CS 103X: Discrete Structures Homework Assignment 7 — Solutions
... are two numbers i, j in this sample such that i|j. Solution For 1 ≤ i ≤ n, define a subset Ai of {1, 2, . . . , 2n} as follows: Ai = {x ∈ A : ∃a ∈ N : x = (2i − 1) × 2a }. The set Ai includes the i-th odd number and the products of that number with powers of 2, up to 2n. These sets have the property ...
... are two numbers i, j in this sample such that i|j. Solution For 1 ≤ i ≤ n, define a subset Ai of {1, 2, . . . , 2n} as follows: Ai = {x ∈ A : ∃a ∈ N : x = (2i − 1) × 2a }. The set Ai includes the i-th odd number and the products of that number with powers of 2, up to 2n. These sets have the property ...
Full-text PDF - American Mathematical Society
... Statement (4) is completely routine and (1) is essentially Theorem 5.5A of [1]. Statements (2) and (3) are established in §2. Throughout this note all spaces should be assumed to be CW-complexes of finite type. As a corollary to the statements (l)-(4) above, one sees that the number #[AaA; A] of pro ...
... Statement (4) is completely routine and (1) is essentially Theorem 5.5A of [1]. Statements (2) and (3) are established in §2. Throughout this note all spaces should be assumed to be CW-complexes of finite type. As a corollary to the statements (l)-(4) above, one sees that the number #[AaA; A] of pro ...
Solution
... A Carmichael number is a composite number n satisfying xn−1 ≡ 1 (mod n) for all x ∈ Z× n . Carmichael number are exactly those numbers for which there exist no Fermat witnesses, although they are not prime. The Fermat test will always answer ‘probably prime’ for these numbers, in spite of the fact t ...
... A Carmichael number is a composite number n satisfying xn−1 ≡ 1 (mod n) for all x ∈ Z× n . Carmichael number are exactly those numbers for which there exist no Fermat witnesses, although they are not prime. The Fermat test will always answer ‘probably prime’ for these numbers, in spite of the fact t ...
ODE Lecture Notes, Section 5.3
... To prove a solution of P x y Q x y R x y 0 y p x y q x y can be written as ...
... To prove a solution of P x y Q x y R x y 0 y p x y q x y can be written as ...
Lucas` square pyramid problem revisited
... infinitely many congruent numbers in each residue class modulo 8 (and, in particular, infinitely many squarefree congruent numbers, congruent to 1, 2, 3, 5, 6 and 7 modulo 8). We can generalize this as follows: ...
... infinitely many congruent numbers in each residue class modulo 8 (and, in particular, infinitely many squarefree congruent numbers, congruent to 1, 2, 3, 5, 6 and 7 modulo 8). We can generalize this as follows: ...
MEI Conference 2009 Proof
... 12. Every positive integer can be written in the form a 2 + b 2 − c 2 where a, b and c are integers 13. For any polynomial equation x n + an −1 x n −1 + ... + a2 x 2 + a1 + a0 = 0 where all the coefficients are integers, if any roots are rational numbers then they must be integers. 14. An equilatera ...
... 12. Every positive integer can be written in the form a 2 + b 2 − c 2 where a, b and c are integers 13. For any polynomial equation x n + an −1 x n −1 + ... + a2 x 2 + a1 + a0 = 0 where all the coefficients are integers, if any roots are rational numbers then they must be integers. 14. An equilatera ...
1998 - CEMC - University of Waterloo
... Since the total amount of floor covered when the rugs do not overlap is 200 m 2 and the total covered when they do overlap is 140 m 2 , then 60 m 2 of rug is wasted on double or triple layers. Thus, a + b + c + 2 k = 60 (2). Subtract equation (1) from equation (2) to get 2 k = 36 and solve for k = 1 ...
... Since the total amount of floor covered when the rugs do not overlap is 200 m 2 and the total covered when they do overlap is 140 m 2 , then 60 m 2 of rug is wasted on double or triple layers. Thus, a + b + c + 2 k = 60 (2). Subtract equation (1) from equation (2) to get 2 k = 36 and solve for k = 1 ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".