Irrational numbers
... A real number that can be expressed as ba with b 6= 0 is termd as a rational number. A real number that is not rational is said to be an irrational. ...
... A real number that can be expressed as ba with b 6= 0 is termd as a rational number. A real number that is not rational is said to be an irrational. ...
1 - University of Kent
... of the iterates belong to the ring Z[x±1 0 , . . . , xN −1 ]; as a consequence, if all the initial values are 1 (or ±1), then each term of the sequence is an integer. Such sequences were popularized by Gale [8, 9], and subsequently Fomin and Zelevinsky found a useful technique - the Caterpillar Lemm ...
... of the iterates belong to the ring Z[x±1 0 , . . . , xN −1 ]; as a consequence, if all the initial values are 1 (or ±1), then each term of the sequence is an integer. Such sequences were popularized by Gale [8, 9], and subsequently Fomin and Zelevinsky found a useful technique - the Caterpillar Lemm ...
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... evaluated at x = b provide special sequences {un}. Of course, fn(1) = Fn, the Fibonacci numbers 0 , 1 , 1, 2, 3, 5, •••, and fn(2) = Pn, the Pell numbers 0 , 1 , 2, 5,12, 29, —.. Divisibility properties of the Fibonacci polynomials [1] and properties of the Pell numbers and the general sequences {fn ...
... evaluated at x = b provide special sequences {un}. Of course, fn(1) = Fn, the Fibonacci numbers 0 , 1 , 1, 2, 3, 5, •••, and fn(2) = Pn, the Pell numbers 0 , 1 , 2, 5,12, 29, —.. Divisibility properties of the Fibonacci polynomials [1] and properties of the Pell numbers and the general sequences {fn ...
Here - Dartmouth Math Home
... “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be supe ...
... “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be supe ...
Primality testing: variations on a theme of Lucas
... “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be supe ...
... “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be supe ...
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... Divide the positive integers into three disjoint subsets A - {^4^}, B - {B^} s and C = {Ck} by examining the smallest term Tk used in the unique Zeckendorf representation in terms of Tribonacci numbers. Let n e A if k = 2 mod 3, n e B if k E 3 mod 3, and n e C if k = 1 mod 3. The numbers An, Bn, and ...
... Divide the positive integers into three disjoint subsets A - {^4^}, B - {B^} s and C = {Ck} by examining the smallest term Tk used in the unique Zeckendorf representation in terms of Tribonacci numbers. Let n e A if k = 2 mod 3, n e B if k E 3 mod 3, and n e C if k = 1 mod 3. The numbers An, Bn, and ...
Revisiting a Number-Theoretic Puzzle: The Census
... but my youngest daughter does,” then this answer confirms that she does have a youngest daughter, which points to the triple {6, 6, 1}. Asking the last question then only tells us that the census taker is choosing between two triples, but the mother’s appropriate answer settles the dilemma. Accordin ...
... but my youngest daughter does,” then this answer confirms that she does have a youngest daughter, which points to the triple {6, 6, 1}. Asking the last question then only tells us that the census taker is choosing between two triples, but the mother’s appropriate answer settles the dilemma. Accordin ...
Catalan Numbers, Their Generalization, and Their Uses
... (p - 1)u' (by the inductive hypothesis). Hence v-p + l<(p1)(u- 1),sov<(p1)u. This c o m p l e t e s the i n d u c t i o n in the ~ - d i r e c t i o n . Figure 7, which gives a special, but not particular, case, may be helpful in following the argument. (ii) Let ~ be a good path to (k, (p - 1)k). We ...
... (p - 1)u' (by the inductive hypothesis). Hence v-p + l<(p1)(u- 1),sov<(p1)u. This c o m p l e t e s the i n d u c t i o n in the ~ - d i r e c t i o n . Figure 7, which gives a special, but not particular, case, may be helpful in following the argument. (ii) Let ~ be a good path to (k, (p - 1)k). We ...
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... Proof: We have z(l, j) = Fj+l for all j > 1, so that row 1 of Z is determined by z(l, 1) = 1 and / Assume k > 1 and that (2) holds for all y > 1, for all / < k. Write the Zeckendorf representation of z(k +1,1) as z(k +1,1) = Z^=1 chFh+l, noting that the following conditions hold: (i) q = l; (ii) ch ...
