CSE 215: Foundations of Computer Science Recitation
... Unit 9: Direct Proofs Involving Divisibility and the Quotient-Remainder Theorem 7. Prove that for all integers a, b and c, if a|b and a|c then a|(b + c). Proof: Suppose a, b and c are any integers such that a|b and a|c. We must show that a|(b + c). By definition of divides, b = ar and c = as for so ...
... Unit 9: Direct Proofs Involving Divisibility and the Quotient-Remainder Theorem 7. Prove that for all integers a, b and c, if a|b and a|c then a|(b + c). Proof: Suppose a, b and c are any integers such that a|b and a|c. We must show that a|(b + c). By definition of divides, b = ar and c = as for so ...
Second Proof: Every Positive Integer is a Frobenius
... Lemma 2.2 If n is an even number, then there exist a, b, c such that N (a, b, c) = n. Proof. If a = b, then N (a, a, c) = N (a, c) = (a − 1)(c − 1) by Sylvester’s formula in [6]. Let n = 2k where k is any positive integer. Set a = b = 2 and c = 2k + 1, then we get N (2, 2, 2k + 1) = n by (1). Lemma ...
... Lemma 2.2 If n is an even number, then there exist a, b, c such that N (a, b, c) = n. Proof. If a = b, then N (a, a, c) = N (a, c) = (a − 1)(c − 1) by Sylvester’s formula in [6]. Let n = 2k where k is any positive integer. Set a = b = 2 and c = 2k + 1, then we get N (2, 2, 2k + 1) = n by (1). Lemma ...
CHAP01 Divisibility
... suppose that numbers whose absolute value is smaller than |n| can be factorised into primes. If n is prime then h = 1 and p1 = n. If n is composite then n = ab for some numbers a, b where |a| and |b| are bigger than 1. Since |a| and |b| are smaller than |n| it follows by the strong principle of indu ...
... suppose that numbers whose absolute value is smaller than |n| can be factorised into primes. If n is prime then h = 1 and p1 = n. If n is composite then n = ab for some numbers a, b where |a| and |b| are bigger than 1. Since |a| and |b| are smaller than |n| it follows by the strong principle of indu ...
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 ...
Keys GEO SY13-14 Openers 4-3
... How do I use similarity and congruence to find numerical relationships among triangle parts? Objective(s) Students will be able to (SWBAT) establish the congruence or non-congruence of two geometric figures. SWBAT establish the similarity or non-similarity of two geometric figures. SWBAT fin ...
... How do I use similarity and congruence to find numerical relationships among triangle parts? Objective(s) Students will be able to (SWBAT) establish the congruence or non-congruence of two geometric figures. SWBAT establish the similarity or non-similarity of two geometric figures. SWBAT fin ...
series with non-zero central critical value
... the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is a modular elliptic curve and, if L(E, 1) 6= 0, then the rank of E is 0. Thus, if f has the property that a positive proportio ...
... the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is a modular elliptic curve and, if L(E, 1) 6= 0, then the rank of E is 0. Thus, if f has the property that a positive proportio ...
Equidistribution and Primes - Princeton Math
... (1) I have chosen to talk on this topic because I believe it has a wide appeal and also there have been some interesting developments in recent years on some of these classical problems. The questions that we discuss are generalizations of the twin prime conjecture; that there are infinitely many pri ...
... (1) I have chosen to talk on this topic because I believe it has a wide appeal and also there have been some interesting developments in recent years on some of these classical problems. The questions that we discuss are generalizations of the twin prime conjecture; that there are infinitely many pri ...
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"".