Number Theory Course 1. Let ? and ? be integers. Show that if ? + ?...

Number Theory Course

1. Let ? and ? be integers. Show that if ? + ? is odd, then ? and ? are of the opposite parity:
a. By giving a proof by contradiction

2. Let n be a natural number. Prove that (?+ 1)2 −1 is even if and only if n is even. You may use any suitable proof technique.

3. Using a proof by induction, show that 5 | 6? − 1, for all natural numbers ? .

Expert Solution

Colby Messinger answered 6 months ago

Question in graph theory: 1. Let (a1,a2,a3,...an) be a sequence of integers. Given that the sum...

Question in graph theory: 1. Let (a1,a2,a3,...an) be a sequence of integers. Given that the sum of all integers = 2(n-1) Write an algorithm that, starting with a sequence (a1,a2,a3,...an) of positive integers, either constructs a tree with this degree sequence or concludes that none is possible.

Number Theory Show that 18! = -1 (mod 437 = 19x23)

a) How many integers in between 1 and 106 have an even number of divisors? Show...

a) How many integers in between 1 and 106 have an even number of divisors? Show work proving your answer. Express your answer in prime factorized form. b) With proof, determine all integer solutions to the following equation: 1935x + 2322y = 177

For all integers n > 2, show that the number of integer partitions of n in...

For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.

Let x, y be integers, and n be a natural number. Prove that x ^(2n) −...

Let x, y be integers, and n be a natural number. Prove that x ^(2n) − y ^(2n) is divisible by x + y

Let A be an array of non-decreasing N integers. Write an algorithm that returns the number...

Let A be an array of non-decreasing N integers. Write an algorithm that returns the number of pairs of integers in A that are equal. Analyze your algorithm thoroughly. Your analysis should include a thorough examination of both the best and the worst-case scenarios. This includes a description of what the best and worst cases would look like. You should include both space and time in your analysis, but keep in mind that space refers to “extra space,” meaning in...

Let us choose seven arbitrary distinct positive integers, not exceeding 24. Show that there will be...

Let us choose seven arbitrary distinct positive integers, not exceeding 24. Show that there will be at least two subsets chosen from these seven numbers with equal total sums. (Keep in mind that sets, and hence subsets, have no repeated elements.) Hint: How many subsets can you form altogether? What is the largest total sum of such a subset?

Number Theory 1

What is the quickest way to determine if a number is a perfect square?

Let P be the uniform probability on the integers from 1 to 99. Let B be...

Let P be the uniform probability on the integers from 1 to 99. Let B be the subset of numbers which have the digit 3. Let A be the subset of even numbers. What is P(A), P(B)? What is P(A|B)? P(B|A)?

Let G be a graph whose vertices are the integers 1 through 8, and let the...

Let G be a graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: vertex adjacent vertices 1 (2, 3, 4) 2 (1, 3, 4) 3 (1, 2, 4) 4 (1, 2, 3, 6) 5 (6, 7, 8) 6 (4, 5, 7) 7 (5, 6, 8) 8 (5, 7) Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the...

Question