j | = 1, or i = j ∈ {1, m} ( ...
Solutions to Practice Final 1 1. (a) What is φ(20 100) where φ is
... we see that 4x3 − 3x = 43 . therefore 16x3 − 12x = 3. If x is constructible then so is y = 2x which must satisfy 2y 3 − 6y = 3, 2y 3 − 6y − 3 = 0. this is a cubic polynomial with rational coefficients. If it has a constructible root it must have a rational one. Suppose pq is a rational root of 2y 3 ...
... we see that 4x3 − 3x = 43 . therefore 16x3 − 12x = 3. If x is constructible then so is y = 2x which must satisfy 2y 3 − 6y = 3, 2y 3 − 6y − 3 = 0. this is a cubic polynomial with rational coefficients. If it has a constructible root it must have a rational one. Suppose pq is a rational root of 2y 3 ...
SOME ASYMPTOTIC FORMULAS IN THE THEORY OF NUMBERS(`)
... The factor d defined by (1.4) is the largest divisor Q*(n) of n contained in L. If T is a nonvacuous subset of L, then one may define, analogous to (1.2), Presented to the Society, March 29,1961; receivedby the editors August 21,1962. 0) This research was supported in part by the National ScienceFou ...
... The factor d defined by (1.4) is the largest divisor Q*(n) of n contained in L. If T is a nonvacuous subset of L, then one may define, analogous to (1.2), Presented to the Society, March 29,1961; receivedby the editors August 21,1962. 0) This research was supported in part by the National ScienceFou ...
29(1)
... for the pn . It is immediate that the two sequences coincide for n = 0, 1, so they must be the same for all n. Like Proposition 1, this is equivalent to a well-known statement about the Un. (b) Lemma 1 implies that 2 cos(tu/n) is a root of p for t = 1, 2, . .., n - 1 and, since the cosine is strictl ...
... for the pn . It is immediate that the two sequences coincide for n = 0, 1, so they must be the same for all n. Like Proposition 1, this is equivalent to a well-known statement about the Un. (b) Lemma 1 implies that 2 cos(tu/n) is a root of p for t = 1, 2, . .., n - 1 and, since the cosine is strictl ...
complete lecture notes in a pdf file - Mathematics
... can constitute either entire or part of a third year project in mathematics. Occasionally, there are also “computer projects” for students who are able to program. The aim of Book III is to introduce an axiomatic approach to set theory. Notice however that we do not include all axioms of set theory: ...
... can constitute either entire or part of a third year project in mathematics. Occasionally, there are also “computer projects” for students who are able to program. The aim of Book III is to introduce an axiomatic approach to set theory. Notice however that we do not include all axioms of set theory: ...
Full text
... The authors define the notion of an independent Pythagorean number and they prove that there exist infinitely many primitive Pythagorean numbers that are not independent (Theorem 10, p. 40). According to that definition (Definition 2, p. 40), a Pythagorean number is called independent if it cannot b ...
... The authors define the notion of an independent Pythagorean number and they prove that there exist infinitely many primitive Pythagorean numbers that are not independent (Theorem 10, p. 40). According to that definition (Definition 2, p. 40), a Pythagorean number is called independent if it cannot b ...
Modular Arithmetic Basics (1) The “floor” function is defined by the
... Modular Arithmetic Basics (1) The “floor” function is defined by the formula bxc := ( the greatest integer less than or equal to x). This is also known as “the greatest integer function,” and in old texts is denoted by (whole) brackets. Examples: b3.789c = 3; b−3.789c = −4. (2) The “greatest common ...
... Modular Arithmetic Basics (1) The “floor” function is defined by the formula bxc := ( the greatest integer less than or equal to x). This is also known as “the greatest integer function,” and in old texts is denoted by (whole) brackets. Examples: b3.789c = 3; b−3.789c = −4. (2) The “greatest common ...
Some simple continued fraction expansions for an infinite product
... fraction is a continued fraction in which a0 is an integer, all the partial numerators are equal to 1 and each partial denominator is a positive integer. We recall (see for example [2, Theorem 14]) that every positive irrational real number has ...
... fraction is a continued fraction in which a0 is an integer, all the partial numerators are equal to 1 and each partial denominator is a positive integer. We recall (see for example [2, Theorem 14]) that every positive irrational real number has ...
Large gaps between consecutive prime numbers
... asymptotically as 1 + O(1/p2 )). As mentioned previously, this quantity will appear in two separate places in the proof of Theorem 2, but these two occurrences will eventually cancel each other out. The Hardy-Littlewood conjecture is still out of reach of current technology. Note that even the much ...
... asymptotically as 1 + O(1/p2 )). As mentioned previously, this quantity will appear in two separate places in the proof of Theorem 2, but these two occurrences will eventually cancel each other out. The Hardy-Littlewood conjecture is still out of reach of current technology. Note that even the much ...
22(2)
... 0 < i < k - 1, are all distinct because a/3 is not a root of unity. If a and 3 a r e rational integers, then the numbers 0 < i < ^ - 1, certainly satisfy a monic polynomial of degree k over ak-i-zgz^ the rational integers. If a and 3 are irrational, then a and 3 are conjugate in the algebraic number ...
... 0 < i < k - 1, are all distinct because a/3 is not a root of unity. If a and 3 a r e rational integers, then the numbers 0 < i < ^ - 1, certainly satisfy a monic polynomial of degree k over ak-i-zgz^ the rational integers. If a and 3 are irrational, then a and 3 are conjugate in the algebraic number ...
Chromatic Graph Theory
... Frank Plumpton Ramsey (1903–1930) was a British philosopher, economist, and mathematician. Ramsey’s first major work was his 1925 paper “The Foundations of Mathematics”, in which he intended to improve upon Principia Mathematica by Bertrand Russell and Alfred North Whitehead. He presented his second ...
... Frank Plumpton Ramsey (1903–1930) was a British philosopher, economist, and mathematician. Ramsey’s first major work was his 1925 paper “The Foundations of Mathematics”, in which he intended to improve upon Principia Mathematica by Bertrand Russell and Alfred North Whitehead. He presented his second ...
On absolutely normal and continued fraction normal
... determines further digits in the expansion of x in integer bases and in its continued fraction. We require that the choice contributes to the two forms of normality but without revisiting the previous digits. For this we need to control, at each step of the construction, the lengths of the new subin ...
... determines further digits in the expansion of x in integer bases and in its continued fraction. We require that the choice contributes to the two forms of normality but without revisiting the previous digits. For this we need to control, at each step of the construction, the lengths of the new subin ...
CS103X: Discrete Structures Homework Assignment 2: Solutions
... – The new number must be a difference of two numbers already on the board, which are themselves linear combinations of 1729 and 1211 by assumption. And a difference of linear combinations is another linear combination: difference of linear combinations of x and y can be expressed as a1 x + b1y − a2 ...
... – The new number must be a difference of two numbers already on the board, which are themselves linear combinations of 1729 and 1211 by assumption. And a difference of linear combinations is another linear combination: difference of linear combinations of x and y can be expressed as a1 x + b1y − a2 ...