Remainder Theorem
... The Factor Theorem is powerful because it can be used to find the roots of polynomial equations. Example 3: Is x 4 a factor of 3x 3 x 2 20 x 5 ? For this question we need to find out if dividing 3x 3 x 2 20 x 5 by x 4 leaves a remainder. If the remainder is 0, then x 4 is a factor ...
... The Factor Theorem is powerful because it can be used to find the roots of polynomial equations. Example 3: Is x 4 a factor of 3x 3 x 2 20 x 5 ? For this question we need to find out if dividing 3x 3 x 2 20 x 5 by x 4 leaves a remainder. If the remainder is 0, then x 4 is a factor ...
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... We let Fn represent the nth Fibonacci number. In [2] and [3] we find relationships between the Fibonacci numbers and their associated matrices. The purpose of this paper is to develop relationships between the generalized Fibonacci numbers and the permanent of a (0,1)-matrix. The kgeneralizedFibonac ...
... We let Fn represent the nth Fibonacci number. In [2] and [3] we find relationships between the Fibonacci numbers and their associated matrices. The purpose of this paper is to develop relationships between the generalized Fibonacci numbers and the permanent of a (0,1)-matrix. The kgeneralizedFibonac ...
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... A remark on §(jri) and h(jri) is needed: these functions are convenient to use; however, if for some natural number 777 > 1 there exist A(m) and B(m) such that for every 777-integer a, aA{jn) - aB^ = 0 (mod 777), then .4(777) could be used in place of (j)(777) + h(m) , and 5(77?) in place of h(jn) . ...
... A remark on §(jri) and h(jri) is needed: these functions are convenient to use; however, if for some natural number 777 > 1 there exist A(m) and B(m) such that for every 777-integer a, aA{jn) - aB^ = 0 (mod 777), then .4(777) could be used in place of (j)(777) + h(m) , and 5(77?) in place of h(jn) . ...
Application to Stirling numbers
... r_i is the color of the ith vertex. We have m constraints of the form 0 < r_i r_j for each such pair i,j. Each of these pi ...
... r_i is the color of the ith vertex. We have m constraints of the form 0 < r_i r_j for each such pair i,j. Each of these pi ...
Non-associative normed algebras and hurwitz
... and y is isomorphic to R, C or Q. This subalgebra is of course a (pre-)Zilbert algebra with identity. In [4], however, it was observed t h a t the "usual" norms for R, C and Q are the only ones making them hilbert algebras with identity. Since these norms all satisfy (ii), the given norm, restricted ...
... and y is isomorphic to R, C or Q. This subalgebra is of course a (pre-)Zilbert algebra with identity. In [4], however, it was observed t h a t the "usual" norms for R, C and Q are the only ones making them hilbert algebras with identity. Since these norms all satisfy (ii), the given norm, restricted ...
The Probability that a Random - American Mathematical Society
... Every odd prime must pass this test. Moreover, Monier [3] and Rabin [4] have shown that if n > 1 is an odd composite, then the probability that it is a strong probable prime to a random base 6, 1 < 6 < n — 1, is less than 4. Let Pi{x) denote the same probability as P(x), except that (iii) is changed ...
... Every odd prime must pass this test. Moreover, Monier [3] and Rabin [4] have shown that if n > 1 is an odd composite, then the probability that it is a strong probable prime to a random base 6, 1 < 6 < n — 1, is less than 4. Let Pi{x) denote the same probability as P(x), except that (iii) is changed ...
Elementary Number Theory
... The algorithm is based on the following simple observation: Lemma 6.1. If a and b are not both zero, then gcd(a, b) = gcd(a, b − a) Proof. Clearly any common divisor of a and b is also a divisor of a − b. Furthermore, since b = a + (b − a), any common divisor of a and b − a is also a divisor of b. W ...
... The algorithm is based on the following simple observation: Lemma 6.1. If a and b are not both zero, then gcd(a, b) = gcd(a, b − a) Proof. Clearly any common divisor of a and b is also a divisor of a − b. Furthermore, since b = a + (b − a), any common divisor of a and b − a is also a divisor of b. W ...
I(k-1)
... order, merge them into a single sorted list. Merge(a,b): 1. If a is empty, return b 2. If b is empty, return a 3. Otherwise, if the first element of b comes before the first element of a, return a list with the first element of b at the front and the result of Merge(a, rest of b) as the rest 4. Othe ...
... order, merge them into a single sorted list. Merge(a,b): 1. If a is empty, return b 2. If b is empty, return a 3. Otherwise, if the first element of b comes before the first element of a, return a list with the first element of b at the front and the result of Merge(a, rest of b) as the rest 4. Othe ...
Weyl`s equidistribution theorem
... Consider the unit circle T in the Euclidean plane. If it is rotated like a stationary wheel, in an anti-clockwise direction by 45 degrees, then the 'spoke of the wheel' joining the centre (0, 0) to the point (1, 0) (call it v) gets mapped to the spoke joining the centre to the point (Cos 45, Sin 45) ...
