Practice questions for Exam 1
... Thus, 3x − 1 = r/s where r = 3p − q is an integer and s = q is an integer, with s 6= 0. Therefore, 3x − 1 is an odd number. We now show “if 3x − 1 is rational, then x is rational”. If 3x − 1 is rational, then 3x − 1 = p/q for some integers p and q with q 6= 0. So we have, p q ...
... Thus, 3x − 1 = r/s where r = 3p − q is an integer and s = q is an integer, with s 6= 0. Therefore, 3x − 1 is an odd number. We now show “if 3x − 1 is rational, then x is rational”. If 3x − 1 is rational, then 3x − 1 = p/q for some integers p and q with q 6= 0. So we have, p q ...
An Ancient Diophantine Equation with applications to Numerical
... positive rational numbers of Eq. (8) as stated by Scott and Styer. The aforementioned theorem of Poincare and Hurwitz does not give any method of generating the infinitely many rational points in the neighbourhood of the point A, and hence is of no help in finding solutions of Eq. (6) in positive rati ...
... positive rational numbers of Eq. (8) as stated by Scott and Styer. The aforementioned theorem of Poincare and Hurwitz does not give any method of generating the infinitely many rational points in the neighbourhood of the point A, and hence is of no help in finding solutions of Eq. (6) in positive rati ...
1 Warming up with rational points on the unit circle
... { A ∈ M2 (C) : A · A∗ = 1} otherwise known at the group of special unitary matrices and denoted SU2 (C). In fact, this is a topologically smooth isomorphism of groups S3 ∼ = SU2 which furthermore smoothly preserves the multiplicative and inversion structure of the groups. Furthermore, combining thes ...
... { A ∈ M2 (C) : A · A∗ = 1} otherwise known at the group of special unitary matrices and denoted SU2 (C). In fact, this is a topologically smooth isomorphism of groups S3 ∼ = SU2 which furthermore smoothly preserves the multiplicative and inversion structure of the groups. Furthermore, combining thes ...
Full text
... much to our knowledge of the theory of numbers. These polynomials are of basic importance in several parts of analysis and calculus of finite differences , and have applications in various fields such as statistics, numerical analysiss and so on. In recent years, the Eulerian numbers and certain gen ...
... much to our knowledge of the theory of numbers. These polynomials are of basic importance in several parts of analysis and calculus of finite differences , and have applications in various fields such as statistics, numerical analysiss and so on. In recent years, the Eulerian numbers and certain gen ...
A1-A3 - Tufts
... • doesn’t suck if the error is less than 1/q; • it’s pretty efficient if less than 1/10q; • it’s super efficient if less than 1/q 2 ; • it’s double super efficient if less than 1/2q 2 ; • it’s crazy efficient if less than 1/100q 2 ; and • insanely efficient if less than 1/q 3 . Passing these tests i ...
... • doesn’t suck if the error is less than 1/q; • it’s pretty efficient if less than 1/10q; • it’s super efficient if less than 1/q 2 ; • it’s double super efficient if less than 1/2q 2 ; • it’s crazy efficient if less than 1/100q 2 ; and • insanely efficient if less than 1/q 3 . Passing these tests i ...
Pythagorean triples and rational geometry
... Properties. If (a, b, c) is a Pythagorean triple, then so is (na, nb, nc), for any n ∈ Z. Conversely, if d is a common divisor of a, b, c, then ( ad , db , dc ) is a Pythagorean triple. We say that such triples are equivalent. A Pythagorean triple is primitive if a, b, c have no non-trivial common d ...
... Properties. If (a, b, c) is a Pythagorean triple, then so is (na, nb, nc), for any n ∈ Z. Conversely, if d is a common divisor of a, b, c, then ( ad , db , dc ) is a Pythagorean triple. We say that such triples are equivalent. A Pythagorean triple is primitive if a, b, c have no non-trivial common d ...
TEICHIB`S STRONG LAW OF LARGE NUMBERS IN GENERAL
... D. Szy n a.1 and A. K u c z m a s z e wska, Note on C11ung:Teicher type conditions for the strong ILIM: pf Inrge I I N H I ~ L ' I . in ~ N Hilbert space, Probability Theory on Vector Spaces 111, Lublin 1983. Lecture Notes in Mathem:~tics, V 1080. Springer-Verjag. 1984, p. 299-305. . H. Teicher, S ~ ...
... D. Szy n a.1 and A. K u c z m a s z e wska, Note on C11ung:Teicher type conditions for the strong ILIM: pf Inrge I I N H I ~ L ' I . in ~ N Hilbert space, Probability Theory on Vector Spaces 111, Lublin 1983. Lecture Notes in Mathem:~tics, V 1080. Springer-Verjag. 1984, p. 299-305. . H. Teicher, S ~ ...
Pythagorean Theorem
... buildings. One will be smaller than the other, however, both ramps must be proportional in a 3:1 ratio. Two measurements are provided below. What are the measurements of the other sides? ...
... buildings. One will be smaller than the other, however, both ramps must be proportional in a 3:1 ratio. Two measurements are provided below. What are the measurements of the other sides? ...
Class Slides - UNL Math Department
... the theorem? b) Represent a and a0 in terms of other variables. ...
