Primes, Composites and Integer Division
... where the pi ’s are prime numbers, the qi ’s are integers greater than 0, and the above decomposition of n in terms of the pi ’s and qi ’s is unique. Remark. We will give the proof of this theorem later in the course. But, hopefully, most have seen elementary algorithms to express large integers in ...
... where the pi ’s are prime numbers, the qi ’s are integers greater than 0, and the above decomposition of n in terms of the pi ’s and qi ’s is unique. Remark. We will give the proof of this theorem later in the course. But, hopefully, most have seen elementary algorithms to express large integers in ...
Pigeonhole Principle - Department of Mathematics
... ..., 2k} whose base 2 representation has precisely three 1 s. (a) Prove that, for eachpositive integer m, there exists at least one positive integer k such thatf(k)=m. (b) Determine all positive integers m for which there exists exactly one k with ...
... ..., 2k} whose base 2 representation has precisely three 1 s. (a) Prove that, for eachpositive integer m, there exists at least one positive integer k such thatf(k)=m. (b) Determine all positive integers m for which there exists exactly one k with ...
Binomial Coefficients, Congruences, Lecture 3 Notes
... (Definition) Binomial Coefficient: If α ∈ C and k is a non-negative integer, ...
... (Definition) Binomial Coefficient: If α ∈ C and k is a non-negative integer, ...
Unique Factorization
... Definition The integers are the numbers . . . , −2, −1, 0, 1, 2, 3, . . . . We denote this set of numbers by Z. Definition A prime number is a positive integer greater than 1 whose only positive divisors are 1 and itself. Example We learned how to factor integers in elementary school: 120 = 23 · 3 · ...
... Definition The integers are the numbers . . . , −2, −1, 0, 1, 2, 3, . . . . We denote this set of numbers by Z. Definition A prime number is a positive integer greater than 1 whose only positive divisors are 1 and itself. Example We learned how to factor integers in elementary school: 120 = 23 · 3 · ...
Lecture 11: the Euler φ-function In the light of the previous lecture
... In the light of the previous lecture, we shall now look in more detail at the function defined there. Let n ≥ 1 be a natural number. Recall that we defined φ(n) to be the number of natural numbers 1 ≤ m ≤ n and coprime to n. This function is called Euler’s φ-function. For example, let’s calculate φ( ...
... In the light of the previous lecture, we shall now look in more detail at the function defined there. Let n ≥ 1 be a natural number. Recall that we defined φ(n) to be the number of natural numbers 1 ≤ m ≤ n and coprime to n. This function is called Euler’s φ-function. For example, let’s calculate φ( ...
Homework 5 (=Exam Practice)
... Homework 5 (=Exam Practice) The exam will cover everything we did so far: induction, binomial theorem, divisibility, division algorithm, gcd, euclidean algorithm, fundamental theorem of arithmetic, prime numbers, congruences, divisibility tests, chinese remainder theorem, Fermat’s theorem, pseudopri ...
... Homework 5 (=Exam Practice) The exam will cover everything we did so far: induction, binomial theorem, divisibility, division algorithm, gcd, euclidean algorithm, fundamental theorem of arithmetic, prime numbers, congruences, divisibility tests, chinese remainder theorem, Fermat’s theorem, pseudopri ...
The Impossibility of Trisecting an Angle with Straightedge and
... and compass. That is, given an angle θ , give a procedure using only straightedge and compass that will construct the angle θ 3 in a finite number of steps. This problem, which dates to around 400 B.C., fascinated mathematicians and amateurs over the centuries and many “solutions” have been proposed ...
... and compass. That is, given an angle θ , give a procedure using only straightedge and compass that will construct the angle θ 3 in a finite number of steps. This problem, which dates to around 400 B.C., fascinated mathematicians and amateurs over the centuries and many “solutions” have been proposed ...
ON THE DIVISIBILITY OF THE CLASS NUMBER OF
... 3. Application. To show that there exist infinitely many fields each with class number divisible by g, we proceed as follows. Theorem 1 shows that there are at least (1/25) 3^ with class number divisible by g. Let gt = gχ be such that the class number of none of these fields is divisible by g t Then ...
... 3. Application. To show that there exist infinitely many fields each with class number divisible by g, we proceed as follows. Theorem 1 shows that there are at least (1/25) 3^ with class number divisible by g. Let gt = gχ be such that the class number of none of these fields is divisible by g t Then ...
LECTURE 14: LINEAR SYSTEMS AND EQUILIBRIUM SOLUTIONS
... is given by det(A) = ad − bc. c d Theorem 1. Let A be a 2 × 2 matrix as above. Then det(A) 6= 0 if and only if the system of equations (1) has a unique solution. This is also true for n × n matrices with the corresponding ~ /dt = A · Y~ has only the origin as its system of equations. Moreover, det(A ...
... is given by det(A) = ad − bc. c d Theorem 1. Let A be a 2 × 2 matrix as above. Then det(A) 6= 0 if and only if the system of equations (1) has a unique solution. This is also true for n × n matrices with the corresponding ~ /dt = A · Y~ has only the origin as its system of equations. Moreover, det(A ...
composite and prime numbers
... infinite number of primes generated by P1=8n+1. On searching, we have not been able to find any primes for S1 for all integer values of n between n=17 and n=3000. One is thus led to the conjecture that2n+1 is a composite number for all positive integer values of n not equal to 1, 2, 4, 8, or16 . Or ...
... infinite number of primes generated by P1=8n+1. On searching, we have not been able to find any primes for S1 for all integer values of n between n=17 and n=3000. One is thus led to the conjecture that2n+1 is a composite number for all positive integer values of n not equal to 1, 2, 4, 8, or16 . Or ...
CS 173: Discrete Structures, Fall 2011 Homework 3
... When writing your proofs, be sure to use the definitions of key concepts (e.g. divisible) as presented in class. Also one goal of this problem set is to practice certain proof techniques. So be sure to use the proof technique specified by the problem instructions, even if there might be other ways t ...
... When writing your proofs, be sure to use the definitions of key concepts (e.g. divisible) as presented in class. Also one goal of this problem set is to practice certain proof techniques. So be sure to use the proof technique specified by the problem instructions, even if there might be other ways t ...
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"".