Integers and division
... Theorem: If n is a composite then n has a prime divisor less than or equal to n . Proof: • If n is composite, then it has a positive integer factor a such that 1 < a < n by definition. This means that n = ab, where b is an integer greater than 1. • Assume a > √n and b > √n. Then ab > √n√n = n, which ...
... Theorem: If n is a composite then n has a prime divisor less than or equal to n . Proof: • If n is composite, then it has a positive integer factor a such that 1 < a < n by definition. This means that n = ab, where b is an integer greater than 1. • Assume a > √n and b > √n. Then ab > √n√n = n, which ...
1 - Columbia Math Department
... Now we can conclude another proof of Proposition 1.4 using this lemma. If all the Fermat numbers were relatively prime, then each must be divisible by a different prime from all the others. So, pn ≤ n Fn = 22 + 1. Thus, we have π(x) > log log x. Challenge 1. Find other elementary proofs of the prime ...
... Now we can conclude another proof of Proposition 1.4 using this lemma. If all the Fermat numbers were relatively prime, then each must be divisible by a different prime from all the others. So, pn ≤ n Fn = 22 + 1. Thus, we have π(x) > log log x. Challenge 1. Find other elementary proofs of the prime ...
Problem Solving in Math (Math 43900) Fall 2013
... Solution: From the recurrence in the first part, we get an = Fn+2 , so Fn counts the number of 0-1 strings of length n − 2 with no two consecutive 1’s. We can count such strings by first deciding on k, the number of 1’s, and by the second part, the number of n−1−k . Summing over k we get the resul ...
... Solution: From the recurrence in the first part, we get an = Fn+2 , so Fn counts the number of 0-1 strings of length n − 2 with no two consecutive 1’s. We can count such strings by first deciding on k, the number of 1’s, and by the second part, the number of n−1−k . Summing over k we get the resul ...
On the Number of Prime Numbers less than a Given Quantity
... modern applications of prime numbers in areas such as physics, cryptography, and coding theory make any information that can be obtained about them of great value. In this paper we are interested in the distribution of prime numbers in M, where M ⊆ N is a given infinite set. This is a fundamental qu ...
... modern applications of prime numbers in areas such as physics, cryptography, and coding theory make any information that can be obtained about them of great value. In this paper we are interested in the distribution of prime numbers in M, where M ⊆ N is a given infinite set. This is a fundamental qu ...
ünivalence of continued fractions and stieltjes transforms1
... Stieltjes transforms; these theorems depend upon the well known fact that (cf. [2]2) a function/(z) analytic in a convex region R is univalent in R if there exists a real constant a for which eiaf'(z) is nowhere real in R. §3 contains similar results for functions represented by classes of continued ...
... Stieltjes transforms; these theorems depend upon the well known fact that (cf. [2]2) a function/(z) analytic in a convex region R is univalent in R if there exists a real constant a for which eiaf'(z) is nowhere real in R. §3 contains similar results for functions represented by classes of continued ...
Probabilistic proofs of existence of rare events, Springer Lecture
... k ≥ 1 there are tournaments in which for every set of k players there is one who beats them all. The proof given in [Er] actually shows that for every fixed k if the number n of players is sufficiently large then almost all tournaments with n players satisfy this property, i.e., the probability that ...
... k ≥ 1 there are tournaments in which for every set of k players there is one who beats them all. The proof given in [Er] actually shows that for every fixed k if the number n of players is sufficiently large then almost all tournaments with n players satisfy this property, i.e., the probability that ...
M328K Final Exam Solutions, May 10, 2003 1. “Bibonacci” numbers
... 1. “Bibonacci” numbers. The Bibonacci numbers b1 , b2 , . . . are defined by b1 = 1, b2 = 1, and, for n > 2, bn = bn−1 + 2bn−2 . a) Prove that, for all positive integers n, bn ≤ 2n−1 . The proof is by generalized induction. It is true for n = 1 and n = 2. Now suppose it is true for all n up to k − 1 ...
... 1. “Bibonacci” numbers. The Bibonacci numbers b1 , b2 , . . . are defined by b1 = 1, b2 = 1, and, for n > 2, bn = bn−1 + 2bn−2 . a) Prove that, for all positive integers n, bn ≤ 2n−1 . The proof is by generalized induction. It is true for n = 1 and n = 2. Now suppose it is true for all n up to k − 1 ...
Chinese Reminder Theorem
... to obtain an answer for each prime factor power of m. The advantage is that it is often easier to analyze congruences mod primes (or mod prime powers) than to work with composite numbers. Example. Here is an example. Find a solution to 13x ≡ 1 (mod 70). The two methods of solution are worthy of care ...
... to obtain an answer for each prime factor power of m. The advantage is that it is often easier to analyze congruences mod primes (or mod prime powers) than to work with composite numbers. Example. Here is an example. Find a solution to 13x ≡ 1 (mod 70). The two methods of solution are worthy of care ...
Number Theory Begins - Princeton University Press
... This will become a recurring theme as we continue our study of number theory. Just as we did with square numbers we will assign various traits to numbers and speak of there being prime numbers, regular numbers, perfect numbers, triangular numbers, Fibonacci numbers, Mersenne numbers—the list goes on ...
... This will become a recurring theme as we continue our study of number theory. Just as we did with square numbers we will assign various traits to numbers and speak of there being prime numbers, regular numbers, perfect numbers, triangular numbers, Fibonacci numbers, Mersenne numbers—the list goes on ...
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"".