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Chapter 1: Problem 24

The three integers \(a=3, b=4, c=5\) in the Pythagorean triple (3,4,5) form an arithmetic progression: that is, \(c-b=b-a\). Find all Pythagorean triples \((a, b, c)\) which form an arithmetic progression \(-\) that is, for which \(c-b=b-a\)

Short Answer

Expert verified
The Pythagorean triples that also form an arithmetic progression are of the form \((3k, 4k, 5k)\) for positive integers \(k\).

Step by step solution

01

Understanding the condition for an arithmetic progression

In the problem, we are given that the three integers \(a\), \(b\), and \(c\) form an arithmetic progression. This means the difference between consecutive terms is constant. Mathematically, this is expressed as \(c - b = b - a\). Rearrange this equation to find a relationship between these terms: \(c = 2b - a\).
02

Using the properties of a Pythagorean triple

In any Pythagorean triple \((a, b, c)\), the numbers abide by the equation \(a^2 + b^2 = c^2\). To find such triples that also form an arithmetic progression, we must ensure both conditions (the progression condition and the Pythagorean condition) are satisfied.
03

Substitute the arithmetic progression condition into the Pythagorean identity

Replace \(c\) in the Pythagorean condition with \(2b - a\) derived from the arithmetic progression condition: \(a^2 + b^2 = (2b - a)^2\).
04

Simplifying the equation

Expand \((2b - a)^2\): \((2b - a)^2 = 4b^2 - 4ab + a^2\). Substitute back into the Pythagorean identity: \(a^2 + b^2 = 4b^2 - 4ab + a^2\).
05

Cancel terms and solve for relations between \(a\) and \(b\)

Subtract \(a^2\) from both sides: \(b^2 = 4b^2 - 4ab\). Rearrange this equation as: \(3b^2 = 4ab\).
06

Find a general expression for possible values

Rearrange the equation \(3b^2 = 4ab\) to find \(b\frac{3}{4} = a\), indicating that for every \(4k\), there exists a corresponding \(3k\) such that \(a = 3k\) and \(b = 4k\) for positive integers \(k\). Thus, \(c = 2(4k) - 3k = 5k\).
07

Verifying the solution through substitution and recognition of pattern

As observed, the general solution matches the pattern: any multiple of the smallest Pythagorean triple (3, 4, 5). For instance, set \(k=1\) yields \((3, 4, 5)\), and for any positive integer \(k\), \((3k, 4k, 5k)\) form a Pythagorean triple as well as an arithmetic progression.

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Key Concepts

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

Arithmetic Progression
Arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is known as the common difference. For instance, in the sequence 3, 7, 11, 15, each number is 4 more than the previous one. This concept is also crucial when dealing with Pythagorean triples that form an arithmetic progression.
This happens when the difference between the largest number and the middle number is equal to the difference between the middle number and the smallest number. Mathematically, for three numbers \(a\), \(b\), and \(c\) to be in arithmetic progression, the condition \(c - b = b - a\) should hold true. For example, in the well-known Pythagorean triple (3, 4, 5), the sequence is in arithmetic progression since \(5 - 4 = 4 - 3\). This concept underlies one of the key requirements in finding such triples.
Integer Sequences
Integer sequences are arrangements of numbers where each number in the sequence is a whole number. These sequences can follow various patterns or rules. Arithmetic progression forms a specific type of integer sequence, where each number is consistently spaced from its neighbors by a fixed integer amount. This consistency is crucial when exploring mathematical problems like the ones involving Pythagorean triples.
Pythagorean triples themselves are integer sequences that fit the equation \(a^2 + b^2 = c^2\). By adding the arithmetic progression condition, these triples are restricted to those for which \(c - b = b - a\) holds, giving birth to sequences like (3, 4, 5). Such sequences are not just mathematical artifacts; they are integral to number theory and hold keys to unlocking further mathematical exploration and understanding.
Mathematical Problem Solving
Mathematical problem solving involves a structured approach to finding solutions to mathematical puzzles. This often includes understanding the problem, identifying relevant concepts, and applying logical steps to reach a solution. In solving the problem of finding all Pythagorean triples that form an arithmetic progression, this method can be clearly seen.
Here, understanding the problem means knowing that both the Pythagorean condition \(a^2 + b^2 = c^2\) and the arithmetic progression condition \(c - b = b - a\) must be satisfied simultaneously. Then, finding a relationship between terms using substitution helps in simplifying complexities, as shown when substituting \(c = 2b - a\) and simplifying the resulting equation. Solving these equations uncovers patterns like (3k, 4k, 5k), where \(k\) is a positive integer, evolving from basic reasoning to elegant general solutions in problem solving.
Elementary Number Theory
Elementary number theory is a branch of mathematics focused on properties and relationships of numbers, particularly integers. It provides the tools and frameworks needed to understand and solve problems involving number sequences and patterns. Pythagorean triples and arithmetic progressions fall under this domain.
The exploration of Pythagorean triples forming an arithmetic progression is a practical application of elementary number theory. It involves manipulating equations and recognizing patterns to identify all sets of numbers satisfying both the Pythagorean condition and the arithmetic sequence condition. Recognizing the solution as being any multiple of the smallest Pythagorean triple (3, 4, 5) not only demonstrates number-theoretic principles but also emphasizes the elegance and interconnectedness of mathematical concepts.

