1 Fermat.- 1.1 Fermat and his “Last Theorem.” Statement of the theorem. History of its discovery..- 1.2 Pythagorean triangles. Pythagorean triples known to the Babylonians 1000 years before Pythagoras..- 1.3 How to find Pythagorean triples. Method based on the fact that the product of two relatively prime numbers can be a square only if both factors are squares..- 1.4 The method of infinite descent..- 1.5 The casen= 4 of the Last Theorem. In this case the proof is a simple application of infinite descent. General theorem reduces to the case of prime exponents..- 1.6 Fermat’s one proof. The proof that a Pythagorean triangle cannot have area a square involves elementary but very ingenious arguments..- 1.7 Sums of two squares and related topics. Fermat’s discoveries about representations of numbers in the form n = x2+ kn2 for k =1, 2, 3. The different pattern when k = 5..- 1.8 Perfect numbers and Fermat’s theorem. Euclid’s formula for perfect numbers leads to the study of Mersenne primes 2n ? 1 which in turn leads to Fermat’s theorem ap ? a ? 0 mod p. Proof of Fermat’s theorem. Fermat numbers. The false conjecture that 232 + 1 is prime..- 1.9 Pell’s equation. Fermat’s challenge to the English. The cyclic method invented by the ancient Indians for the solution of Ax2+ 1=y2 for given nonsquare A. Misnaming of this equation as “Pell’s equation” by Euler. Exercises: Proof that Pell’s equation always has an infinity of solutions and that the cyclic method produces them all..- 1.10 Other number-theoretic discoveries of Fermat. Fermat’s legacy of challenge problems and the solutions of these problems at the hands of Lagrange, Euler, Gauss, Cauchy, and others..- 2 Euler.- 2.1 Euler and the case n = 3. Euler never published a correct proof that x3+y3?z3 but this theorem can be proved using his techniques..- 2.2 Euler’s proof of the casen = 3. Reduction of Fermat’s Last Theorem in the case n = 3 to the statement that p2+ 3q2 can be a cube (p and q relatively prime) only if there exist a and b such that p = a3 –9ab2, q = 3a2b — 3b3..- 2.3 Arithmetic of surds. The condition for p2+ 3q2 to be a cube can be written simply as $$p + y\sqrt { - 3} = {\left( {a + b\sqrt { - 3} } \right)^3}$$, that is, $$p + q\sqrt { - 3} $$ is a cube. Euler’s fallacious proof, using unique factorization, that this condition is necessary for p2 + 3q2 = cube..- 2.4 Euler on sums of two squares. Euler’s proofs of the basic theorems concerning representations of numbers in the forms x2+y2 and x2+ 3y2. Exercises: Numbers of the form x2+ 2y2..- 2.5 Remainder of the proof whenn = 3. Use of Euler’s techniques to prove x3+y3?z3..- 2.6 Addendum on sums of two squares. Method for solving p = x2+y2 when p is a prime of the form 4n + 1. Solving p = x2+3y2 and p = x2+ 2y2..- 3 From Euler to Kummer.- 3.1 Introduction. Lagrange, Legendre, and Gauss. 3.2 Sophie Germain’s.- theorem. Sophie Germain. Division of Fermat’s Last Theorem into two cases, Case I (x,y, and z relatively prime to the exponent p) and Case II (otherwise). Sophie Germain’s theorem is a sufficient condition for Case I. It easily proves Case I for all small primes..- 3.3 The casen= 5. Proof that x5+y5?z5. The joint achievement of Dirichlet and Legendre. General technique is like Euler’s proof that x3+y3?z3 except that p2 –5q2 a fifth power implies $$p + q\sqrt 5 = {\left( {a + b\sqrt 5 } \right)^5}$$ only under the additional condition 5|q..- 3.4 The casesn = 14 andn = 7. These proofs, by Dirichlet and Lamé respectively, are not explained here. To go further and prove Fermat’s Last Theorem for larger exponents clearly requires new techniques. Exercise: Dirichlet’s proof of the case n = 14..- 4 Kummer’s theory of ideal factors.- 4.1 The events of 1847. Lamé’s “proof” of Fermat’s Last Theorem. Liouville’s objection. Cauchy’s attempts at a proof. Kummer’s letter to Liouville. Failure of unique factorization. Kummer’s new theory of ideal complex numbers..