Elementary Number Theory: Proofs and Integers

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Why is it important to try to prove a discovered statement is true?

It guarantees the statement is universally accepted.

It confirms the statement is genuine. (correct)

It often leads to discovering that the statement is false. (correct)

It is just a formal requirement in mathematics.

What must you understand to evaluate the truth or falsity of a mathematical statement?

Common misconceptions related to the topic.

The historical context of mathematical theories.

The meanings of all terms in the statement. (correct)

The background of the statement's discoverer.

Which of the following integers is considered prime?

7 (correct)

How would you classify the integer 0 based on its evenness or oddness?

Even Signup and view all the answers

What is the outcome of the statement that every integer greater than 1 is either prime or composite?

True for all integers greater than 1 Signup and view all the answers

To prove a statement of the form ∃x ∈ D such that Q(x), what must you find?

At least one x in D that makes Q(x) true. Signup and view all the answers

What does the integer 1 qualify as in terms of prime and composite classification?

Neither prime nor composite Signup and view all the answers

Which of the following correctly describes how mathematicians handle definitions?

They define terms carefully and precisely. Signup and view all the answers

What must be shown to prove that m + n is even?

m and n can be expressed as 2r and 2s respectively. Signup and view all the answers

Which of the following is an important guideline in writing a proof?

Begin with a clear statement of the theorem. Signup and view all the answers

What should be marked to signify the start of a proof?

Proof Signup and view all the answers

Which of the following is not a recommended style for writing proofs?

Using incomplete sentences. Signup and view all the answers

What does it mean for a proof to be self-contained?

It should explain the meaning of every variable used. Signup and view all the answers

How should assumptions be described in a proof?

Preface them with 'Suppose' or 'Assume'. Signup and view all the answers

Why is it important to keep the reader informed about each statement in a proof?

To ensure they know when something is assumed or established. Signup and view all the answers

What is the purpose of introducing new variables during a proof?

To reference known quantities that can be used later. Signup and view all the answers

What is the first step when presented with a statement to be proved?

Ask yourself whether you believe the statement to be true. Signup and view all the answers

Which representation is used for even integers in the proof process?

2k for integer k. Signup and view all the answers

What is the most effective method for proving a universal statement?

Generalizing from the generic particular Signup and view all the answers

What conclusion needs to be shown to complete the proof that the sum of two even integers is even?

The sum of the two integers, m and n, is even. Signup and view all the answers

What does the formal restatement of the problem involve?

Showing that two integers being even implies their sum is even. Signup and view all the answers

In the context of universal statements, what does the term 'generic particular' refer to?

Any number regardless of size Signup and view all the answers

What is the outcome if you correctly apply the method of generalizing from the generic particular to a statement of the form 'If P(x), then Q(x)?'

You have proved the statement using direct proof Signup and view all the answers

Which law of algebra is used to demonstrate that the sum of two even integers is even?

Distributive Law Signup and view all the answers

What must be known to connect the starting point of the proof to the conclusion?

What it means for an integer to be even. Signup and view all the answers

To prove a universal statement for all elements in set D using the generic particular method, what must you assume about x?

x is an arbitrary but specific choice from D Signup and view all the answers

What is the meaning of 'arbitrarily chosen' in the context of proving statements about even integers?

Any even integers can be selected, regardless of their specific values. Signup and view all the answers

What is the critical condition for the truth of an if-then statement 'If P(x), then Q(x)'?

P(x) must always imply Q(x) Signup and view all the answers

In the mathematical trick described, what is the final result, regardless of the chosen initial number?

The result is always 7 Signup and view all the answers

What does m and n being 'particular but arbitrarily chosen integers' imply in a proof?

They could represent any even integers, enhancing generality in the proof. Signup and view all the answers

Which of the following statements about the method of direct proof is true?

It relies on showing P(x) leads to Q(x) for any generic choice of x Signup and view all the answers

What does proving that the sum of any two even integers is even illustrate?

A specific case of universal statements Signup and view all the answers

What did Pierre de Fermat claim about the equation $x^n + y^n = z^n$ for integers $n$ greater than or equal to 3?

There are no positive integer solutions. Signup and view all the answers

What was the nature of Fermat's proof for his last theorem?

No proof was found among his papers. Signup and view all the answers

How successful has the Goldbach conjecture been in demonstrating evidence for even integers?

It has been shown true for all even integers up to $10^{18}$. Signup and view all the answers

What did Leonhard Euler conjecture about the equation $a^4 + b^4 + c^4 = d^4$?

It has only trivial whole number solutions. Signup and view all the answers

What significant counterexample disproved Euler's conjecture about fourth powers?

$95,800^4 + 217,519^4 + 414,560^4 = 422,481^4$. Signup and view all the answers

Which mathematician is known for proposing the Goldbach conjecture?

Christian Goldbach Signup and view all the answers

What conclusion can be drawn from the historical attempts to prove Fermat's last theorem?

Many mathematicians have failed to find either proof or counterexample. Signup and view all the answers

What characterization can be made about problems like the Goldbach conjecture in number theory?

They often yield false results despite being plausible. Signup and view all the answers

What is required to prove that an existential statement is false?

You must prove its negation is true. Signup and view all the answers

Which of the following best describes the shape of the proof that G is connected?

Assume G is complete and bipartite, conclude G is connected. Signup and view all the answers

Which statement represents the negation of 'There is a positive integer n such that n^2 + 3n + 2 is prime'?

