Functions and Equations - Iowa State University Department of
... All multiples of 2 are even numbers, that is, they end in 0,2,4,6, or 8. The sum of the digits in all multiples of 3 is a lesser multiple of 3. All multiples of 5 end in 0 or 5. All multiples of 6 are even numbers and the sum of the digits is a lesser multiple of 3. A number is a multiple of 8 if th ...
... All multiples of 2 are even numbers, that is, they end in 0,2,4,6, or 8. The sum of the digits in all multiples of 3 is a lesser multiple of 3. All multiples of 5 end in 0 or 5. All multiples of 6 are even numbers and the sum of the digits is a lesser multiple of 3. A number is a multiple of 8 if th ...
Asymptotic Expansions of Central Binomial Coefficients and Catalan
... We would like to thank the anonymous referee who brought to our attention the existence of the manuscript [10] where similar problems are treated. D. Kessler and J. Schiff proved that expansion mentioned above contains only odd powers of n + 14 (for the central binomial coefficient) and n + 43 (for ...
... We would like to thank the anonymous referee who brought to our attention the existence of the manuscript [10] where similar problems are treated. D. Kessler and J. Schiff proved that expansion mentioned above contains only odd powers of n + 14 (for the central binomial coefficient) and n + 43 (for ...
Math 240 - Allan Wang
... A proposition or statement is an assertion which is either definitely true or definitely false Proposition typically denoted with letters, conventionally P, Q, R, … and are called atoms Propositional calculus is a language for expressing complex statements, together with a set of rules for deciding ...
... A proposition or statement is an assertion which is either definitely true or definitely false Proposition typically denoted with letters, conventionally P, Q, R, … and are called atoms Propositional calculus is a language for expressing complex statements, together with a set of rules for deciding ...
EXTREMAL EFFECTIVE DIVISORS OF BRILL
... Nevertheless, the main result of this paper shows that the above question has a negative answer. Theorem 1.1. For every n ≥ 6, there exist extremal effective divisors in M1,n that are different from the Da ’s. See Theorems 3.3, 4.4 and 5.4 for a more precise statement. Let us explain our method. For ...
... Nevertheless, the main result of this paper shows that the above question has a negative answer. Theorem 1.1. For every n ≥ 6, there exist extremal effective divisors in M1,n that are different from the Da ’s. See Theorems 3.3, 4.4 and 5.4 for a more precise statement. Let us explain our method. For ...
CHAPTER 10 Mathematical Induction
... usually a very simple statement, the basis step is often quite easy to do. The second step (2) is called the inductive step. In the inductive step direct proof is most often used to prove S k ⇒ S k+1 , so this step is usually carried out by assuming S k is true and showing this forces S k+1 to be tr ...
... usually a very simple statement, the basis step is often quite easy to do. The second step (2) is called the inductive step. In the inductive step direct proof is most often used to prove S k ⇒ S k+1 , so this step is usually carried out by assuming S k is true and showing this forces S k+1 to be tr ...
some cosine relations and the regular heptagon
... We will see that all three of the above are easy to derive. The next two expressions were originally discovered by the famous Indian mathematician Ramanujan [3]. We will derive them by a (possibly new) method found by the second author of this paper. ...
... We will see that all three of the above are easy to derive. The next two expressions were originally discovered by the famous Indian mathematician Ramanujan [3]. We will derive them by a (possibly new) method found by the second author of this paper. ...
PDF 72K - UCSD CSE
... Hence m + n equals twice an integer, and so by de nition of even, m + n is even [as was to be shown]. 15. This problem is solved in the book. 18. Start of proof: Suppose x is any [particular but arbitrarily chosen] real number such that x > 1. [We must show that x2 > x.] ...
... Hence m + n equals twice an integer, and so by de nition of even, m + n is even [as was to be shown]. 15. This problem is solved in the book. 18. Start of proof: Suppose x is any [particular but arbitrarily chosen] real number such that x > 1. [We must show that x2 > x.] ...
Notes on Algebraic Numbers
... argument shows that its eigenvalues are algebraic numbers. For the converse suppose that α is an algebraic integer, so that there exist b1 , b2 . . . , bn ∈ Z with αn + b1 αn−1 + · · · + bn−1 α + bn = 0. We can rewrite this as αn = −b1 αn−1 − · · · − bn−1 α − bn . ...
... argument shows that its eigenvalues are algebraic numbers. For the converse suppose that α is an algebraic integer, so that there exist b1 , b2 . . . , bn ∈ Z with αn + b1 αn−1 + · · · + bn−1 α + bn = 0. We can rewrite this as αn = −b1 αn−1 − · · · − bn−1 α − bn . ...
Transcendence of Various Infinite Series Applications of Baker’s Theorem and
... where ζq is the primitive qth root of unity, e2πi/q . As we see in equation (2.3), we can relate special values of the digamma function to logarithms of algebraic numbers. This is where Baker’s theory enters into this work and allows us to conclude transcendence properties after we find closed forms ...
