The sums of the reciprocals of Fibonacci polynomials and Lucas

... © 2012 Zhengang and Wenpeng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor ...

... © 2012 Zhengang and Wenpeng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor ...

on highly composite and similar numbers

... For highly composite numbers we have the same upper limit, but we know that the number of highly composite numbers exceeds (log x)1+c. It would be interesting to know the exact order in both cases. In the theory of colossally abundant numbers the most interesting question is whether the quotient of ...

... For highly composite numbers we have the same upper limit, but we know that the number of highly composite numbers exceeds (log x)1+c. It would be interesting to know the exact order in both cases. In the theory of colossally abundant numbers the most interesting question is whether the quotient of ...

THE DISTRIBUTION OF PRIME NUMBERS Andrew Granville and K

... distribution of prime numbers, though the contents of that book mostly stem out of Riemann’s seminal idea.8 In this text we rework the basic results on the distribution of primes to be a consequence of results on the distribution of mean values of multiplicative functions, stemming mostly from the f ...

... distribution of prime numbers, though the contents of that book mostly stem out of Riemann’s seminal idea.8 In this text we rework the basic results on the distribution of primes to be a consequence of results on the distribution of mean values of multiplicative functions, stemming mostly from the f ...

Elementary Real Analysis - ClassicalRealAnalysis.info

... 4. To develop many of the topics that the authors feel all students of mathematics should know. There are now many texts that address some or all of these objectives. These books range from ones that do little more than address objective (1) to ones that try to address all four objectives. The books ...

... 4. To develop many of the topics that the authors feel all students of mathematics should know. There are now many texts that address some or all of these objectives. These books range from ones that do little more than address objective (1) to ones that try to address all four objectives. The books ...

The Fibonacci Numbers

... Theorem (Sum of first few Fibonacci numbers.) F1 + F2 + F3 + · · · + Fn = Fn+2 − 1 Theorem (Sum of first few EVEN Fibonacci numbers.) F2 + F4 + F6 + · · · + F2n = F2n+1 − 1 Theorem (Sum of first few ODD Fibonacci numbers.) F1 + F3 + F5 + · · · + F2n−1 = F2n Theorem (Sum of first few SQUARES of Fibon ...

... Theorem (Sum of first few Fibonacci numbers.) F1 + F2 + F3 + · · · + Fn = Fn+2 − 1 Theorem (Sum of first few EVEN Fibonacci numbers.) F2 + F4 + F6 + · · · + F2n = F2n+1 − 1 Theorem (Sum of first few ODD Fibonacci numbers.) F1 + F3 + F5 + · · · + F2n−1 = F2n Theorem (Sum of first few SQUARES of Fibon ...

On some polynomial-time primality algorithms

... In 1801 Gauss said: “The problem of distinguishing prime numbers from composite numbers is one of the most fundamental and important in arithmetic. It has remained as a central question in our subject from ancient times to this day, and yet still fascinates and frustrates us all 1 ”. This problem ha ...

... In 1801 Gauss said: “The problem of distinguishing prime numbers from composite numbers is one of the most fundamental and important in arithmetic. It has remained as a central question in our subject from ancient times to this day, and yet still fascinates and frustrates us all 1 ”. This problem ha ...

Student_Solution_Chap_09

... b. Euler’s phi-function, φ(n), which is sometimes called the Euler’s totient function, finds the number of integers that are both smaller than n and relatively prime to n. 5. We discussed two versions of Fermat’s little theorem. The first version says that if p is a prime and a is an integer such th ...

... b. Euler’s phi-function, φ(n), which is sometimes called the Euler’s totient function, finds the number of integers that are both smaller than n and relatively prime to n. 5. We discussed two versions of Fermat’s little theorem. The first version says that if p is a prime and a is an integer such th ...

Characterstics of Ternary Semirings

... [9] Madhusudhana Rao. D. and Srinivasa Rao. G., Special Elements of a Ternary Semirings, International Journal of Engineering Research and Applications, Vol. 4, Issue 11 ...

... [9] Madhusudhana Rao. D. and Srinivasa Rao. G., Special Elements of a Ternary Semirings, International Journal of Engineering Research and Applications, Vol. 4, Issue 11 ...

Intermediate Math Circles November 25, 2009 Diophantine

... The next two are harder, but if you can, find a rule for. . . (6) When a number is divisible by 8. (7) When a number is divisible by 7. (Hint: the rule has something to do with breaking the number down into the ones digit, and then the number made up of all the other digits: 1031 = 103 and 1: Now, c ...

... The next two are harder, but if you can, find a rule for. . . (6) When a number is divisible by 8. (7) When a number is divisible by 7. (Hint: the rule has something to do with breaking the number down into the ones digit, and then the number made up of all the other digits: 1031 = 103 and 1: Now, c ...

ON THE DISTRIBUTION OF EXTREME VALUES

... function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We al ...

... function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We al ...

Extremely Abundant Numbers and the Riemann Hypothesis

... since 10080 ∈ SA. Therefore, n belongs to SA. Next, motivating our construction of XA numbers, we will establish the first interesting result of the paper. Theorem 6. If there is any counterexample to Robin’s inequality (1), then the least one is an XA number. Proof. By doing some computer calculati ...

... since 10080 ∈ SA. Therefore, n belongs to SA. Next, motivating our construction of XA numbers, we will establish the first interesting result of the paper. Theorem 6. If there is any counterexample to Robin’s inequality (1), then the least one is an XA number. Proof. By doing some computer calculati ...

Section2.1notesall

... Division algorithm: Let m be a positive integer ( m 0 ) and let b be any integer. Then there is exactly one pair of integers q (called the quotient) and r (called the remainder) such that b qm r where 0 r m . A number of primary interest in this class will be the remainder r that we obtain ...

