slides03 - Duke University
... • Suppose that x+y were not irrational. Then x+y would be rational, so integers i,j: x+y = i/j. So, let is and js be any such integers where x+y = is/ js. • Now, with all these things named, we can start seeing what happens when we put them together. • So, we have that (ix/jx) + y = (is/js). • Obs ...
... • Suppose that x+y were not irrational. Then x+y would be rational, so integers i,j: x+y = i/j. So, let is and js be any such integers where x+y = is/ js. • Now, with all these things named, we can start seeing what happens when we put them together. • So, we have that (ix/jx) + y = (is/js). • Obs ...
On Existence of Infinitely Many Primes of The Form x2+1
... Are there infinitely many primes of the form x2+1? It has been an unsolved problem in mathematics. Landau listed it as one of four basic problems about primes at ICM 1912[1]. H. Iwaniec showed that there are infinitely many numbers of the form n2+1 with at most two prime factors in 1987[2]. A theor ...
... Are there infinitely many primes of the form x2+1? It has been an unsolved problem in mathematics. Landau listed it as one of four basic problems about primes at ICM 1912[1]. H. Iwaniec showed that there are infinitely many numbers of the form n2+1 with at most two prime factors in 1987[2]. A theor ...
Math 3: Unit 1 – Reasoning and Proof Inductive, Deductive
... If Diego goes shopping, then he will buy a pretzel. If the mall is open, then Angela and Diego will go shopping. If Angela goes shopping, then she will buy pizza. The mall is open ...
... If Diego goes shopping, then he will buy a pretzel. If the mall is open, then Angela and Diego will go shopping. If Angela goes shopping, then she will buy pizza. The mall is open ...
7. Prime Numbers Part VI of PJE
... “Interesting” problems concerning primes. The following are conjectures and are all examples of problems that can be simply stated yet for which the answers are as yet unknown. 1) Goldbach’s Conjecture, Is every even integer n ≥ 4 the sum of two primes? ...
... “Interesting” problems concerning primes. The following are conjectures and are all examples of problems that can be simply stated yet for which the answers are as yet unknown. 1) Goldbach’s Conjecture, Is every even integer n ≥ 4 the sum of two primes? ...
Special Products – Blue Level Problems In
... In Exercises 11 - 13, Rick Claims that if you multiply four consecutive integers together and add 1, you always get a perfect square. 11. Show that Rick’s statement is true if the smallest of the integers is 2. 12. Show that Rick’s statement is true if the smallest of the integers is 3. 13. Suppose ...
... In Exercises 11 - 13, Rick Claims that if you multiply four consecutive integers together and add 1, you always get a perfect square. 11. Show that Rick’s statement is true if the smallest of the integers is 2. 12. Show that Rick’s statement is true if the smallest of the integers is 3. 13. Suppose ...
Mathematical Reasoning (Part III)
... then the proof begins by assuming the existence of a counterexample of this statement. Therefore, the proof might begin with: – Assume, to the contrary, that there exists some element x ∈ D for which P (x) is true and Q(x) is false. or – By contradiction, assume, that there exists an element x ∈ D s ...
... then the proof begins by assuming the existence of a counterexample of this statement. Therefore, the proof might begin with: – Assume, to the contrary, that there exists some element x ∈ D for which P (x) is true and Q(x) is false. or – By contradiction, assume, that there exists an element x ∈ D s ...
Fermat Numbers: A False Conjecture Leads to Fun and
... his passion was mathematics. He shone in arithmetic (which in its more advanced form, is what we call number theory today), but made seminal contributions in other parts of mathematics as well, and even in physics. Great mathematicians, and Fermat was squarely in that league, are characterized by de ...
... his passion was mathematics. He shone in arithmetic (which in its more advanced form, is what we call number theory today), but made seminal contributions in other parts of mathematics as well, and even in physics. Great mathematicians, and Fermat was squarely in that league, are characterized by de ...
