Fermat’s Last Theorem
Early History
The Babylonians (1800-1650 B.C.)- tables for the case n = 2; general formula?
Pythagoras, Plato (ca. 550 B.C., ca. 400 B.C.)- general formula for the case n = 2
Diophantus of Alexandria (ca. 250 A.D.)- wrote the Arithmetica (13 books, 9 survived)
Pierre de Fermat (1601 - 1665)- formulated FLT, proved the case n = 4
Leonard Euler (1707 - 1783)- the case n = 3 (1753)
Carl Friederich Gauss (1777 - 1855)- filled in ”details” (gap) in Euler’s proof
Gabriel Lame (1795 - 1870)- the case n = 5 (1839);- attempted a general proof using Z[ζ] (1847)
Peter Gustav Lejeune-Dirichlet (1805 - 1859)- the cases n = 7 (1828) and n = 14 (1832)
Ernst Edward Kummer (1810 - 1893)- FLTp is true for all regular primes p (1847/50)- criterion for determining regular primes (1851);- in particular, he found (1874) that of the 37 primes p < 163,
only 8 are irregular (= not regular):37, 59, 67, 101, 103, 131, 149, 157··· ··· ··· ··· ··· ··· ··· ··· ··· ···1976: FLTn is true for n < 125, 000 (S. Wagstaff + computer)12. Observatio Domini Petri de Fermat
Cubum autem in duos cubos, aut quadrato-quadratum in duos
quadrato-quadratos, et generaliter nullam in infinitum ulta
quadratum postestatem in duos ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem sane dexteri.
Hanc marginis exiguitas non caperet.
On the other hand it is impossible to separate a cube into twocubes, or a biquadrate into two biquadrates, or generally any
power except a square into two powers with the same exponent.
I have discovered a truly marvellous proof of this, which however
the margin is not large enough to contain.
[Translation: T. Heath]23. A Basic Principle
Problem: Find all the integer solutions (x, y, z) ∈ Z3 of theDiophantine equation(1)

F (x, y, z) = 0,

where F ∈ Z[x, y, z] is an integral polynomial.
Examples: 1) Fermat polynomial:
F (x, y, z) = Fn(x, y, z) = xn + y n − z n.
2) Elliptic curve:
F (x, y, z) = Fa,b(x, y, z) = y 2z − x3 + axz 2 + bz 3,
where a, b ∈ Z and ∆(Fa,b) = 16(4a3 + 27b3) 6= 0.
Easier Problem: For each prime number p, solve the congruence(2)
F (x, y, z) ≡ 0 (mod p);
this is a finite problem (for each p), for we need to check only p3
values. In particular, the number of solutions
Np∗(F ) = #{(x, y, z) ∈ (Z/pZ)3 : F (x, y, z) ≡ 0 (mod p)}= #{(x, y, z) ∈ Z3 : 0 ≤ x, y, z < p and p|F (x, y, z)}≤ p3 < ∞.
Put: Np(F ) = (Np∗(F ) − 1)/(p − 1)
= #of essentially distinct solutions of (2).
Question: Do these numbers shed any light on the solutionsof (1)?
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Basic (Conjectural) Principle: the sequence of numbers(3)

defap(F ) = (p + 1) − Np(F ),as p → ∞,should determine the nature of the solutions of (1).
For elliptic curves, this principle assumes the form of two very
precise conjectures which have been partly verified:(TWS)–Conjecture: - due to Y. Taniyama (1955), A. Weil(1967), and G. Shimura (1971)(B/SwD)–Conjecture: - B. Birch, H.P.F. Swinnerton–Dyer(1960’s)
Theorem 1 (Kolyvagin(1988), Murty-Murty(1991) Let
E : Fa,b(x, y, z) = 0 be an elliptic curve satifying (TWS).
Then the sequence ap(E) = p + 1 − Np(Fa,b), p → ∞, determines a (“computable”) constant LE (1) ∈ R. If
LE (1) 6= 0,
then the equation Fa,b(x, y, z) = 0 has only finitely many integral solutions (x, y, z) ∈ Z3 with gcd(x, y, z) = 1, and thesecan be explicitly calculated.
Note. The above theorem constitutes an explicit algorithm
which has been implemented on a MAPLE package called APECS.
Example (Frey). The above leads to a computer proof (atrue proof!) of FLT3 and FLT4, using only four short computercommands.
44. The TWS–Conjecture
To state this conjecture, we need two concepts:
1) The conductor N = NE of an elliptic curve E: this is apositive integer
N | ∆a,b
which is closely related to ∆a,b (and is explicitly computable).
2) The space S(N ) = S2(Γ0(N )) of modular forms of level N :this consists of (complex-valued) functions of the form
∞Xf (z) =an(f )q n, with q = e2πiz ,n=1

where the an(f ) ∈ C and the sum converges for Im(z) > 0;
these are to satisfy certain additional properties such as the ruleaz + bf= (cz + d)2f (z),cz + d
where a, b, c, d ∈ Z are any integers with ad − bc = 1 and N |c.
Properties: 1) S(N ) is a finite-dimensional C-vector space ofNdimension gN := dimC S(N ) ≈ 12.
2) Each f ∈ S(N ) is uniquely described by its first 2gN ≈ N6
Fourier coefficients a1(f ), . . . , a2gN (f ).
3) S(N ) has a distinguished C-basis H(N ) = H +(N ) ∪H −(N ). The functions in H +(N ) are called newforms, thosein H −(N ) oldforms. For each N , these forms are explicitly computable (and have been computed for N ≤ 106).
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Conjecture (TWS): For every elliptic curve E of conductorP
N , there is a (unique) newform f (z) =an(f )q n ∈ H +(N ) oflevel N such that(4)ap(E) = ap(f ),for all primes p 6 | N .

Theorem (Shimura, 1971). For each f ∈ H +(N ) withintegral Fourier coefficients there is an elliptic curve E (ofconductor N ) such that (4) holds.
Theorem (Wiles, 1995). Conjecture (TWS) is true if NE
is squarefree.

65. TWSss ⇒ FLTSuppose FLTp is false: there exist a, b, c ∈ Z with abc 6= 0 suchthatap + bp = cp.
We may suppose (w.l.o.g.) that 2|a and that p ≥ 5. Consider
the elliptic curvey 2z = x(x − apz)(x + bpz),
called a Frey curve. Then:1) ∆ = (abc)2p
2) NE is squarefree (since 16|ap).
Thus, by Wiles’s theorem, there is an f = fE ∈ H +(NE ) suchthat (4) holds.
Claim: Such an fE does not exist!
Theorem (“Lowering the Level” - Ribet, 1991)
Suppose f = fE ∈ H +(N ) is a newform of level N . Fora fixed prime number p > 3 let Mp denote the product ofthe prime numbers q > 2 such that p|exptq (∆E ). Then there
exists g ∈ H +(N/Mp) such thatan(g) ≡ an(f ) (mod p),for all n ≥ 1 with gcd(n, N ) = 1.

Conclusion. Apply this to fE as above. Then by 1) we obtainthat Mp = N2 , so by Ribet’s theorem there is a newform g ∈H +(2). But this is impossible since dim S(2) = 0.
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