For all positive integers n, n^2 + 3n + 2 is not prime. Signup and view all the answers

What mathematical expression was factored to show that n^2 + 3n + 2 is not prime?

(n + 1)(n + 2) Signup and view all the answers

Why are n + 1 and n + 2 noted to be greater than 1 in the proof?

To confirm that n^2 + 3n + 2 is composite. Signup and view all the answers

In formal proof, what does the existence of an object in the domain imply?

The object satisfies the hypothesis of the statement. Signup and view all the answers

What is the conclusion drawn from the proof regarding the graph G?

G is both bipartite and connected. Signup and view all the answers

Which concept is critical when generalizing from a particular instance in the proof process?

Proving a universal statement. Signup and view all the answers

Study Notes

Elementary Number Theory and Methods of Proof

This chapter covers fundamental concepts in number theory and the methods for proving mathematical statements in that field.
Discovery and proof are interwoven throughout the problem-solving process.
Attempting to justify why a statement is true can reveal its falsehood.

Section 4.1: Direct Proof and Counterexample I: Introduction

A fundamental familiarity with basic algebraic laws is assumed.
Properties of equality (reflexive, symmetric, transitive) are used.
Integers are closed under addition, subtraction, and multiplication (this essentially means that the result of those operations on integers is also an integer).
Most integer quotients are not integers.

Definitions

An integer is even if it can be expressed as twice another integer.
An integer is odd if it can be expressed as twice another integer plus 1.

Example 1: Even and Odd Integers

0 is an even integer.
-301 is an odd integer.
The product of two integers is even.
The sum of an even and odd number is odd.
Every integer is either even or odd.

Definitions (continued)

Prime numbers are positive integers greater than 1 with only positive factors of 1 and itself.
Composite numbers are positive integers greater than 1 that are not prime (i.e., they can be written as a product of two smaller positive integers).

Example 2: Prime and Composite Numbers

1 is not a prime number.
Every integer greater than 1 is either prime or composite.
The first six prime numbers are 2, 3, 5, 7, 11, and 13.
The first six composite numbers are 4, 6, 8, 9, 10, and 12.

Proving Existential Statements

An existential statement is true if at least one object in a domain satisfies the property.
Constructive proofs find the object explicitly.
Nonconstructive proofs show that an object exists without finding it.

Example 3: Constructive Proofs of Existence

There exists an even integer that can be expressed as the sum of two prime numbers in two different ways.

Proving Existential Statements (continued)

Nonconstructive proofs demonstrate existence by showing that assuming there is no such object leads to a contradiction.

Disproving Universal Statements by Counterexample

A universal statement is proven false by finding a specific case (counterexample) that violates it.
The process involves identifying a case where the hypothesis is true, but the conclusion is false.

Example 4: Disproof by Counterexample

The statement "for all real numbers a and b, if a²=b², then a = b" is false due to counterexamples involving negative numbers.

Proving Universal Statements

Universal statements often use exhaustion (checking all) to prove a statement true across a finite domain.
Generalizing from the generic particular is commonly used for universal statements covering an infinite domain.

Example 5: The Method of Exhaustion

The statement "for all even integers n, from 4 to 26, n can be written as a sum of two primes" is provable using a method of exhaustion.

Example 6: Generalizing from the Generic Particular

An example demonstrating a mathematical "trick" that depends on the idea of generalizing from the generic particular, where each step is shown algebraically.

Proving Universal Statements (continued)

Direct proof: show that the hypothesis implies the conclusion by using known or established properties
Direct proof is based on the logical rule of implication. The fact that the only way an if-then statement is false is if the hypothesis is true and the conclusion is false, is leveraged to prove the statement is true.

Example 7: A Direct Proof of Theorem

Proving that the sum of any two even integers is even using direct proof. This includes a formal restatement and steps in the proof.

Directions for Writing Proofs of Universal Statements

Guidelines for crafting clear mathematical proofs, including detailed explanations and justifications for each step.

Example 8: Identifying the "Starting Point" and the "Conclusion to be Shown"

Identifying the starting point and conclusion to be shown in a universal statement using an example with graph theory.

Showing That an Existential Statement is False

To prove an existential statement is false, prove that its negation, a universal statement, is true.

Example 9: Disproving an Existential Statement

Showing that there is no positive integer n such that n² + 3n + 2 is prime. The negation is proven universal, using a particular but arbitrarily chosen integer, showing that the resulting expression is always a product of integers that are greater than 1.

Conjecture, Proof, and Disproof

Historical context of conjectures, including Fermat's last theorem and Goldbach's conjecture.
Examples of conjectures that were later proven false (Euler's conjecture).

Common Mistakes in Writing Proofs

Avoiding mistakes like arguing only from examples, using the same variable for different values, jumping to conclusions, circular reasoning, or conflating what is known with what is to be shown.
Avoid imprecisely using the word "if" when "because" is intended.
Understand that just because a statement works for a few cases doesn't mean it is true always.

Getting Proofs Started

After grasping the method of direct proof, the starting points and conclusions can be established even from theorems not immediately understood. The proof's structure can be determined from the statement's linguistic construction.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Explore fundamental concepts in elementary number theory and the methods of proof used in mathematical reasoning. This quiz examines the properties of even and odd integers as well as the essential techniques for direct proof and counterexamples. Test your understanding of integers and their operations in the context of number theory.