A question on linear independence of square roots Martin Klazar1
... As for the references, inspection of the memory and (manual!) search of the library first revealed that the textbooks by Hlawka, Schoißengaier and Taschner [4, Exercise 7 to Chapter 2] and Laczkovich [5, Exercises 4.2 and 4.3] contain the original question as an exercise. They give no references but ...
... As for the references, inspection of the memory and (manual!) search of the library first revealed that the textbooks by Hlawka, Schoißengaier and Taschner [4, Exercise 7 to Chapter 2] and Laczkovich [5, Exercises 4.2 and 4.3] contain the original question as an exercise. They give no references but ...
4-3: Alternating Series, and the Alternating Series Theorem
... down to a value as n → ∞ (if you’re curious, the above series converges to ln 2 which we’ll prove later in the semester). • Definition: A series is called Pan alternating series if the terms alternate in sign. That is, an alternating series is a series of the form (−1)k+1 ak where ak > 0 for all k. ...
... down to a value as n → ∞ (if you’re curious, the above series converges to ln 2 which we’ll prove later in the semester). • Definition: A series is called Pan alternating series if the terms alternate in sign. That is, an alternating series is a series of the form (−1)k+1 ak where ak > 0 for all k. ...
Rational Numbers, Divisibility and the Quotient Remainder Theorem
... • If n is a composite integer, then n has a prime divisor less than or equal to the square root of n • Show that 899 is composite • Proof ...
... • If n is a composite integer, then n has a prime divisor less than or equal to the square root of n • Show that 899 is composite • Proof ...
Answer sheet for 2012 final exam
... any continuous map of the 2-dimensional disk D2 to itself has a fixed point. You may assume π1 S 1 = Z. Solution: See page 32 of Hatcher. ...
... any continuous map of the 2-dimensional disk D2 to itself has a fixed point. You may assume π1 S 1 = Z. Solution: See page 32 of Hatcher. ...
NUMBER THEORY 1. Divisor Counting Theorem 1. A number is a
... The method used in these proofs can be generalized to create a better primality test than the one described in corollary 1. Theorem 11 (Existence of factorizations). Every composite number can be expressed as a product of two or more prime numbers. Proof. If n is composite, it can be expressed as a ...
... The method used in these proofs can be generalized to create a better primality test than the one described in corollary 1. Theorem 11 (Existence of factorizations). Every composite number can be expressed as a product of two or more prime numbers. Proof. If n is composite, it can be expressed as a ...
m5zn_8a0e185bfba5c83
... The proof is established by showing that an object a with P(a) is true must exist without explicitly demonstrating one. Proofs by contradiction are usually used in such cases. Example: Let x1,x2,..,xn be positive integers such that their average is m. prove that there exists xi such that xi≥m ...
... The proof is established by showing that an object a with P(a) is true must exist without explicitly demonstrating one. Proofs by contradiction are usually used in such cases. Example: Let x1,x2,..,xn be positive integers such that their average is m. prove that there exists xi such that xi≥m ...
ENGG 2440A: Discrete Mathematics for Engineers Homework 2 The
... • Case 1: a is friends with at least 4 other people in the collection. • Case 2: a is a stranger to at least 4 other people in the collection. One of these two cases must hold. Let’s discuss Case 1. If all the people who are friends with a are strangers among themselves, this is a group of 4 strange ...
... • Case 1: a is friends with at least 4 other people in the collection. • Case 2: a is a stranger to at least 4 other people in the collection. One of these two cases must hold. Let’s discuss Case 1. If all the people who are friends with a are strangers among themselves, this is a group of 4 strange ...
Prime Factors of Cyclotomic Class Numbers
... To search for the prime factors of Pd, we therefore try as divisors of Pd only the numbers in the arithmetic progression 2xdx + 1 (x — 1, 2, 3, . . . ). The first such divisor is either a prime or a power of a prime. After removing all such factors below some limit, an attempt can be made to represe ...
... To search for the prime factors of Pd, we therefore try as divisors of Pd only the numbers in the arithmetic progression 2xdx + 1 (x — 1, 2, 3, . . . ). The first such divisor is either a prime or a power of a prime. After removing all such factors below some limit, an attempt can be made to represe ...