1+1 + ll + fl.lfcl + M
... The present paper is concerned with establishing convergence criteria of this general type. The principal result is given in Theorem 3.1. The method to be used is the following. Denote the nth. approximant of (1.1) by An/Bn. Conditions on the numbers an are determined which imply that the approximan ...
... The present paper is concerned with establishing convergence criteria of this general type. The principal result is given in Theorem 3.1. The method to be used is the following. Denote the nth. approximant of (1.1) by An/Bn. Conditions on the numbers an are determined which imply that the approximan ...
1.4 The set of Real Numbers: Quick Definition and
... Around 500 BC, Pythagora knew that not every quantity could be expressed as a rational number. Consider for example a right triangle in which the length of the sides around the right angle is 1. Then, the length x of the hypotenuse satis…es 12 + 12 = x2 . In other words x2 = 2. What is the number x ...
... Around 500 BC, Pythagora knew that not every quantity could be expressed as a rational number. Consider for example a right triangle in which the length of the sides around the right angle is 1. Then, the length x of the hypotenuse satis…es 12 + 12 = x2 . In other words x2 = 2. What is the number x ...
(pdf)
... Ramsey theory is named after Frank Plumpton Ramsey, an English mathematician, philosopher, and economist, who worked at Cambridge and died in 1930. He intended for his original paper proving the infinite version of Ramsey’s Theorem, published posthumously in 1930, to have applications to mathematica ...
... Ramsey theory is named after Frank Plumpton Ramsey, an English mathematician, philosopher, and economist, who worked at Cambridge and died in 1930. He intended for his original paper proving the infinite version of Ramsey’s Theorem, published posthumously in 1930, to have applications to mathematica ...
A SIMPLE RULE TO DISTINGUISH PRIME FROM COMPOSITE
... composite numbers will be distinguished from each other. In theorem (2), it was proved that if x and y are odd natural numbers, then, x and y can be defined using m and n. KEYWORDS: Prime Numbers, Composite Numbers ...
... composite numbers will be distinguished from each other. In theorem (2), it was proved that if x and y are odd natural numbers, then, x and y can be defined using m and n. KEYWORDS: Prime Numbers, Composite Numbers ...
Full text
... where (x)m = x(x − 1) · · · (x − m + 1) denotes the falling factorial. If j is a nonnegative integer, Sj,r (s, a) converges for r, s, a ∈ C such that <(s) > <(r) and <(a) > −j; when r ∈ Z+ it has poles of order j + 1 at s = 1, 2, .., r and of order at most j at nonpositive integers s. When j = 0 we ...
... where (x)m = x(x − 1) · · · (x − m + 1) denotes the falling factorial. If j is a nonnegative integer, Sj,r (s, a) converges for r, s, a ∈ C such that <(s) > <(r) and <(a) > −j; when r ∈ Z+ it has poles of order j + 1 at s = 1, 2, .., r and of order at most j at nonpositive integers s. When j = 0 we ...
Divisibility Tests and Factoring
... Divisibility Tests and Factoring • There are simple tests for divisibility by small numbers such as 2, 3, 5, 7, and 9. These tests involve performing operations on the decimal representation of the number to be tested for divisibility. • Fermat factorization attempts to factor a number by representi ...
... Divisibility Tests and Factoring • There are simple tests for divisibility by small numbers such as 2, 3, 5, 7, and 9. These tests involve performing operations on the decimal representation of the number to be tested for divisibility. • Fermat factorization attempts to factor a number by representi ...
userfiles/SECTION F PROOF BY CONTRADICTION
... construct a proof by contradiction In this section we prove two famous results from ancient Greek mathematics using proof by contradiction. They are ‘ 2 is an irrational number’ and ‘there are an infinite number of primes’. Both these results belong to an area of mathematics called number theory. Th ...
... construct a proof by contradiction In this section we prove two famous results from ancient Greek mathematics using proof by contradiction. They are ‘ 2 is an irrational number’ and ‘there are an infinite number of primes’. Both these results belong to an area of mathematics called number theory. Th ...
2.9.2 Problems P10 Try small prime numbers first. p p2 + 2 2 6 3 11
... The condition in the problem is the same as saying that every prime number p divides m. This can happen only if m = 0, that is, only if n = 1. Hence n = 1 is the only positive integer with the given property. P15 We try to find a pattern. ...
... The condition in the problem is the same as saying that every prime number p divides m. This can happen only if m = 0, that is, only if n = 1. Hence n = 1 is the only positive integer with the given property. P15 We try to find a pattern. ...
Mathematical Analysis and Proof. Edition No. 2 Brochure
... Mathematical Analysis and Proof. Edition No. 2 Description: ...
... Mathematical Analysis and Proof. Edition No. 2 Description: ...
F.Y. B.Sc. - Mathematics
... No. of Credits: 2 The aim of this course is to show certain applications of continuous and differentiable functions that students studied in the first semester. Hence the first two sections are continuation of the first semester course. The next section is integration. It is expected that student sh ...
