Fermat’s Last Theorem (n = 3 and n = 4)

History

After the French mathematician Pierre de Fermat died in the seventeenth century, a note written by Fermat was discovered in which he conjectured that there were no nonzero integer solutions to the equation xⁿ + yⁿ = zⁿ for n > 2.  This proposition became known as Fermat’s Last Theorem. 

Fermat also wrote that he had “discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain,” but no proof could be found in any of his other notes.  At the time, though, this seemed trivial, because surely some other mathematician would find a proof for it.  But as years turned to decades, and decades turned to centuries, no general proof could be found.

The proof for Fermat’s Last Theorem became a Holy Grail for mathematics.  Like the Holy Grail, some mathematicians spent their lives trying to find a proof but came back empty-handed.  And like the Holy Grail, other mathematicians doubted its existence.  Perhaps Fermat’s Last Theorem was incorrect and so couldn’t be proved (although no counter-example could be found, either).  Perhaps Fermat didn’t actually have a proof, or perhaps he thought he had a proof but then realized later that it was incorrect and so never published it.  On the other hand, perhaps Fermat had a proof but there have been no mathematicians as great as him since to re-create his proof.  In any event, by the beginning of the twentieth century finding a general proof for Fermat’s Last Theorem became sweeter than just mathematical notoriety, because an industrialist named Wolfskehl bequeathed a prize of 100,000 German Marks to the first mathematician who could prove it.

Fermat’s Last Theorem: The Holy Grail of Mathematics

In the meantime, specific proofs were found for Fermat’s Last Theorem.  A proof from Fermat himself was discovered for the specific case of n = 4 (that there were no nonzero integer solutions to the equation x⁴ + y⁴ = z⁴).  Euler proved Fermat’s Last Theorem for n = 3 in 1770, over a hundred years after Fermat’s death.  Dirichlet and Legendre proved Fermat’s Last Theorem for n = 5 in 1825, nearly two hundred years after Fermat’s death. 

More and more specific proofs were found for greater and greater exponents, but it was not until 1995, over three hundred years after Fermat’s death, that Wiles published a general proof that included all exponents greater than two.  His proof is over one hundred pages long, and uses modern methods and techniques not available to Fermat (and beyond the scope of this article), so most mathematicians agree that this is probably not the proof Fermat had in mind back in the seventeenth century (if he even had one).

Wiles

Difficulties

One of the reasons why Fermat’s Last Theorem is so difficult to prove is because it has to be true for a certain set of numbers but false for another set of numbers.  First of all, Fermat’s Last Theorem only applies for nonzero integer solutions.  However, integer solutions to xⁿ + yⁿ = zⁿ do exist for all exponents n as long as one of x, y, or z is equal to zero, for example, 1³ + 0³ = 1³ or 2⁴ + 0⁴ = 2⁴.

Secondly, Fermat’s Last Theorem also only applies for exponents greater than two.  Nonzero integer solutions to xⁿ + yⁿ = zⁿ do exist for n = 1 and n = 2.  When n = 1, the formula becomes x + y = z, which has infinitely many nonzero integer solutions, for example, 3 + 4 = 7 or 5 + 12 = 17.  When n = 2, the formula becomes x² + y² = z², which is the Pythagorean Theorem, and since it can be proved that all solutions to the Pythagorean Theorem are x = p² – q², y = 2pq, and y = p² + q² for any p and q (see here for a proof), an infinitely many combinations of nonzero integers p and q can be picked to generate nonzero integer solutions.  For example, if p = 2 and q = 1, then x = p² – q² = 2² – 1² = 3, y = 2pq = 2(2)(1) = 4, and z = p² + q² = 2² + 1² = 5, so x² + y² = z² is 3² + 4² = 5².  Or for another example, if p = 3 and q = 2, then x = p² – q² = 3² – 2² = 5, y = 2pq = 2(3)(2) = 12, and z = p² + q² = 3² + 2² = 13, so x² + y² = z² is 5² + 12² = 13².

Fermat’s Last Theorem for n = 3

Recall from above that Euler proved Fermat’s Last Theorem for n = 3 (that there are no nonzero integer solutions to x³ + y³ = z³) in 1770.  A few different ways to prove Fermat’s Last Theorem for n = 3 have been found, and a full proof, which is loosely based on the proof found on Freeman’s blog, can be found here. 

Summarizing the proof briefly, first assume that there are positive integer solutions to x³ + y³ = z³.  Then there must exist a cube in the form of 2p(p² + 3q²) where p and q are relatively prime and where the greatest common factor of 2p and p² + 3q² is 1 or 3.  In either case, you can use the fact that all odd factors of p² + 3q² must also have that same form (proof for that here) to prove that there exists smaller positive integers x’, y’, and z’ in which x’³ + y’³ = z’³.  Since we can apply the same argument infinitely to obtain smaller and smaller positive integers, there is a contradiction by the method of infinite descent, and so we reject our assumption that there are positive solutions to x³ + y³ = z³.  We can also reject that there are any negative solutions to x³ + y³ = z³ because the formula can be rearranged to an all positive integer solution still in the same form.  Since there are no positive or negative solutions to x³ + y³ = z³, there are no nonzero integer solutions to x³ + y³ = z³.

Fermat’s Last Theorem for n = 4

Also recall from above that a proof for n = 4 (that there are no nonzero integer solutions to x⁴ + y⁴ = z⁴) was given by Fermat himself.  The specific case for n = 4 is actually easier than proving the specific case for n = 3, because a fourth power is a square of a square, so a Pythagorean solution can be found twice in such a way to produce a contradiction once again by the method of infinite descent.  A full proof for Fermat’s Last Theorem for n = 4 can be found here.

Fermat’s Last Theorem for n ≥ 5

Proving Fermat’s Last Theorem for n = 3 proves Fermat’s Last Theorem for n for all multiples of 3, and likewise proving Fermat’s Last Theorem for n = 4 proves Fermat’s Last Theorem for n for all multiples of 4.  For example, to prove that there are no nonzero integer solutions for n for all multiples of 3, or n = 3m, one can simply express x^3m + y^3m = z^3m as (x^m)³ + (y^m)³ = (z^m)³, a special case for n = 3; and to prove that there are no nonzero integer solutions for n for all multiples of 4, or n = 4m, one can simply express x^4m + y^4m = z^4m as (x^m)⁴ + (y^m)⁴ = (z^m)⁴, a special case of n = 4.  So a general proof for Fermat’s Last Theorem only requires a proof for n of all prime numbers greater than two.

It is tempting to use the proofs for n = 3 or n = 4 and apply them to the proofs for higher exponents, but unfortunately both proofs for n = 3 and n = 4 exploit properties that are unique for its exponent.  The proof for n = 3 uses a unique property that all odd factors of p² + 3q² must also have that same form.  But using the same logic as the proof for n = 3 on n = 5, we would need all odd factors of p⁴ + 10p²q² + 5q⁴ to have the same form, but unfortunately this is simply not true, and the same can be said for higher exponents.  Furthermore, the proof for n = 4 uses the unique property that a fourth power is a square of a square, which is not true of any primes greater than two.  So a specific proof for any prime n ≥ 5 must use a completely different (and more difficult) approach altogether.

Conclusion

Even though Fermat’s Last Theorem was proved by Wiles in 1995, some mysteries remain.  Did Fermat really have a legitimate general proof for his theorem, or was he mistaken?  Will a general proof be found that is shorter or uses a traditional approach?  Perhaps it will take another three centuries to answer these questions.  If only Fermat’s margin was bigger!