... Consider the unit circle T in the Euclidean plane. If it is rotated like a stationary wheel, in an anti-clockwise direction by 45 degrees, then the 'spoke of the wheel' joining the centre (0, 0) to the point (1, 0) (call it v) gets mapped to the spoke joining the centre to the point (Cos 45, Sin 45) ...
Chapter 1 Introduction to prime number theory
... with a not so difficult proof based on complex analysis alone and avoiding functional analysis. In this course, we prove the Tauberian theorem via Newman’s method, and deduce from this the Prime Number Theorem as well as the Prime Number Theorem for arithmetic progressions (see below). ...
... with a not so difficult proof based on complex analysis alone and avoiding functional analysis. In this course, we prove the Tauberian theorem via Newman’s method, and deduce from this the Prime Number Theorem as well as the Prime Number Theorem for arithmetic progressions (see below). ...
Sample Chapter
... ii. Induction Hypothesis: It is assumed that S is true for all values greater than 2 and less than k. Therefore, numbers between 2 and k are either prime or can be written as product of two or more prime numbers. iii. Induction Step: It is to be proved that (k +1) is prime or can be written as produ ...
... ii. Induction Hypothesis: It is assumed that S is true for all values greater than 2 and less than k. Therefore, numbers between 2 and k are either prime or can be written as product of two or more prime numbers. iii. Induction Step: It is to be proved that (k +1) is prime or can be written as produ ...
Marianthi Karavitis - Stony Brook Math Department
... are equivalent to the original function once they were reduced. For example, 1/(x1) would be in the same equivalence class as (x+1)/( -1). Questions: 1) Thinking back to my example of last names for equivalence classes, what other types of equivalence classes can you think of that you encounter ever ...
... are equivalent to the original function once they were reduced. For example, 1/(x1) would be in the same equivalence class as (x+1)/( -1). Questions: 1) Thinking back to my example of last names for equivalence classes, what other types of equivalence classes can you think of that you encounter ever ...
n! r!(nr)! - Carnegie Mellon School of Computer Science
... ck, the coefficient of Xk, is the number of paths with exactly k X’s. ...
... ck, the coefficient of Xk, is the number of paths with exactly k X’s. ...
Full text
... Engineers often increase the reliability of a system by making the conditions under which it fails more stringent. An example from reliability theory is what is called a "consecutive-fc-out-of-n:.F system" [1]. This is a system of n independent, linearly ordered components, each of which operates (f ...
... Engineers often increase the reliability of a system by making the conditions under which it fails more stringent. An example from reliability theory is what is called a "consecutive-fc-out-of-n:.F system" [1]. This is a system of n independent, linearly ordered components, each of which operates (f ...
Lower Bounds for Relatively Prime Amicable Numbers of Opposite
... 1. Introduction. More than 1000 pairs of amicable numbers have been discovered to date (see [5] and the bibliography in [1]). Each of these pairs has a greatest common divisor which exceeds one, and the members of each pair are of the same parity. In [3] Kanold has shown that if m and n are relative ...
... 1. Introduction. More than 1000 pairs of amicable numbers have been discovered to date (see [5] and the bibliography in [1]). Each of these pairs has a greatest common divisor which exceeds one, and the members of each pair are of the same parity. In [3] Kanold has shown that if m and n are relative ...
Weeks 9 and 10 - Shadows Government
... the pigeonhole principle one box must contain at least 2 + 1 = 3 people. Suppose the friends box contains b, c and d (and possibly others). If any two of {b, c, d} know each other, say b and c then {a, b, c} forms a set of mutual friends. But if none of {b, c, d} know each other then they form a set ...
... the pigeonhole principle one box must contain at least 2 + 1 = 3 people. Suppose the friends box contains b, c and d (and possibly others). If any two of {b, c, d} know each other, say b and c then {a, b, c} forms a set of mutual friends. But if none of {b, c, d} know each other then they form a set ...
Over Chapter 1
... of the second, while the unemployment rate of the first is less than the unemployment rate of the second. El Paso has a greater population than Maverick, while El Paso has a lower unemployment rate than Maverick. ...
... of the second, while the unemployment rate of the first is less than the unemployment rate of the second. El Paso has a greater population than Maverick, while El Paso has a lower unemployment rate than Maverick. ...
Over Chapter 1
... of the second, while the unemployment rate of the first is less than the unemployment rate of the second. El Paso has a greater population than Maverick, while El Paso has a lower unemployment rate than Maverick. ...
... of the second, while the unemployment rate of the first is less than the unemployment rate of the second. El Paso has a greater population than Maverick, while El Paso has a lower unemployment rate than Maverick. ...
Combinatorial properties of the numbers of tableaux of bounded
... by M.Aigner in [1] correspond bijectively to the integers counting standard Young tableaux of a given shape with at most 2 columns. Firstly, we arrange the entries of the Ballot Matrix in a new lower triangular matrix A in such a way that the entries of the n-th row count standard Young tableaux wi ...
... by M.Aigner in [1] correspond bijectively to the integers counting standard Young tableaux of a given shape with at most 2 columns. Firstly, we arrange the entries of the Ballot Matrix in a new lower triangular matrix A in such a way that the entries of the n-th row count standard Young tableaux wi ...