... No. of Credits: 2 The aim of this course is to show certain applications of continuous and differentiable functions that students studied in the first semester. Hence the first two sections are continuation of the first semester course. The next section is integration. It is expected that student sh ...
Abstract:
... loops linked together (Hopf link). Here, a number will be found which we call the Gauss’ linking number. Once again, the physics method of deriving Biôt-Savart law is shown, for further comparison. Finally, topology is called into action to reduce any 2 component link into a series of simple Hopf li ...
... loops linked together (Hopf link). Here, a number will be found which we call the Gauss’ linking number. Once again, the physics method of deriving Biôt-Savart law is shown, for further comparison. Finally, topology is called into action to reduce any 2 component link into a series of simple Hopf li ...
Problems
... Prove this theorem, by showing that for every natural number n we can find a rational number p/q with q < n + 1 and |α − p/q| < 1/nq. For this, let cn = nα −bnαc for any n. Now the n+1 numbers c1 , c2 , . . . , cn+1 lie in the unit interval. Divide the interval into n equal parts. By the doocot prin ...
... Prove this theorem, by showing that for every natural number n we can find a rational number p/q with q < n + 1 and |α − p/q| < 1/nq. For this, let cn = nα −bnαc for any n. Now the n+1 numbers c1 , c2 , . . . , cn+1 lie in the unit interval. Divide the interval into n equal parts. By the doocot prin ...
Document
... • Questions: Are there infinitely many of these primitive ones? Can we figure out a way to manufacture them? ...
... • Questions: Are there infinitely many of these primitive ones? Can we figure out a way to manufacture them? ...
The ABC Conjecture
... xn + y n = z n has no solutions if n > 2 and, underneath that, I have a truly marvelous proof of this proposition which this margin is too narrow to contain. This simple statement set off a three century search for a proof of this proposition, as the conjecture that would go on to be called Fermat’s ...
... xn + y n = z n has no solutions if n > 2 and, underneath that, I have a truly marvelous proof of this proposition which this margin is too narrow to contain. This simple statement set off a three century search for a proof of this proposition, as the conjecture that would go on to be called Fermat’s ...
NON-NORMALITY OF CONTINUED FRACTION PARTIAL
... where the positive integer digits ai are known as the partial quotients. Unfortunately, for almost all x E (0,1) the partial quotients form an unbounded set of integers. It is there fore necessary to define what is meant by normality in such a case. One possible way is to consider the partial quoti ...
... where the positive integer digits ai are known as the partial quotients. Unfortunately, for almost all x E (0,1) the partial quotients form an unbounded set of integers. It is there fore necessary to define what is meant by normality in such a case. One possible way is to consider the partial quoti ...
Full text
... Remarks: This implies that if p is a non-defective odd prime, then p = 5 or p = 3 or 7 (mod 20). While it is easily seen that 2, 3, 5 and 7 are non-defective, the author has not been able to find any other non-defective primes. From Theorems 2 and 6, we have Theorem 7. If n > 1 is non-defective, the ...
... Remarks: This implies that if p is a non-defective odd prime, then p = 5 or p = 3 or 7 (mod 20). While it is easily seen that 2, 3, 5 and 7 are non-defective, the author has not been able to find any other non-defective primes. From Theorems 2 and 6, we have Theorem 7. If n > 1 is non-defective, the ...
Reasoning with Quantifiers
... Example (Goldbach’s conjecture): Prove that every even integer greater than 2 is the sum of two primes. (We can’t use the method of exhaustion…the domain is infinite). We suspect this statement is true since it is true for every even integer checked to date. Goldbach’s conjecture has been shown to b ...
... Example (Goldbach’s conjecture): Prove that every even integer greater than 2 is the sum of two primes. (We can’t use the method of exhaustion…the domain is infinite). We suspect this statement is true since it is true for every even integer checked to date. Goldbach’s conjecture has been shown to b ...
Theorem (Infinitude of Prime Numbers).
... Note: This is not a formal proof of the theorem. This is just the general strategy of the proof illustrated with a particular example. There are other ways of proving this theorem, this is just one strategy. The strategy of the proof is to show that given an arbitrary integer n we can always find a ...
... Note: This is not a formal proof of the theorem. This is just the general strategy of the proof illustrated with a particular example. There are other ways of proving this theorem, this is just one strategy. The strategy of the proof is to show that given an arbitrary integer n we can always find a ...
Abstract Representation: Your Ancient Heritage
... written in a way that is easy for other humans (and yourself) to understand Similarly, good proofs should be easy to understand. Although the formal proof does not require certain explanatory sentences (e.g., “the idea of this proof is basically X”), good proofs usually do ...
... written in a way that is easy for other humans (and yourself) to understand Similarly, good proofs should be easy to understand. Although the formal proof does not require certain explanatory sentences (e.g., “the idea of this proof is basically X”), good proofs usually do ...