... the theorem? b) Represent a and a0 in terms of other variables. ...
The Repeated Sums of Integers
... through (n − 1, m), and vice versa. That means the number of paths from A to B will be the sum of all the paths from A to (n, m − 1) and from A to (n − 1, m). In other words, P (n, m) = P (n, m − 1) + P (n − 1, m). Repeating the same argument, we can see that the number of paths from A to (n − 1, m) ...
... through (n − 1, m), and vice versa. That means the number of paths from A to B will be the sum of all the paths from A to (n, m − 1) and from A to (n − 1, m). In other words, P (n, m) = P (n, m − 1) + P (n − 1, m). Repeating the same argument, we can see that the number of paths from A to (n − 1, m) ...
ON THE ERD¨OS-STRAUS CONJECTURE
... that Theorem 1 can be applied and turn that term into a sum of only two unit fractions. Indeed, for k = 7l + r, we get that n = 24k + 1 ≡ 0 (mod 7) if r = 2, 2(3k + 1) + 1 ≡ 0 (mod 7) if r = 3, n + 1 = 2(12k + 1) ≡ 0 (mod 7) if r = 4, and n + 2 = 24k + 3 ≡ 0 (mod 7) if r = 6. Calculating the residue ...
... that Theorem 1 can be applied and turn that term into a sum of only two unit fractions. Indeed, for k = 7l + r, we get that n = 24k + 1 ≡ 0 (mod 7) if r = 2, 2(3k + 1) + 1 ≡ 0 (mod 7) if r = 3, n + 1 = 2(12k + 1) ≡ 0 (mod 7) if r = 4, and n + 2 = 24k + 3 ≡ 0 (mod 7) if r = 6. Calculating the residue ...
ON DIOPHANTINE APPROXIMATIONS^)
... Scott [24] showed that if we restrict the fractions p/q to be from any one of the three classes (i) p, q both odd, (ii) p odd, q even, or (iii) p even, q odd, then there are infinitely many such p/q satisfying (1). Other proofs of this result have been given by Robinson [22], Oppenheim [20] and Kuip ...
... Scott [24] showed that if we restrict the fractions p/q to be from any one of the three classes (i) p, q both odd, (ii) p odd, q even, or (iii) p even, q odd, then there are infinitely many such p/q satisfying (1). Other proofs of this result have been given by Robinson [22], Oppenheim [20] and Kuip ...
(pdf)
... Elegant in its simplicity, this equation can be applied to advanced topics in mathematics and physics. Beginning math students, or just someone with a curious mind, will find it an accessible introduction to number theory. The equation itself has a rich history apart from the general theory of Dioph ...
... Elegant in its simplicity, this equation can be applied to advanced topics in mathematics and physics. Beginning math students, or just someone with a curious mind, will find it an accessible introduction to number theory. The equation itself has a rich history apart from the general theory of Dioph ...
B. The Binomial Theorem
... In Pascal’s triangle, each number (for n > 0) is the sum of the two adjacent numbers in the line above. In terms of binomial coefficients, this constuction is ...
... In Pascal’s triangle, each number (for n > 0) is the sum of the two adjacent numbers in the line above. In terms of binomial coefficients, this constuction is ...
Section 5-3b - Austin Mohr
... you are explaining the concept to another student in the course). If the problem requires computation, make sure that it is written neatly and arranged so that another person can follow your work. Before Class 1. Read my paraphrase of Monday’s class. 2. (∗) Mimic the proof of the Divisibility Rule f ...
... you are explaining the concept to another student in the course). If the problem requires computation, make sure that it is written neatly and arranged so that another person can follow your work. Before Class 1. Read my paraphrase of Monday’s class. 2. (∗) Mimic the proof of the Divisibility Rule f ...
Prime Numbers and the Convergents of a Continued Fraction
... represented in decimal format. Continued fractions are found useful in many different areas of mathematics and science. Since ancient times they have played an important role in the approximation to real numbers by rational numbers, using convergents. In 1939 P. Erdos and K. Mahler showed that there ...
... represented in decimal format. Continued fractions are found useful in many different areas of mathematics and science. Since ancient times they have played an important role in the approximation to real numbers by rational numbers, using convergents. In 1939 P. Erdos and K. Mahler showed that there ...
A Tour of Formal Verification with Coq:Knuth`s Algorithm for Prime
... There is no relation between the length of a program and the difficulty of its proof of correctness. Very long programs performing elementary tasks could be trivial to prove correct, while short programs relying on some very deep properties could be much harder. Highly optimized programs usually bel ...
... There is no relation between the length of a program and the difficulty of its proof of correctness. Very long programs performing elementary tasks could be trivial to prove correct, while short programs relying on some very deep properties could be much harder. Highly optimized programs usually bel ...
Full text
... good book on number theory. One such book is An Introduction to the Theory of Numbers by Niven andZuckerman, published by John Wiley and Sons, Inc.) EULER'S FAMOUS RECURRENCE FORMULA is usually written as ...
... good book on number theory. One such book is An Introduction to the Theory of Numbers by Niven andZuckerman, published by John Wiley and Sons, Inc.) EULER'S FAMOUS RECURRENCE FORMULA is usually written as ...
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