... Proof: We have z(l, j) = Fj+l for all j > 1, so that row 1 of Z is determined by z(l, 1) = 1 and / Assume k > 1 and that (2) holds for all y > 1, for all / < k. Write the Zeckendorf representation of z(k +1,1) as z(k +1,1) = Z^=1 chFh+l, noting that the following conditions hold: (i) q = l; (ii) ch ...
On the introductory notes on Artin`s Conjecture
... 5. Corollaries of the result by Heath-Brown An n-tuple of integer numbers x1 , . . . , xn is said to be multiplicative dependent (the numbers Q are then called multiplicative dependent) if there exist a1 , . . . , an integers, not all zero, such that n1 xai i = 1. Multiplicative independent means no ...
... 5. Corollaries of the result by Heath-Brown An n-tuple of integer numbers x1 , . . . , xn is said to be multiplicative dependent (the numbers Q are then called multiplicative dependent) if there exist a1 , . . . , an integers, not all zero, such that n1 xai i = 1. Multiplicative independent means no ...
SectionModularArithm..
... Notation for Special Sets Recall that a set is a collection of objects enclosed in braces. The objects in the sets are call elements. If a is an element of a set, we write a S . For example, 2 {1, 2, 3, 4} but 5 1, 2, 3, 4} . Sets can have both a finite and an infinite number of elements. The ...
... Notation for Special Sets Recall that a set is a collection of objects enclosed in braces. The objects in the sets are call elements. If a is an element of a set, we write a S . For example, 2 {1, 2, 3, 4} but 5 1, 2, 3, 4} . Sets can have both a finite and an infinite number of elements. The ...
A PROBABILISTIC INTERPRETATION OF A SEQUENCE RELATED
... expressed in terms of the classical Gegenbauer polynomials C n 2 . The coefficients a n are also generalized to a family of numbers {a n (µ)} with parameter µ. The special cases µ = 0 and µ = ± 12 are discussed in detail. Section 2 produces a recurrence for {a n } from which the facts that a n is in ...
... expressed in terms of the classical Gegenbauer polynomials C n 2 . The coefficients a n are also generalized to a family of numbers {a n (µ)} with parameter µ. The special cases µ = 0 and µ = ± 12 are discussed in detail. Section 2 produces a recurrence for {a n } from which the facts that a n is in ...
2.1 inductive reasoning and conjecture ink.notebook
... Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 7. If points A, B, and C are collinear, then AB + BC = AC. ...
... Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 7. If points A, B, and C are collinear, then AB + BC = AC. ...
CSCI 2610 - Discrete Mathematics
... factor (divisor) that is less than or equal to √n Proof: if n is composite, we know it has a factor a with 1 < a < n. IOW n = ab for some b > 1. So, either a ≤ √n or b ≤ √n (note, if a > √n and b > √n then ab > n, nope). OK, both a and b are divisors of n, and n has a positive divisor not exceeding ...
... factor (divisor) that is less than or equal to √n Proof: if n is composite, we know it has a factor a with 1 < a < n. IOW n = ab for some b > 1. So, either a ≤ √n or b ≤ √n (note, if a > √n and b > √n then ab > n, nope). OK, both a and b are divisors of n, and n has a positive divisor not exceeding ...
Lecture notes on cryptography and RSA.
... where n > 2 and odd. There are two very obvious solutions to this: x ≡ 1 and x ≡ −1. Are there other solutions? It turns out that if n is prime, or a power of a prime, then these are the only solutions; this fact we won’t prove. It’s also true that if n is not a prime or power of a prime (but still ...
... where n > 2 and odd. There are two very obvious solutions to this: x ≡ 1 and x ≡ −1. Are there other solutions? It turns out that if n is prime, or a power of a prime, then these are the only solutions; this fact we won’t prove. It’s also true that if n is not a prime or power of a prime (but still ...
Proof - Rose
... natural numbers. The simplest and most familiar is base-10, which is used in everyday life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than expone ...
... natural numbers. The simplest and most familiar is base-10, which is used in everyday life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than expone ...
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... We next state a simple rule that enables one to calculate the polynomial χG (x, y) recursively. In what follows, G\e denotes the graph obtained by removing the edge e from G, and for a subgraph H of G, the graph G\H is gotten from G by removing H and all the edges of G that are adjacent to vertices ...
... We next state a simple rule that enables one to calculate the polynomial χG (x, y) recursively. In what follows, G\e denotes the graph obtained by removing the edge e from G, and for a subgraph H of G, the graph G\H is gotten from G by removing H and all the edges of G that are adjacent to vertices ...