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Chapter 3: Problem 21

Pythagorean Triples. Three natural numbers \(a, b,\) and \(c\) with \(a

Short Answer

Expert verified
In summary, there are two theorems related to Pythagorean triples: 1. The only Pythagorean triple consisting of three consecutive natural numbers is \((3, 4, 5)\). 2. The only Pythagorean triple in the form of \(m, m+7\), and \(m+8\) is \((5, 12, 13)\).

Step by step solution

01

(Part A: Consecutive Natural Numbers)

We need to find all Pythagorean triples in the form of consecutive natural numbers, i.e., \(m, m+1,\) and \(m+2\). Thus, for each such triple \((m, m+1, m+2)\), we need to satisfy the condition \(m^2 + (m+1)^2 = (m+2)^2\). Let's break down the equation and see if we can find any patterns or relationships: \[(m^2 + (m+1)^2) = (m+2)^2\] \[m^2 + m^2 + 2m + 1 = m^2 + 4m + 4\] Combine terms: \[2m^2 + 2m + 1 = m^2 + 4m + 4\] Subtract \(m^2 + 4m + 1\) from both sides to get a quadratic equation: \[m^2 - 2m - 3 = 0\] Now, we'll find the potential values for m by factoring the quadratic equation: \[(m - 3)(m + 1) = 0\] This gives us two possible solutions for \(m\): 1. \(m = 3\): In this case, the triple would be \((3, 4, 5)\). Checking our condition, we can see that it holds true: \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\). 2. \(m = -1\): In this case, the triple would be \((-1, 0, 1)\). However, since we're looking for natural numbers, this solution is not valid. So, there is only one Pythagorean triple consisting of three consecutive natural numbers: \((3, 4, 5)\). The theorem states that any consecutive natural numbers that form a Pythagorean triple must be \((3, 4, 5)\).
02

(Part B: m, m+7, and m+8)

Now, we are trying to find all Pythagorean triples in the form \(m, m+7,\) and \(m+8\). To satisfy Pythagorean triple conditions, we need to find \(m\) for which the equation \(m^2 + (m+7)^2 = (m+8)^2\) holds true. Let's break down the equation and simplify it: \[(m^2 + (m+7)^2) = (m+8)^2\] \[m^2 + m^2 + 14m + 49 = m^2 + 16m + 64\] Combine terms: \[2m^2 + 14m + 49 = m^2 + 16m + 64\] Subtract \(m^2 + 16m + 49\) from both sides to get a quadratic equation: \[m^2 - 2m - 15 = 0\] Now, we'll find the potential values of \(m\) by factoring the quadratic equation: \[(m - 5)(m + 3) = 0\] This gives us two possible solutions for \(m\): 1. \(m = 5\): In this case, the triple would be \((5, 12, 13)\). Checking our condition, we can see it holds true: \(5^2 + 12^2 = 25 + 144 = 169 = 13^2\). 2. \(m = -3\): In this case, the triple would be \((-3, 4, 5)\). However, since we're looking for natural numbers, this solution is not valid. So, there is only one Pythagorean triple satisfying the condition \(m, m+7,\) and \(m+8\): \((5, 12, 13)\). The theorem states that any three natural numbers in the form of \(m, m+7\), and \(m+8\) that form a Pythagorean triple must be \((5, 12, 13)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Proof
In mathematics, a mathematical proof is an inferential argument for a mathematical statement, showing that the statement in question is logically derived from a set of axioms and known truths. When proving the existence of Pythagorean triples, such as those formed by consecutive natural numbers, proofs often involve demonstrating that specific conditions can be met by potential solutions.

For instance, in the case of finding consecutive natural numbers that form a Pythagorean triple, we introduce variables and perform algebraic manipulations to derive a quadratic equation. The solutions to this equation, if they exist, will reveal potential candidates for the original Pythagorean triple. A proof is considered complete and valid only if it logically demonstrates the theorem in question, accounting for all possibilities and using accepted mathematical operations.
Consecutive Natural Numbers
The term consecutive natural numbers refers to a sequence of numbers where each number is one more than the previous one, starting from any particular natural number. For example, 5, 6, and 7 are consecutive natural numbers. Such numbers are extremely useful in various mathematical concepts, including the formation of Pythagorean triples.

