EULER’S THEOREM 1. Introduction
... sequence, the 12 numerators fall into two sets of size 6: {1, 3, 4, 9, 10, 12} and {2, 5, 6, 7, 8, 11}. Is there some significance to these two sets of numbers? More generally, can you explain how many digit sequences (up to shifting) will occur among all the reduced fractions with a given denominat ...
... sequence, the 12 numerators fall into two sets of size 6: {1, 3, 4, 9, 10, 12} and {2, 5, 6, 7, 8, 11}. Is there some significance to these two sets of numbers? More generally, can you explain how many digit sequences (up to shifting) will occur among all the reduced fractions with a given denominat ...
Geodesics, volumes and Lehmer`s conjecture Mikhail Belolipetsky
... manifolds we come to the questions of whether such sequences do exist at all and if yes, then what can we say about the isosystolic properties of corresponding ...
... manifolds we come to the questions of whether such sequences do exist at all and if yes, then what can we say about the isosystolic properties of corresponding ...
THE DISTRIBUTION OF LEADING DIGITS AND UNIFORM
... observations classified into 20 data types published by Benford (1938). Close to 30% of the data Benford presents comes from arithmetical sequences. The chisquared statistic for goodness of fit of Benford's arithmetical sequences to the model (1.1) is greater than 440 on 8 degrees of freedom. This s ...
... observations classified into 20 data types published by Benford (1938). Close to 30% of the data Benford presents comes from arithmetical sequences. The chisquared statistic for goodness of fit of Benford's arithmetical sequences to the model (1.1) is greater than 440 on 8 degrees of freedom. This s ...
Inductive Reasoning
... Start with 640. Divide by 2 to find the next number. Discuss with your neighbor……what are the next 5 numbers in the list? ...
... Start with 640. Divide by 2 to find the next number. Discuss with your neighbor……what are the next 5 numbers in the list? ...
The Asymptotic Density of Relatively Prime Pairs and of Square
... Pick a positive integer at random. What is the probability of it being even? As stated, this question is not well posed, because there is no uniform probability measure on the set N of positive integers. However, what one can do is fix a positive integer n, and choose a number uniformly at random fr ...
... Pick a positive integer at random. What is the probability of it being even? As stated, this question is not well posed, because there is no uniform probability measure on the set N of positive integers. However, what one can do is fix a positive integer n, and choose a number uniformly at random fr ...
A Simple Proof that e is Irrational
... In 1882, German mathematician Ferdinand von Lindemann (1852-1939) proved that π is transcendental, putting an end to nearly 2500 years of conjecture. By his proof he showed that π transcends the power of algebra to display it in its totality; i.e., π cannot be expressed in any finite series of arith ...
... In 1882, German mathematician Ferdinand von Lindemann (1852-1939) proved that π is transcendental, putting an end to nearly 2500 years of conjecture. By his proof he showed that π transcends the power of algebra to display it in its totality; i.e., π cannot be expressed in any finite series of arith ...
1. Introduction - DML-PL
... then we say that G is τ -partitionable. The following conjecture, known as the Path Partition Conjecture, is stated in [1], [9] and [3] and studied in [2], [6] and [7]. Conjecture 1. Every graph is τ -partitionable. We shall show that, if the Path Partition Conjecture is true, the following conjectu ...
... then we say that G is τ -partitionable. The following conjecture, known as the Path Partition Conjecture, is stated in [1], [9] and [3] and studied in [2], [6] and [7]. Conjecture 1. Every graph is τ -partitionable. We shall show that, if the Path Partition Conjecture is true, the following conjectu ...
1 Introduction 2 Sets 3 The Sum Principle
... Example: The function f : R → R defined by f (x) = x3 is one-to-one. But the function g : R → R defined by g(x) = x2 is not one-to-one. In these terms. we can view a list of k elements from a set B (called a k-element permuation), is a one-to-one function from the set K = {1, . . . , k} to B. Suppo ...
... Example: The function f : R → R defined by f (x) = x3 is one-to-one. But the function g : R → R defined by g(x) = x2 is not one-to-one. In these terms. we can view a list of k elements from a set B (called a k-element permuation), is a one-to-one function from the set K = {1, . . . , k} to B. Suppo ...
Approximation to real numbers by algebraic numbers of
... |(L · L · Ld · . . . · L2 )| < C |x| it gives, that they are contained in finite number of hyperplanes. Now we will prove, that every hyperplane may contain only −d finite number of solutions even of the first inequality - |L(x)| < c1 |x| . Lemma 5.2.1. For every linear form L(x) = β0 x0 +. . .+βd x ...
... |(L · L · Ld · . . . · L2 )| < C |x| it gives, that they are contained in finite number of hyperplanes. Now we will prove, that every hyperplane may contain only −d finite number of solutions even of the first inequality - |L(x)| < c1 |x| . Lemma 5.2.1. For every linear form L(x) = β0 x0 +. . .+βd x ...
