LECTURE 10: THE INTEGERS
... Suppose you start with $5 and you pay $1 each day for a cup of coffee. Suppose your credit is good at the store! After the first day you have $4 After the second day you have $3 After the third day you have $2 After the fourth day you have $1 After the fifth day you have $0 After the sixth day you o ...
... Suppose you start with $5 and you pay $1 each day for a cup of coffee. Suppose your credit is good at the store! After the first day you have $4 After the second day you have $3 After the third day you have $2 After the fourth day you have $1 After the fifth day you have $0 After the sixth day you o ...
A REVERSE SIERPI´NSKI NUMBER PROBLEM 1. introduction A
... A Sierpiński number k > 0 is an odd number such that k · 2n + 1 is composite for all integers n > 0. Waclaw Sierpiński, in 1960, proved that there are infinitely many such numbers [12] but found no exact values. (This is a dual of a problem of Euler that Erdös solved in 1950 [6].) In 1962 John Se ...
... A Sierpiński number k > 0 is an odd number such that k · 2n + 1 is composite for all integers n > 0. Waclaw Sierpiński, in 1960, proved that there are infinitely many such numbers [12] but found no exact values. (This is a dual of a problem of Euler that Erdös solved in 1950 [6].) In 1962 John Se ...
Natural Numbers: The counting numbers starting at 1: {1, 2, 3,
... greater than or equal to (closed circle or brackets) < less than (open circle or parentheses) less than or equal to (closed circle or brackets) For example: graph ...
... greater than or equal to (closed circle or brackets) < less than (open circle or parentheses) less than or equal to (closed circle or brackets) For example: graph ...
Graded assignment three
... P14: Use the arithmetic properties (or any of the previously established properties such as Proposition 1 on p. 12) of the integers to prove the following: Let a and b be integers. Prove that (a)·b (ab) . This is #4 in the section 1.2 textbook exercises, p. 13. Hints: You need to show that ab ...
... P14: Use the arithmetic properties (or any of the previously established properties such as Proposition 1 on p. 12) of the integers to prove the following: Let a and b be integers. Prove that (a)·b (ab) . This is #4 in the section 1.2 textbook exercises, p. 13. Hints: You need to show that ab ...
Lecture 5. Introduction to Set Theory and the Pigeonhole Principle
... More challenging problems. We now show how to make a point disappear. Proposition 3. The closed unit interval [0, 1] is equinumerous with the half closed unit interval [0, 1). Proof. Define the function f : [0, 1] → [0, 1) as follows: ...
... More challenging problems. We now show how to make a point disappear. Proposition 3. The closed unit interval [0, 1] is equinumerous with the half closed unit interval [0, 1). Proof. Define the function f : [0, 1] → [0, 1) as follows: ...
Week 5 Chapter 4 CheckPoint Complete the CheckPoint and post to
... .., 96, 98, 100. Every other number is a multiple of 4 and therefore is divisible by 4. Therefore one half of the set is divisible by 4. b. Incorrect. Think about 12. Both 2 and 4 are factors of 12 and therefore divide 12 however 8 isn't a factor of 12. c. Correct. Because 12 divides the number N, a ...
... .., 96, 98, 100. Every other number is a multiple of 4 and therefore is divisible by 4. Therefore one half of the set is divisible by 4. b. Incorrect. Think about 12. Both 2 and 4 are factors of 12 and therefore divide 12 however 8 isn't a factor of 12. c. Correct. Because 12 divides the number N, a ...
GETTING STARTED ON INEQUALITIES
... some algebra reduce this to the fact that the square of a difference of two numbers is non-negative. (3) A piece of wire 40 cm long is bent to form the contour of a rectangle. a) Find (with proof!) the largest possible area of a rectangle thus formed. b) The rectangle thus formed is inscribed in a c ...
... some algebra reduce this to the fact that the square of a difference of two numbers is non-negative. (3) A piece of wire 40 cm long is bent to form the contour of a rectangle. a) Find (with proof!) the largest possible area of a rectangle thus formed. b) The rectangle thus formed is inscribed in a c ...
Full text
... It is known that there a r e infinitely many solutions of the equation A3 + B3 + C 3 = D3 in positive integers (see Shanks [5, p. 157]). Here as a simple application of (6) we shall construct certain sets of non-trivial positive integral solutions of the 2-sided 3-cube equation ...
... It is known that there a r e infinitely many solutions of the equation A3 + B3 + C 3 = D3 in positive integers (see Shanks [5, p. 157]). Here as a simple application of (6) we shall construct certain sets of non-trivial positive integral solutions of the 2-sided 3-cube equation ...