... Division algorithm: Let m be a positive integer ( m 0 ) and let b be any integer. Then there is exactly one pair of integers q (called the quotient) and r (called the remainder) such that b qm r where 0 r m . A number of primary interest in this class will be the remainder r that we obtain ...

On Visibility Graphs — Upper Bounds and

... bar visibility graphs with thickness at least 23 (k + 1) . Finally, we give a method to count the number of edges of a semi-bar k-visibility graph based on the structure of its semi-bar k-visibility representation. This method is inspired by skyscraper problems. Skyscraper puzzles are grids of integ ...

... bar visibility graphs with thickness at least 23 (k + 1) . Finally, we give a method to count the number of edges of a semi-bar k-visibility graph based on the structure of its semi-bar k-visibility representation. This method is inspired by skyscraper problems. Skyscraper puzzles are grids of integ ...

On prime factors of integers which are sums or shifted products by

... logarithms of algebraic numbers, see [32]. Győry and Sárközy [17] proved that the conjecture holds in the special case that at least one of the numbers a, b, c, a/b, b/c, a/c has bounded prime factors. This work, later refined by Bugeaud and Luca [3], depends on a result of Evertse [12] on the nu ...

... logarithms of algebraic numbers, see [32]. Győry and Sárközy [17] proved that the conjecture holds in the special case that at least one of the numbers a, b, c, a/b, b/c, a/c has bounded prime factors. This work, later refined by Bugeaud and Luca [3], depends on a result of Evertse [12] on the nu ...

1 Introduction and Preliminaries

... maintains up-to-date information. Eric Brier (1998, unpublished) showed that there are infinitely many numbers which are simultaneously Sierpiński and Riesel numbers. His example had 41 digits, and Yves Gallot (2002, unpublished) later found an example with 27 digits. We note here a smaller example ...

... maintains up-to-date information. Eric Brier (1998, unpublished) showed that there are infinitely many numbers which are simultaneously Sierpiński and Riesel numbers. His example had 41 digits, and Yves Gallot (2002, unpublished) later found an example with 27 digits. We note here a smaller example ...

4-RSA

... • Problem: We have a number of objects, but we do not know exactly how many. If we count them by threes we have two left over. If we count them by fives we have three left over. If we count them by sevens we have two left over. How many objects are there? ...

... • Problem: We have a number of objects, but we do not know exactly how many. If we count them by threes we have two left over. If we count them by fives we have three left over. If we count them by sevens we have two left over. How many objects are there? ...

Section2.2notesall

... The common prime factors of two numbers can be used to find the greatest common divisor of two numbers, which we define next. Definition: The greatest common divisor of two natural numbers a and b , denoted as gcd( a, b) , is the largest natural number that divides a and b with no remainder. Elemen ...

... The common prime factors of two numbers can be used to find the greatest common divisor of two numbers, which we define next. Definition: The greatest common divisor of two natural numbers a and b , denoted as gcd( a, b) , is the largest natural number that divides a and b with no remainder. Elemen ...

Mod p - Math.utah.edu

... discriminant to guide us to whether or not b2 − 4ac has a square root mod p. There is an amazing result, known as Quadratic Reciprocity, that helps us tell whether numbers have square roots mod p, but this would take us too far afield. We will just be satisified here with seeing what happens one pri ...

... discriminant to guide us to whether or not b2 − 4ac has a square root mod p. There is an amazing result, known as Quadratic Reciprocity, that helps us tell whether numbers have square roots mod p, but this would take us too far afield. We will just be satisified here with seeing what happens one pri ...

Congruence Notes for Math 135

... Notation: We could develop the following section formally for an integer made up of an arbitrary number of digits (using sigma notation), but for simplicity we’ll write, e.g., abcde for the number whose digits are a, b, c, d, e, i.e. the number 104 · a + 103 · b + 102 · c + 10 · d + e. Remember that ...

... Notation: We could develop the following section formally for an integer made up of an arbitrary number of digits (using sigma notation), but for simplicity we’ll write, e.g., abcde for the number whose digits are a, b, c, d, e, i.e. the number 104 · a + 103 · b + 102 · c + 10 · d + e. Remember that ...

An introduction to the Smarandache Square

... Case 1. According to the theorem 7 Ssc(n)=n and Ssc(n+1)=n+1 that implies that Ssc(n)<>Ssc(n+1) Case 2. Without loss of generality let's suppose that: n = pa ⋅ q b n + 1 = p a ⋅ qb + 1 = sc ⋅ t d where p,q,s and t are distinct primes. According to the theorem 4: Ssc( n) = Ssc ( p a ⋅ q b ) = p odd ( ...

... Case 1. According to the theorem 7 Ssc(n)=n and Ssc(n+1)=n+1 that implies that Ssc(n)<>Ssc(n+1) Case 2. Without loss of generality let's suppose that: n = pa ⋅ q b n + 1 = p a ⋅ qb + 1 = sc ⋅ t d where p,q,s and t are distinct primes. According to the theorem 4: Ssc( n) = Ssc ( p a ⋅ q b ) = p odd ( ...

nicely typed notes

... elements in each set are related. We call these sets equivalence classes. Equivalence classes partition the set on which the equivalence relation is defined into a collection of sets that have the following properties. 1. No two of the equivalence classes have members in common. (We call such sets d ...

... elements in each set are related. We call these sets equivalence classes. Equivalence classes partition the set on which the equivalence relation is defined into a collection of sets that have the following properties. 1. No two of the equivalence classes have members in common. (We call such sets d ...