High-Performance Implementations on the Cell Broadband Engine
... Use 128-bit multiplication + fast reduction modulo pe For US$ 10, 000 buy 33 PS3s [1] T. Güneysu, C. Paar, and J. Pelzl. Special-purpose hardware for solving the elliptic curve discrete logarithm problem. ACM Transactions on Reconfigurable Technology and Systems, 1(2):1-21, 2008. [2] S. Kumar, C. P ...
... Use 128-bit multiplication + fast reduction modulo pe For US$ 10, 000 buy 33 PS3s [1] T. Güneysu, C. Paar, and J. Pelzl. Special-purpose hardware for solving the elliptic curve discrete logarithm problem. ACM Transactions on Reconfigurable Technology and Systems, 1(2):1-21, 2008. [2] S. Kumar, C. P ...
The Pentagonal Number Theorem and All That
... I do not see. This can be shown in a most pleasant investigation, together with tranquil pastime and the endurance of pertinacious labor, all three of which I lack.” Euler mentions the theorem many more times over the next few years, in letters we do possess to Niklaus Bernoulli, Christian Goldbach, ...
... I do not see. This can be shown in a most pleasant investigation, together with tranquil pastime and the endurance of pertinacious labor, all three of which I lack.” Euler mentions the theorem many more times over the next few years, in letters we do possess to Niklaus Bernoulli, Christian Goldbach, ...
Cauchy Sequences
... (yn ) is also a Cauchy sequence. One can think of a divergent Cauchy sequence like (yn ) as a “hole-detector,” because, so to speak, it converges to a hole—to a position where there ought to be a number but there em isn’t a number. Fortunately, in the case of R, the existence of lub’s and glb’s ensu ...
... (yn ) is also a Cauchy sequence. One can think of a divergent Cauchy sequence like (yn ) as a “hole-detector,” because, so to speak, it converges to a hole—to a position where there ought to be a number but there em isn’t a number. Fortunately, in the case of R, the existence of lub’s and glb’s ensu ...
Pythagorean Theorem
... • Step 1: Write the formula a2 + b2 = c2 • Step 2: Substitute known values 82 + 152 = c2 • Step 3: Solve for the missing variable, in this case c. We are not done yet… We have found c2, but not just plain c. We were told to solve for x, not c. So we should replace the c with an x. ...
... • Step 1: Write the formula a2 + b2 = c2 • Step 2: Substitute known values 82 + 152 = c2 • Step 3: Solve for the missing variable, in this case c. We are not done yet… We have found c2, but not just plain c. We were told to solve for x, not c. So we should replace the c with an x. ...
7.5 The Converse of the Pythagorean Theorem
... equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8. G.6 Explain a proof of the Pythagorean Theorem and its converse. 8. G.7 Apply the Pythagorean Theorem t ...
... equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8. G.6 Explain a proof of the Pythagorean Theorem and its converse. 8. G.7 Apply the Pythagorean Theorem t ...
On a conjecture of Chowla and Milnor
... On a Conjecture of Chowla and Milnor Sanoli Gun, M. Ram Murty, and Purusottam Rath Abstract. In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of L-functions associated to periodic function ...
... On a Conjecture of Chowla and Milnor Sanoli Gun, M. Ram Murty, and Purusottam Rath Abstract. In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of L-functions associated to periodic function ...
Homework for Chapter 1 and 2 scanned from the textbook (4th ed)
... context of some mathematical system. For example, in Section 3 when we discussed a conjecture related to prime numbers, the natural context of that discussion was the positive integers. In Example 3.7 when talking about odd and even numbers, the context was the set of all integers. Very often a theo ...
... context of some mathematical system. For example, in Section 3 when we discussed a conjecture related to prime numbers, the natural context of that discussion was the positive integers. In Example 3.7 when talking about odd and even numbers, the context was the set of all integers. Very often a theo ...
Lecture 3: January 14 3.1 Primality Testing (continued)
... The previous section proved that there exists an assignment satisfying at least 78 m clauses without telling how to construct such a solution. In fact, the probability of hitting one may be vanishingly small. In this section, we show that the randomized algorithm which assigns random values to varia ...
... The previous section proved that there exists an assignment satisfying at least 78 m clauses without telling how to construct such a solution. In fact, the probability of hitting one may be vanishingly small. In this section, we show that the randomized algorithm which assigns random values to varia ...
Math `Convincing and Proving` Critiquing `Proofs` Tasks
... Attempt 3 is the basis for a very elegant proof, but there are some holes and unnecessary jumps in it at present. The first is that it is always dangeraous to argue from adiagram because the diagram does not show all possible cases. In this example, the shaded area is clearly greater than 0 , but w ...
... Attempt 3 is the basis for a very elegant proof, but there are some holes and unnecessary jumps in it at present. The first is that it is always dangeraous to argue from adiagram because the diagram does not show all possible cases. In this example, the shaded area is clearly greater than 0 , but w ...