In our exercise, by representing three consecutive natural numbers as m, m+1, and m+2, we dramatically simplify the process of identifying and proving the potential for these numbers to form a Pythagorean triple. It is the concise pattern of consecutive numbers that enables us to derive a single quadratic equation whose solutions will dictate the validity of the triple as a whole.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in one variable that can be expressed in the general form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Quadratic equations are integral in finding Pythagorean triples, as they often arise when equating the sum of squares of two natural numbers to the square of another.

In our exercise, to find the Pythagorean triples, we simplify the given conditions and arrive at a quadratic equation. Solving it is a two-step process: First, we factorize the equation or use the quadratic formula to find the values of m. Second, we test these values against our initial conditions, namely, that we are interested in natural numbers only, dismissing any negative or non-integer solutions. The viable solutions indicate the sets of consecutive natural numbers that may constitute Pythagorean triples.
Number Theory
Number theory is a branch of pure mathematics devoted to the study of the integers and integer-valued functions. This field includes the exploration of numbers that meet specific criteria, like the Pythagorean triples in our exercise. By employing number theory concepts, we examine the properties of natural numbers and their relationships within mathematical structures, like sequences of consecutive integers.

In the context of Pythagorean triples, number theory helps us understand the integers involved in these triplet sets' characteristics. It paves the way for proving the uniqueness or the limitation of solutions with respect to given constraints. Additionally, number theory's comprehensive tools and techniques, including modular arithmetic and the theory of equations, allow for an in-depth analysis of mathematical problems, such as identifying specific types of Pythagorean triples.

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Most popular questions from this chapter

(a) Let \(n \in \mathbb{N}\) and let \(a \in \mathbb{Z}\). Explain why \(n\) divides \(a\) if and only if \(a \equiv 0(\bmod n)\) (b) Let \(a \in \mathbb{Z}\). Explain why if \(a \neq 0(\bmod 3),\) then \(a \equiv 1(\bmod 3)\) or \(a \equiv 2(\bmod 3)\) (c) Is the following proposition true or false? Justify your conclusion. For each \(a \in \mathbb{Z},\) if \(a \neq 0(\bmod 3),\) then \(a^{2} \equiv 1(\bmod 3)\).

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For all integers \(a, b,\) and \(c\) with \(a \neq 0,\) if \(a \mid b,\) then \(a \mid(b c)\). (b) For all integers \(a\) and \(b\) with \(a \neq 0,\) if \(6 \mid(a b),\) then \(6 \mid a\) or \(6 \mid b\). (c) For all integers \(a, b,\) and \(c\) with \(a \neq 0,\) if \(a\) divides \((b-1)\) and \(a\) divides \((c-1),\) then \(a\) divides \((b c-1)\) (d) For each integer \(n,\) if 7 divides \(\left(n^{2}-4\right),\) then 7 divides \((n-2)\). (e) For every integer \(n, 4 n^{2}+7 n+6\) is an odd integer. ? (f) For every odd integer \(n, 4 n^{2}+7 n+6\) is an odd integer. (g) For all integers \(a, b,\) and \(d\) with \(d \neq 0,\) if \(d\) divides both \(a-b\) and \(a+b,\) then \(d\) divides \(a\) (h) For all integers \(a, b,\) and \(c\) with \(a \neq 0,\) if \(a \mid(b c),\) then \(a \mid b\) or \(a \mid c .\)

(a) Prove Part (2) of Proposition 3.23 . For each \(x \in \mathbb{R},|-x|=|x|\). (b) Prove Part (2) of Theorem 3.25 . For all real numbers \(x\) and \(y,|x y|=|x||y|\).

(a) If an integer has a remainder of 6 when it is divided by 7 , is it possible to determine the remainder of the square of that integer when it is divided by \(7 ?\) If so, determine the remainder and prove that your answer is correct. (b) If an integer has a remainder of 11 when it is divided by 12 , is it possible to determine the remainder of the square of that integer when it is divided by \(12 ?\) If so, determine the remainder and prove that your answer is correct. (c) Let \(n\) be a natural number greater than 2. If an integer has a remainder of \(n-1\) when it is divided by \(n\), is it possible to determine the remainder of the square of that integer when it is divided by \(n ?\) If so, determine the remainder and prove that your answer is correct.

In Exercise (15) in Section 3.2, we proved that there exists a real number solution to the equation \(x^{3}-4 x^{2}=7\). Prove that there is no integer \(x\) such that \(x^{3}-4 x^{2}=7\)
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