Algebra

Il termine algebra (dall'arabo الجبر, al-ǧabr che significa "unione", "connessione" o "completamento", ma anche "aggiustare") deriva dal libro del matematico persiano Muḥammad ibn Mūsā al-Ḫwārizmī, intitolato Al-kitāb al-muḫtaṣar fī ḥīsāb al-ǧabr wa l-muqābala ("Compendio sul calcolo per completamento e bilanciamento"), conosciuto anche nella forma breve Al-kitāb al-ǧabr wa l-muqābala, che tratta la risoluzione delle equazioni di primo e di secondo grado.

Number Theory

Number theory is the study of the set of positive whole numbers

1 2 3 4 5 6 7 ...

which are often called the set of natural numbers. We will especially want to study the relationships between different sorts of numbers. Since ancient times, people have separated the natural numbers into a variety of different types. Here are some familiar and not-so-familiar examples:

odd: 1 3 5 7 9 11 ...

even: 2 4 6 8 10 ...

square: 1 4 9 16 25 36 ...

cube: 1 8 27 64 125 .. .

prime: 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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 ...

composite: 4 6 8 9 10 12 14 15 16 ...

1(modulo4): 1 5 9 13 17 21 25 .. .

3 (modulo 4): 3 7 11 15 19 23 27 .. .

triangular: 1 3 6 10 15 21 ...

perfect: 6 28 496 ...

Many of these types of numbers are undoubtedly already known to you. Oth­ers, such as the "modulo 4" numbers, may not be familiar. A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and sim­ilarly for the 3 (modulo 4) numbers. A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. The Fibonacci numbers are created by starting with 1 and 1. Then, to get the next number in the list, just add the previous two. Finally, a number is perfect if the sum of all its divisors, other than itself, adds back up to the original number. Thus, the numbers dividing 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Similarly, the divisors of 28 are 1, 2, 4, 7, and 14, and

We will encounter all these types of numbers, and many others, in our excursion through the Theory of Numbers.

Some Typical Number Theoretic Questions

The main goal of number theory is to discover interesting and unexpected rela­ tionships between different sorts of numbers and to prove that these relationships are true. In this section we will describe a few typical number theoretic problems, some of which we will eventually solve, some of which have known solutions too difficult for us to include, and some of which remain unsolved to this day.

Sums of Squares I. Can the sum of two squares be a square? The answer is clearly "YES"; for example 3² + 4² 5² and 5² + 12² = 13² . These are examples of Pythagorean triples. We will describe all Pythagorean triples in Chapter 2.

Sums of Higher Powers. Can the sum of two cubes be a cube? Can the sum of two fourth powers be a fourth power? In general, can the sum of two nth powers be an nth power? The answer is "NO." This famous problem, called Fermat's Last Theorem, was first posed by Pierre de Fermat in the seventeenth century, but was not completely solved until 1994 by Andrew Wiles. Wiles's proof uses sophisticated mathematical techniques that we will not be able to describe in detail, but in Chapter 30 we will prove that no fourth power is a sum of two fourth powers, and in Chapter 46 we will sketch some of the ideas that go into Wiles' s proof.