Monday, 2 June 2014

Fermat's Last Theorem

Fermat’s Last Theorem

An ancient result states that a triangle with vertices A, B and C with lengths AB = a, BC = b and AC = c is right angled at B iff a2 + b2 = c2. The fact that a2 + b2 = c2 is a necessary condition for the right angle is the famous Pythagoras theorem. However, it is also a sufficient condition for the triangle to be a right angle. In fact, it is known that

a triangle is obtuse at B iff a2 + b2 < c2;
a triangle is right angles at B iff a2 + b2 = c2;
a triangle is acute at B iff a2 + b2 > c2.

In all the above cases the sufficiency can be proved using the law of cosines. An important application of above result is that it can used to construct right angles (say in a construction of a building).

What are all the integer triples a,b,c for which a2 + b2 = c2? For instance, (1, 1, 2) is not a integer triple that solves a2 + b2 = c2. Note that if a integer triple (a,b,c) solves a2 + b2 = c2 then all combinations of (±a,±b,±c) are also integer triple. Therefore, it is enough to look for non-negative integer triples (a,b,c). Further, (0, 1, 1) and (1, 0, 1) is also a trivial integer solution which will not represent a triangle. A positive integer triple (a,b,c) is said to be Pythagorean triple if a2 + b2 = c2. Note that if (a,b,c) is a Pythagorean triple then ka,kb,kc) is also a Pythagorean triple for all k . For instance, (3, 4, 5) is a Pythagorean triple then (6, 8, 10) (multiplied by 2) is also a Pythagorean triple. This corresponds to the case similar right triangles. A Pythagorean triple (a,b,c) is said to be primitve if gcd(a,b,c) = 1, i.e., a,b,c have no common factors, except 1, among them. This is equivalent to saying that gcd(a,b) = 1 because whatever divides LHS will also divide RHS. For instance, (3, 4, 5) and (5, 12, 13) are both primitive Pythagorean triple (PPT).

As we have already noted, there are integer triples that are not PPT. For instance, (1, 2, 3) is not a PPT. Therefore, two natural questions are:

How many PPT’s are there, finite or infinite?
Is it possible to generate all PPT’s?

It turns out that there are infinitely many PPT’s and the answer to second question is in affirmative.

Theorem 1. If (a,b,c) is a PPT then c is always an odd number and, between a and b, one is even and the other odd.

Proof. Since gcd(a,b) = 1 both cannot be even. Suppose both and a and b are odd, then a = 2k + 1 and b = 2l + 1. Therefore a2 and b2 are also odd because square of an odd number is odd (Use (2k + 1)2)!). Therefore a2 + b2 is even because sum of odd numbers is even. Thus c2 is even and, hence c is even. Let c = 2m. Then

4k2 + 4k + 1 + 4l2 + 4l + 1 = 4m2 2(k2 + l2 + k + l) + 1 = 2m2.

Note that LHS is odd and RHS is even which is a contradiction. Therefore, one of a and b is even and the other odd. This implies c2 is odd and, hence, c is odd. □

The above result rules out all triples (a,b,c), with c even, as a possible PPT. It is enough to consider c odd. But that still does not say all odd c are allowed. For instance, there is no choice of (a,b) such that (a,b, 3) is a PPT. In fact, the first PPT is (3, 4, 5).

Also owing to the result above, henceforth in the pair (a,b), we shall always denote the odd number by a and the even number by b. For any PPT (a,b,c), we write a2 = c2 b2 = (c + b)(c b). Note that c b > 0.

Theorem 2. For any PPT (a,b,c), c + b and c b are relatively prime.

