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(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}

Step by step solution

01

Understand Prime Representation as Sum of Squares

For a prime number to be expressible as the sum of two squares, it should be either 2 or congruent to 1 modulo 4. This is directly based on Fermat's theorem on sums of two squares which states that an odd prime \( p \) can be expressed as the sum of two squares if and only if \( p \equiv 1 \pmod{4} \). We begin by listing all primes less than 100.

02

Find Primes < 100

List all prime numbers less than 100: \(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\).

03

Apply Fermat's Theorem to Primes

Check which of these primes is congruent to 1 modulo 4. They are: \(5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97\). Note, the prime 2 is always expressible as the sum \(1^2 + 1^2\).

04

Multiply Two Sums of Squares

To express \( (a^2 + b^2)(c^2 + d^2) \) as another sum of two squares, use the identity: \((a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2\). This shows closure under multiplication, simplifying products of sums of two squares.

05

Check Condition 4k+3 Cannot be Sum of Two Squares

To prove that no integer of the form \(4k+3\) can be written as the sum of two squares, assume \( n = a^2 + b^2 \) is such an integer. The squares \(a^2\) and \(b^2\) can only be 0 or 1 modulo 4, and thus their sum can never be \(3 \pmod{4}\) since cases \((0 + 0)\), \((1 + 0)\), and \((1 + 1)\) yield only \(0, 1, \text{and} \ 2\).

06

List Integers < 100 as Sum of Two Squares

Use the closed set property and previous results to find all integers: \(1^2, 2 = 1^2 + 1^2, 4, 5 = 2^2 + 1^2, 8, 9 = 3^2, 10 = 3^2 + 1^2, 13 = 3^2 + 2^2, 17, 18 = 3^2 + 3^2, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98\).

07

Determine Integer Squares with Integer Coordinates

For an integer \(N\) to form a square with vertices at integer coordinates, \(N\) must be expressible as a sum of two squares. Hence all integers listed in step 6 satisfy this condition.

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Key Concepts

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

Fermat's Theorem

Fermat's theorem on sums of two squares is a fascinating concept in number theory. It lays down conditions under which a prime number can be expressed as the sum of two squares. Fermat's theorem specifically states that an odd prime number, say \( p \), can be written as the sum of two squares if and only if \( p \equiv 1 \pmod{4} \). This means that when you divide \( p \) by 4, the remainder should be 1. For instance, the prime number 13 can be expressed as \( 3^2 + 2^2 \) because 13 \(\equiv\) 1 (mod 4). This theorem does not only apply to primes of the stated congruence property. The prime number 2 also fits into this category, as it can be expressed as \( 1^2 + 1^2 \). When solving problems involving complementing square sums, Fermat's theorem provides a foundational guideline for determining which prime numbers can be expressed in such a manner.

Prime Numbers

Prime numbers are one of the basic building blocks in the study of mathematics, especially within number theory. A prime number is one that is greater than 1 and cannot be precisely divided by any number other than 1 and itself without leaving a remainder. Looking at numbers below 100, we have \( 2, 3, 5, 7, 11, 13, 17, \) among others. These numbers play a crucial role in various mathematical theories and theorems. For example, in the context of expressing numbers as the sum of two squares, knowing the properties of prime numbers is essential. Different properties and congruence relations define when a prime can be expressed as a sum of two squares. Additionally, understanding prime numbers helps in deducing various mathematical forms and in proving theorems related to them, such as Fermat's theorem on sums of two squares.

Modulo Arithmetic

Modulo arithmetic, often termed 'clock arithmetic', is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, the modulus. In a more technical sense, for any integer \( a \), and a positive integer \( n \), the expression \( a \equiv b \pmod{n} \) means that \( a \) and \( b \) leave the same remainder when divided by \( n \). The applications of modulo arithmetic are vast; from simplifying large calculations to forming the base of various mathematical proofs, it provides a backbone for arithmetic operations in specific integer contexts. For example, when determining which prime numbers can be expressed as the sum of two squares, we often look for numbers where \( p \equiv 1 \pmod{4} \). This congruence signifies the role and importance of modulo operations in resolving complex mathematical problems involving integer representations.

Integer Representation

Integer representation refers to different ways in which integers can be expressed or transformed mathematically. In mathematics, integers can often be represented in varying forms to suit specific equations or properties such as quadratic forms, sums, or differences. In the context of the sum of two squares, integers take on a distinct and significant representation. For instance, some integers can be represented in the form \( x^2 + y^2 \), which aligns with principles from number theory, including Fermat's theorem. Integer representation allows mathematicians to explore and understand deeper properties about numbers. It is not just about finding which numbers work but discovering new mathematical phenomena and structures within number theory.