Chapter 5. Mathematics in the time of Descartes and Fermat

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new lines of research.One popular theme in number theory has been that of sums of squares.integers can be written in the formx 2 + y 2 ,Whatwhere x and y are integers, or in the formx 2 + y 2 + z 2 ,where x, y and z are integers, and so on. Bachet (who translated the Arithmetica ofDiophantus into Latin in 1621, a translation much used by Fermat) conjectured thatevery integer n is a sum of four squares,n = x 2 + y 2 + z 2 + w 2 ,where x, y, z and w are integers. This was eventually rigorously proved by Lagrangein 1770’s. Fermat, in a letter to Mersenne in 1640, asserted that any prime p satisfyingp ≡ 1 mod 4 can be written asp = a 2 + b 2 ,for suitable integers a and b. If we assume that 0 < a < b, then a and b are unique. Exactdetails of his proof are not available but he claimed to use his method of infinite descent,which is a powerful method that may be applied to certain Diophantine equations. It is asort of induction in reverse. He probably used the technique in his study of the so-calledFermat problems, such as showing that there are no solutions to the equationx 3 + y 3 = z 3 ,where x, y and z are all non-zero integers. He certainly showed that there are no non-trivialinteger solutions to the equationx 4 + y 4 = z 4 ,using his method of infinite descent.His success in showing the non-existence of non-trivial integral solutions to theseDiophantine equations must have led Fermat to believe that the method of descent couldlikewise prove that there are no non-zero integers x, y and z that satisfyx n + y n = z n ,6
whenever n is an integer greater than 2. He certainly left a note in the margin of his copyof the Arithmetica stating that he had found such a solution. This assertion achievedinternational prominence with the publication of Fermat’s collected works in 1679, whichbrought Fermat’s claim out into the open. Nobody was able to make the method of descentapply in the cases where n > 4, and we must presume that Fermat was wrong in his claim.In fact, attempts to verify the truth of Fermat’s Last Theorem (as the problem becameknown) led to the development of a vast generalization of arithmetic known as algebraicnumber theory, in which the theory of primes and unique factorization is carried over tocertain special types of complex number. Even this was not enough to prove Fermat’sLast Theorem in general, as the complete proof by Andrew Wiles in 1995 used techniquesof algebraic geometry and modular form theory.Fermat considered further problems concerning primes.that every prime p satisfying p ≡ 1 mod 3 is expressible asFor example, he claimedp = a 2 + 3b 2for suitable integers a and b (letter to Pascal, 1654). All of Fermat’s claims are true,and he may well have proved them. However, definitive proofs first appeared with Euler,and Gauss provided the most satisfactory approach to problems of the kind considered byFermat.We will now discuss some of Fermat’s work relating to primes. The Greeks hadinvestigated properties of integers and Euclid’s Elements includes an account of whatthey had discovered. Greek mathematicians had a particular interest in perfect numbers.An integer n > 1 is said to be perfect if the sum of all its positive integer divisors, including1 but excluding n itself, equals n. Thus, for example, 6 is perfect as its divisors less than6 are 1, 2 and 3 and1 + 2 + 3 = 6.Similarly, 28 is perfect, as its divisors are 1, 2, 4, 7 and 14, whose sum is 28.A means of producing even perfect numbers is described in Euclid’s Elements in thefollowing way. Suppose that for some positive integer m, 2 m − 1 is a prime. Then theinteger 2 m−1 (2 m − 1) is perfect. For, if we set 2 m − 1 = p, where p is a prime, it is easyto see that the divisors of n = 2 m−1 p are1, 2, 2 2 , · · · , 2 m−1 , p, 2p, · · · , 2 m−2 p.7
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Page 9 and 10: Using this result, it is easy to se

Chapter 5. Mathematics in the time of Descartes and Fermat

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