Proof. Let x divide both c + b and c b. Then c + b = kx and c b = lx for some k,l . Then 2c = x(k + l) and 2b = x(k l) which means that x divides both 2b and 2c. But gcd(b,c) = 1 and, hence, x = 1 or x = 2. Since x should also divide a2, hence a. But a is odd so x = 1. □

Above result says that a2, a perfect square, is product of two relatively prime numbers c + b and c b. Let c + b = p1x1p kxk and c b = q1y1q lyl be the respective unique prime decomposition. Then a2 = p 1x1p kxkq 1y1q lyl. Since c + b and c b are relatively prime piqj for all i,j. Thus, each xi and yj is even. Therefore, c + b and c b are also perfect squares, say c + b = m2 and c b = n2, for some m,n such that m > n and gcd(m,n) = 1. Consequently, a = mn and, since a is odd, both m and n are odd. So, for every choice of odd positive integer m,n such that m > n and gcd(m,n) = 1, we have a PPT (a,b,c) given by a = mn, b = m2n2 2 and c = m2+n2 2 . The last two equalities are obtained by solving for b and c using c + b = m2 and c b = n2. Since there are infinitely many choices of m,n satisfying above condition there are infinitely many PPT’s.

A generalisation of the Pythagorean triple condition is, for a fixed integer n > 2, seeking triples (a,b,c) such that an + bn = cn. Obviously, there are some trivial solution if (a,b,c) is such that abc = 0. However, it turns out there are no non-trivial solution, i.e., (a,b,c) with abc0

Theorem 3 (Fermat’s Last Theorem). Given an integer n > 2, there are no integer solutions to an + bn = cn with abc0.

Suppose n = kl. Then if the equation an + bn = cn has integer solution then, using (ak)l + (bk)l = (ck)l, (ak,bk,ck) is a solution corresponding to n = l. Therefore, to prove FLT it is enough to prove the result for l n. Note that any n > 2 is:

either n = 2m, for some integer m > 2, which is same as n = 4 × 2k2. Choose l = 4 in this case;
or an odd prime. Choose l = n in this case;
or a muliple of an odd prime. Choose l to be the odd prime.

Thus, it is enough to prove the result for n = 4 and odd prime n. If there is an integer solution to a4 + b4 = c4 then (a,b,c2) is an integer solution to x4 + y4 = z2. So, to prove FLT for n = 4 we show the following theorem:

Theorem 4. There are no integer solution to x4 + y4 = z2 with xyz0.

Proof. Suppose there is a integer triple (x,y,z) such that xyz0 solving x4 + y4 = z2. Without loss of generality, we may assume that x,y,z are all positive, gcd(x,y,z) = 1 and, hence, gcd(x,y) = 1.

(Step 1): Note that (x2,y2,z) is a PPT, since gcd(x2,y2,z) = 1 (if necessary, we rewrite x and y such that x2 is odd and y2 is even). We also know that z is always odd.

(Step 2): There exists odd numbers m,n , such that m > n, gcd(m,n) = 1 and

x2 = mn; y2 = m2 n2 2  and z = m2 + n2 2 .

Then, 2y2 = m2 n2 = (m + n)(m n). Since m and n are odd, both m + n and m n are even and, hence, are not coprime.

(Step 3): Let d > 1 divide both m + n and m n. Then m + n = kd and m n = ld for some k,l . Then 2m = d(k + l) and 2n = d(k l) which means that d divides both 2m and 2n. But gcd(m,n) = 1 therefore d divides 2. Therefore, d = 2. Hence, m + n = 2k, m n = 2l, m = k + l and n = k l. Since gcd(m + n,m n) = 2, gcd(k,l) = 1.

(Step 4): In fact, using the equation of x2, we have x2 + l2 = k2, i.e., (x,l,k) is a PPT. Therefore, l is even and k is odd. Let l = 2. Then gcd(k,) = 1.

(Step 5): Using equation for y2, we get 2y2 = 4kl = 8k, i.e., y2 = 4k. Hence k and are perfect squares, k = u2 and = v2 and gcd(u,v) = 1. Therefore, (x, 2v2,u2) is a PPT.