- 4.2 Cyclotomic integers. Basic definitions and operations. The norm of a cyclotomic integer. The distinction between “prime” and “irreducible.” Division using the norm..- 4.3 Factorization of primesp?1 mod ?. Derivation of necessary and sufficient conditions for a cyclotomic integer to be a prime factor of such a prime p..- 4.4 Computations whenp = 1 mod ?. Explicit factorizations of such primes for small values of p and ?. Kummer’s factorizations for ??19 and p?1000. Impossibility of factorization when ?=23 and p = 47. The idea behind Kummer’s “ideal” prime factors..- 4.5 Periods. The conjugation ?:???? corresponding to a primitive root ? mod ?. A cyclotomic integer is made up of periods of length f if and only if it is invariant under ?e where ef=?-1..- 4.6 Factorization of primesp?1 mod ?. If f is the exponent of p mod ? and if h (?) is any prime factor of p then the periods of length f are all congruent to integers mod h (?). This makes it easy to test cyclotomic integers made up of periods for divisibility by h(a)..- 4.7 Computations whenp ? 1 mod ?. Explicit factorizations for small values of p and ?..- 4.8 Extension of the divisibility test. Testing arbitrary cyclotomic integers— not just those made up of periods—for divisibility by a given prime cyclotomic integer h(?)..- 4.9 Prime divisors. The tests for divisibility by prime factors exist in all cases, even those in which there is no prime factor. This is the basis for the definition of “ideal” prime factors or prime divisors. Inadequacy of Kummer’s original proof of the basic proposition..- 4.10 Multiplicities and the exceptional prime. Definition of the multiplicity with which a prime divisor divides a cyclotomic integer. The one prime divisor (1- ?) of ?..- 4.11 The fundamental theorem. A cyclotomic integer g(?) divides another h(?) if and only if every prime divisor which divides g(?) divides h(?) with multiplicity at least as great..- 4.12 Divisors. Definition of divisors. Notation..- 4.13 Terminology. A divisor is determined by the set of all things that it divides. “Ideals.”.- 4.14 Conjugations and the norm of a divisor. Conjugates of a divisor. Norm of a divisor as a divison and as an integer. There are N(A) classes of cyclotomic integers mod A. The Chinese remainder theorem..- 4.15 Summary..- 5 Fermat’s Last Theorem for regular primes.- 5.1 Kummer’s remarks on quadratic integers. The notion of equivalence of divisors. Kummer’s allusion to a theory of divisors for quadratic integers $$x + y\sqrt D $$ and its connection with Gauss’s theory of binary quadratic forms..- 5.2 Equivalence of divisors in a special case. Analysis of the question “Which divisors are divisors of cyclotomic integers?” in a specific case..- 5.3 The class number. Definition and basic properties of equivalence of divisors. Representative sets. Proof that the class number is finite..- 5.4 Kummer’s two conditions. The types of arguments used to prove Fermat’s Last Theorem for the exponents 3 and 5 motivate the singling out of the primes ? for which (A) the class number is not divisible by ? and (B) units congruent to integers mod ? are ?th powers. Such primes are called “regular.”.- 5.5 The proof for regular primes. Kummer’s deduction of Fermat’s Last Theorem for regular prime exponents. For any unit e(?), the unit e(?)/e(?-1) is of the form ?r..- 5.6 Quadratic reciprocity. Kummer’s theory leads not only to a proof of the famous quadratic reciprocity law but also to a derivation of the statement of the law. Legendre symbols. The supplementary laws..- 6 Determination of the class number.- 6.1 Introduction. The main theorem to be proved is Kummer’s theorem that ? is regular if and only if it does not divide the numerators of the Bernoulli numbers B2, B4, B?-3..- 6.2 The Euler product formula. Analog of the formula for the case of cyclotomic integers. The class number formula is found by multiplying both sides by (s-1) and evaluating the limit as s?1..- 6.3 First steps. Proof of the generalized Euler product formula. The Riemann zeta function..- 6.4 Refor
