Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Paperback, 1977. 3rd Print)

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 $$p + y\sqrt { - 3} = {\left( {a + b\sqrt { - 3} } ight)^3}$$, 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 $$p + q\sqrt 5 = {\left( {a + b\sqrt 5 } ight)^5}$$ only under the additional condition 5 q..- 3.4 The casesn = 14 andn = 7. These proofs, by Dirichlet and Lame 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. Lame'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.