1. Problems and Results in Number Theory
... k integers . Then the necessary and sufficient condition that there should be infinitely many values of n for which all the integers n +a„ i = 1, . . . , k are primes is that for no prime p should the set a,, . . . , a k form a complete set of residues mod p . The condition is clearly necessary ; th ...
... k integers . Then the necessary and sufficient condition that there should be infinitely many values of n for which all the integers n +a„ i = 1, . . . , k are primes is that for no prime p should the set a,, . . . , a k form a complete set of residues mod p . The condition is clearly necessary ; th ...
On the multiplicative properties of arithmetic functions
... where Σt is taken over all k ^ n such that *Bk = 2 and either p1 \ k or p2\k; Σ2 is taken over those k ^ n for which *!?£ > 2 and such that either pt\k or p21 k; and J£8 is taken over the remaining k <; n. By Lemma 3.2, 2Ί *Bk/d(k) = o(w) To estimate J?2 we let A represent the set of all positive in ...
... where Σt is taken over all k ^ n such that *Bk = 2 and either p1 \ k or p2\k; Σ2 is taken over those k ^ n for which *!?£ > 2 and such that either pt\k or p21 k; and J£8 is taken over the remaining k <; n. By Lemma 3.2, 2Ί *Bk/d(k) = o(w) To estimate J?2 we let A represent the set of all positive in ...
![[hal-00574623, v2] Averaging along Uniform Random Integers](http://s1.studyres.com/store/data/019969824_1-25527940ea4f317ec31969269e4745aa-300x300.png)
ON A LEMMA OF LITTLEWOOD AND OFFORD
... We clearly can assume that all the Xi are not less than 1. To every sum ]Qfc=i€fcxfc w e associate a subset of the integers from 1 to n as follows: k belongs to the subset if and only if e&= + 1 . If two sums 2^J»i€^jb and ]Cfc=i*& #& are both in 7, neither of the corresponding subsets can contain t ...
... We clearly can assume that all the Xi are not less than 1. To every sum ]Qfc=i€fcxfc w e associate a subset of the integers from 1 to n as follows: k belongs to the subset if and only if e&= + 1 . If two sums 2^J»i€^jb and ]Cfc=i*& #& are both in 7, neither of the corresponding subsets can contain t ...
A54 INTEGERS 10 (2010), 733-745 REPRESENTATION NUMBERS
... v are adjacent if and only if gcd(f (u)−f (v), r) = 1: we refer to f as a representative labeling of G. Equivalently, G is representable modulo r if there exists an injective map f : V (G) → Zr such that vi is adjacent to vj if and only if f (i) − f (j) is a unit of (the ring) Zr . The representatio ...
... v are adjacent if and only if gcd(f (u)−f (v), r) = 1: we refer to f as a representative labeling of G. Equivalently, G is representable modulo r if there exists an injective map f : V (G) → Zr such that vi is adjacent to vj if and only if f (i) − f (j) is a unit of (the ring) Zr . The representatio ...
1 Sets
... Let S be a set. A multiset M over S is a collection of objects from S that the elements of S can be repeated; the repeated objects are indistinguishable. For instance, the collection {a, a, b, c, c, c} is a multiset of 6 objects over the set {a, b, c}; of course it is also a multiset over the set {a ...
... Let S be a set. A multiset M over S is a collection of objects from S that the elements of S can be repeated; the repeated objects are indistinguishable. For instance, the collection {a, a, b, c, c, c} is a multiset of 6 objects over the set {a, b, c}; of course it is also a multiset over the set {a ...
Alg II 5-7 The Binomial Theorem
... The "coefficients only" column matches the numbers in Pascal's Triangle. Pascal's Triangle is a triangular array of numbers in which the first and last number of each row is 1. Each of the other numbers in the row is the sum of the two numbers above it. ...
... The "coefficients only" column matches the numbers in Pascal's Triangle. Pascal's Triangle is a triangular array of numbers in which the first and last number of each row is 1. Each of the other numbers in the row is the sum of the two numbers above it. ...
2. Primes Primes. • A natural number greater than 1 is prime if it
... (which will be larger than k) such that no natural number less than k and greater than 1 divides n. 2.25. Theorem (Infinitude of Primes Theorem). There are infinitely many prime numbers. 2.26. Question. After you have devised a proof or proofs for 2.25, what were the most clever or most difficult pa ...
... (which will be larger than k) such that no natural number less than k and greater than 1 divides n. 2.25. Theorem (Infinitude of Primes Theorem). There are infinitely many prime numbers. 2.26. Question. After you have devised a proof or proofs for 2.25, what were the most clever or most difficult pa ...
Solutions to Exam 1 Problem 1. Suppose that A and B are sets
... You will not receive maximum points on this problem if you use the fundamental theorem of arithmetic in your proof. Solution 1: Suppose that a is a positive integer such that a3 = 4. Then 2|a3 . Since 2 is a prime and a3 = a(a2 ), 2 divides one of the factors a or a2 of a3 . If 2 divides a2 , then 2 ...
... You will not receive maximum points on this problem if you use the fundamental theorem of arithmetic in your proof. Solution 1: Suppose that a is a positive integer such that a3 = 4. Then 2|a3 . Since 2 is a prime and a3 = a(a2 ), 2 divides one of the factors a or a2 of a3 . If 2 divides a2 , then 2 ...
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"".