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Most popular questions from this chapter

(a) Which of the prime numbers \(<100\) can be written as the sum of two squares? (b) Find an easy way to immediately write \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)\) in the form \(\left(x^{2}+y^{2}\right)\). (This shows that the set of integers which can be written as the sum of two squares is "closed" under multiplication.) (c) Prove that no integer (and hence no prime number) of the form \(4 k+3\) can be written as the sum of two squares. (d) The only even prime number can clearly be written as a sum of two squares: \(2=1^{2}+1^{2}\). Euler \((1707-1783)\) proved that every odd prime number of the form \(4 k+1\) can be written as the sum of two squares in exactly one way. Find all integers \(<100\) that can be written as a sum of two squares. (e) For which integers \(N<100\) is it possible to construct a square of area \(N\), with vertices having integer coordinates? In Problem 25 parts (a) and (d) you had to decide which integers \(<100\) can be written as a sum of two squares as an exercise in mental arithmetic. In part (b) the fact that this set of integers is closed under multiplication turned out to be an application of the arithmetic of norms for complex numbers. Part (e) then interpreted sums of two squares geometrically by using Pythagoras' Theorem on the square lattice. These exercises are worth engaging in for their own sake. But it may also be of interest to know that writing an integer as a sum of two squares is a serious mathematical question \- and in more than one sense. Gauss \((1777-1855),\) in his book Disquisitiones arithmeticae (1801) gave a complete analysis of when an integer can be represented by a 'quadratic form', such as \(x^{2}+y^{2}\) (as in Problem 25) or \(x^{2}-2 y^{2}\) (as in Problem \(\mathbf{5 4}(\mathrm{c})\) in Chapter 2 ). A completely separate question (often attributed to Edward Waring \((1736-1798))\) concerns which integers can be expressed as a \(k^{\text {th }}\) power, or as a sum of \(n\) such powers. If we restrict to the case \(k=2\) (i.e. squares), then: \- When \(n=2,\) Euler \((1707-1783)\) proved that the integers that can be written as a sum of two squares are precisely those of the form $$ m^{2} \times p_{0} \times p_{1} \times p_{2} \times \cdots \times p_{s} $$ where \(p_{0}=1\) or \(2,\) and \(p_{1}

(a)(i) Explain why any integer that is a factor (or a divisor) of both \(m\) and \(n\) must also be a factor of their difference \(m-n,\) and of their sum \(m+n\). (ii) Prove that $$ H C F(m, n)=H C F(m-n, n) $$ (iii) Use this to calculate in your head \(H C F(1001,91)\) without factorising either number. (b)(i) Prove that: \(H C F(m, m+1)=1\). (ii) Find \(H C F(m, 2 m+1)\). (iii) Find \(H C F\left(m^{2}+1, m-1\right)\).

Problem 28 (A folded triangle) The equilateral triangle \(\triangle A B C\) has sides of length \(1 \mathrm{~cm} . \quad D\) and \(E\) are points on the sides \(A B\) and \(A C\) respectively, such that folding \(\triangle A D E\) along \(D E\) folds the point \(A\) onto \(A^{\prime}\) which lies outside \(\triangle A B C\) What is the total perimeter of the region formed by the three single layered parts of the folded triangle (i.e. excluding the quadrilateral with a folded layer on top )?

Write out the first 12 or so powers of 4 : $$ 4,16,64,256,1024,4096,16384,65536, \ldots $$ Now create two sequences: the sequence of final digits: \(4,6,4,6,4,6, \ldots\) the sequence of leading digits: \(4,1,6,2,1,4,1,6, \ldots\) Both sequences seem to consist of a single "block", which repeats over and over for ever. (a) How long is the apparent repeating block for the first sequence? How long is the apparent repeating block for the second sequence? (b) It may not be immediately clear whether either of these sequences really repeats forever. Nor may it be clear whether the two sequences are alike, or whether one is quite different from the other. Can you give a simple proof that one of these sequences recurs, that is, repeats forever? (c) Can you explain why the other sequence seems to recur, and decide whether it really does recur forever?

Let \(\Delta=\operatorname{area}(\triangle A B C)\) (a) Prove that $$ \Delta=\frac{1}{2} \cdot a b \cdot \sin C . $$ (b) Prove that \(4 R \Delta=a b c\).
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