We repeat Step 2 on (x, 2v2,u2). There exists odd numbers M,N , such that M > N and gcd(M,N) = 1 we have

x = MN; 2v2 = M2 N2 2  and u2 = M2 + N2 2 .

We repeat Step 3 on M + N and M N to obtain K and L such that gcd(K,L) = 1, M + N = 2K, M N = 2L, M = K + L and N = K L.

We repeat Step 5 on 2v2 to conclude that v2 = KL and, hence, K = U2 and L = V 2 and gcd(U,V ) = 1.

Using the value of M and N in the equation of u2, we get

u2 = M2 + N2 2 = 2(K2 + L2) 2 = U4 + V 4.

Therefore, (U,V,u) is also a non-trivial integer solution of x4 + y4 = z2. Let us compare the two solutions (x,y,z) and (U,V,u). Note that

z = m2 + n2 2 = (u2 + 2v2)2 + (u2 2v2)2 2 = u4 + 4v4.

This implies that u4 < z, hence u < z. Thus, when we started with a non-trivial solution (x,y,z) we obtained another non-trivial solution (U,V,u) such that 0 < u < z. We can repeat the entire proof for again for (U,V,u) playing the role of (x,y,z) above and can obtain another non-trivial solution U1,V 1,u1) such that 0 < u1 < u < z. This is a contradiction because we cannot carry on infinite number of times as suggested by our proof. Therefore, our assumption on the existence of a non-trivial integer solution is false. □

With all the effort above, it only remains to prove the Fermat’s Last theorem for odd primes.

Theorem 5 (Fermat’s Last Theorem). Given an odd prime p, there are no integer solutions to ap + bp = cp with abc0.

Definition 1. An odd prime number p is called regular if p does not divide the numerator of the Bernoulli number Bk, for all even k p 3. Any odd prime which is not regular is called irregular.

The odd primes 3, 5, 7,, 31 are all regular primes. The first irregular prime is 37. It is an open question: are there infinitely many regular primes? However, it is known that there are infinitely many irregular primes.

Theorem 6 (Kummer, 1850). If p is a regular odd prime then the equation

ap + bp = cp

has no solution in .

Gerhard Frey proved the following result related to FLT:

Theorem 7 (Proved in 1984). Given a odd prime p if there exists a non-trivial solution to the equation

ap + bp = cp

then the following elliptic curve, called Frey’s curve, must exist:

y2 = x(x ap)(x + bp).

So, what is an elliptic curve? Given a,b,c , consider plane curve of the form y2 = x3 + ax2 + bx + c. Let us call it Γ. The discriminant of Γ is defined as

Δ(Γ) := 4a3 + a2b2 4b3 27c2 + 18abc.

The Δ = 0 relates to the case when the curve self-intersects, called singular points. A curve Γ is said to be an elliptic curve if Δ(Γ)0. If Δ < 0 the curve will divide the plane in two connected components and if Δ > 0 it divides the plane into more than two connected components. Elliptic curves are symmetrical about x-axis. The points on the elliptic curve form an abelian group under a suitable binary operation (+). An elliptic curve Γ is modular if there exists a L-function of Γ. In 1955, the Taniyama-Shimura-Weil (TSW) conjectured (also called modularity theorem) states that:

Theorem 8 (Later proved in 2001). Every elliptic curve over the field of rationals is modular.

This means if Frey’s curve existed it must be modular. But, in 1985, Jean-Pierre Serre conjectured (epsilon conjecture) that Frey’s curve is not modular which was later proved by Ken Ribet in 1986. Since Frey’s curve is an elliptic curve it must be modular assuming TSW conjecture. An elliptic curve is semistable if a prime divides its discriminant and Frey’s curve is semistable. In 1994-95 by Andrew Wiles showed that:

Theorem 9 (Proved in 1994). Every semistable elliptic curve over the field of rationals is modular.

This means Frey’s curve cannot exist and, hence, Fermat’s last theorem is true.