By W. L. Ferrar

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Extra info for Algebra: A Text-Book of Determinants, Matrices, and Algebraic Forms

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Hence, we may assume that b a; that is, b a 0. Consider the sequence b > b a > b 2a > b 3a > . q C1/a < 0 (this is just Infinite Descent, described on page 12; in more down-to-earth language, there can be at most b steps before this sequence becomes negative). If we define r D b qa, then b D qa C r . We also have the desired inequalities. Clearly, 0 Ä r . q C 1/a 0, contradicting the definition of q. Let’s prove uniqueness. 15 can be weakened to a; b 2 Z and a ¤ 0; the inequalities for the remainder now read 0 Ä r < jaj.

Fermat’s son edited the next edition of Diophantus, published in 1670; this version contains Fermat’s annotations, including his famous “Last Theorem;” it contained other unproved assertions as well, most true, some not. By the early 1800s, only Fermat’s Last Theorem remained undecided. It became a famous problem, resisting the attempts of mathematicians of the highest order for 350 years, until it was finally proved, in 1995, by Wiles. 2 Diophantus 15 believe that Fermat did not have a correct proof.

Definition. If a and b are integers, then a divides b, denoted by a j b; if there is an integer c with b D ca. We also say that a is a divisor (or a factor) of b, and that b is a multiple of a. 12. Consider some special cases. Every number a divides itself, for a D a 1; similarly, 1 divides every number. Every number a divides 0: taking c D 0, we have 0 D a 0. On the other hand, if 0 divides b, then b D 0, for b D 0 c D 0. Note that 3 j 6, because 6 D 3 2, but 3 − 5 (that is, 3 does not divide 5): even though 5 D 3 35 , the fraction 53 is not an integer.

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