... where ζq is the primitive qth root of unity, e2πi/q . As we see in equation (2.3), we can relate special values of the digamma function to logarithms of algebraic numbers. This is where Baker’s theory enters into this work and allows us to conclude transcendence properties after we find closed forms ...
2. Primes Primes. • A natural number greater than 1 is prime if it
... primes. The prime numbers give us a world of questions to explore. We will prove that there are infinitely many primes, but how are they distributed among the natural numbers? How many primes are there less than a natural number n? How can we find them? How can we use them? These questions and others ...
... primes. The prime numbers give us a world of questions to explore. We will prove that there are infinitely many primes, but how are they distributed among the natural numbers? How many primes are there less than a natural number n? How can we find them? How can we use them? These questions and others ...
Name_______________________________________ Date
... 2. How can I use the Pythagorean Theorem to decide if three lengths would form a right triangle? 3. How can I use the Pythagorean Theorem to find an unknown length in a right triangle? Think about It! What type of angles do you think ancient civilizations were concerned about? Why? _________________ ...
... 2. How can I use the Pythagorean Theorem to decide if three lengths would form a right triangle? 3. How can I use the Pythagorean Theorem to find an unknown length in a right triangle? Think about It! What type of angles do you think ancient civilizations were concerned about? Why? _________________ ...
PDF
... Lemma 1. Let S be a subset of C that contains a nonzero complex number and α ∈ C. Then α is constructible from S if and only if there exists a finite sequence α1 , . . . , αn ∈ C such that α1 is immediately constructible from S, α2 is immediately constructible from S∪{α1 }, . . . , and α is immediat ...
... Lemma 1. Let S be a subset of C that contains a nonzero complex number and α ∈ C. Then α is constructible from S if and only if there exists a finite sequence α1 , . . . , αn ∈ C such that α1 is immediately constructible from S, α2 is immediately constructible from S∪{α1 }, . . . , and α is immediat ...
AN ARITHMETIC FUNCTION ARISING FROM THE DEDEKIND ψ
... less than 2g(k). All elements of Section I are necessarily odd by Theorem 2.1. Section II of class k is the set of elements of class k that are greater than or equal to 2g(k) and strictly less than 2k . Section II of a class may contain both even and odd elements. Finally, Section III of class k is ...
... less than 2g(k). All elements of Section I are necessarily odd by Theorem 2.1. Section II of class k is the set of elements of class k that are greater than or equal to 2g(k) and strictly less than 2k . Section II of a class may contain both even and odd elements. Finally, Section III of class k is ...
An Introduction to The Twin Prime Conjecture
... related primes, both in relation to number theory as a whole, and as specific, well-defined problems. One of the first results of looking at twin primes was the discovery that, aside from (3, 5), all twin primes are of the form 6n ± 1. This comes from noticing that any prime greater than 3 must be of t ...
... related primes, both in relation to number theory as a whole, and as specific, well-defined problems. One of the first results of looking at twin primes was the discovery that, aside from (3, 5), all twin primes are of the form 6n ± 1. This comes from noticing that any prime greater than 3 must be of t ...
ARML Lecture VII - Number Theory
... Whenever the word remainder appears, you should immediately think modulos. Likewise, determining the last few digits of a number should make you consider modulos. The above theorems are merely suppliments to the algebra that can be performed on modular equations, which we outline here. The rules of ...
... Whenever the word remainder appears, you should immediately think modulos. Likewise, determining the last few digits of a number should make you consider modulos. The above theorems are merely suppliments to the algebra that can be performed on modular equations, which we outline here. The rules of ...
2. Primes Primes. • A natural number greater than 1 is prime if it
... primes. The prime numbers give us a world of questions to explore. We will prove that there are infinitely many primes, but how are they distributed among the natural numbers? How many primes are there less than a natural number n? How can we find them? How can we use them? These questions and other ...
... primes. The prime numbers give us a world of questions to explore. We will prove that there are infinitely many primes, but how are they distributed among the natural numbers? How many primes are there less than a natural number n? How can we find them? How can we use them? These questions and other ...
Mathematical Induction - Penn Math
... For more examples of the constructive uses of induction, including the FFT, Quicksorting, finding maximum independent sets in graphs, etc., you can download Chapter 2 of Algorithms and Complexity from ...
... For more examples of the constructive uses of induction, including the FFT, Quicksorting, finding maximum independent sets in graphs, etc., you can download Chapter 2 of Algorithms and Complexity from ...
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... An example of an improvement that can be made on the upper bounds of Ramsey's numbers comes from the case of two colors, which we call red and blue. From the proof, we get the bound ...
... An example of an improvement that can be made on the upper bounds of Ramsey's numbers comes from the case of two colors, which we call red and blue. From the proof, we get the bound ...