THE WAR OF MATHEMATICS THAT LASTED FOR 358 YEARS

- SAURISH GUPTA

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THE LAST THEOREM- Given by Pierre de Fermat in 1637, found along the margin of his copy of ARITHMETICA, written by Diophantus(c.200/214AD-c. 284/298AD), was the following

No three positive integers x,y,z satisfy

x^n + y^n = z^n ,

for any integer n>2.

Although a special case of it was solved by Fermat himself using infinite descent method the theorem was not completely and correctly proved till 1995 when a reserved Englishman, an English mathematician and a ROYAL SOCIETY RESEARCH PROFESSOR at the University of Oxford, SIR ANDREW JOHN WILES proved it. The proof due to Andrew Wiles was a result of centuries of work by dozens of mathematicians. The theorem which Wiles actually proved is far deeper a concept and more mathematical in nature than its corollary i.e FERMAT’S LAST THEOREM. The mathematicians who had actually dived into the problem and had brought out some famous mathematics for the problem were SOPHIE GERMAIN, ERNST KUMMER, and GERHARD FREY. Apart from them there were two other Japanese mathematicians and without them Fermat would still have been on the winning side with his RIDDLE. They were GORO SHIMURA and YUTAKA TANIYAMA.

EULER’S LITTLE CONTRIBUTION- SIR LEONHARD EULER, the Swiss mathematician, physicist, astronomer, geographer, logician and engineer and the most famous mathematician of the 18th century, had proved the FERMAT’S LAST THEOREM true for n=3 with the same method used by Fermat for proving it for n=4, i.e INFINITE DESCENT method.

WORKS OF SOPHIE GERMAIN- In the 19th century, Sophie Germain came up with an interesting plan to attack Fermat’s Last Theorem. She planned to go on with a set of primes, say ‘p’, such that if ‘p’ is prime then 2p+1 is also prime. These primes are known as Germain Primes and recently are of great use in public key cryptography.

Germain’s method roughly meant that there were ‘probably’ no solutions to the equation, x^n + y^n = z^n, for n bring Germain primes because if they had solutions then either x, y or z would be a multiple of n, and this would put a tight restriction on any solutions. In 1925, two mathematicians Adrian- Marie- Legendre and Gustave Lejeune Dirichlet gave success to Germain’s method, independently, by implying her methods to the problem and proving her method to be true.

ERNST KUMMER’S CONTRIBUTION- In the mid 19th century, mathematicians began to explore proof ideas involving factoring the left side (x^p + y^p) in the ring Z[ζp​], where ζp​ is a primitive pth root of unity. This led to a deep study of such rings (and their fields of fractions Q(ζp​), called cyclotomic fields), which was the genesis of modern algebraic number theory. Ernst Kummer, one of the pioneers of this field, identified a class of primes which were amenable to these techniques, which he called regular primes. He was able to prove Fermat's last theorem for regular prime exponent, but could do nothing substantive with irregular primes. It was later proved that there are infinitely many irregular primes (heuristically, the probability of a prime being irregular is roughly 39%), so this approach was doomed to have only limited success.

Kummer's ideas did furnish an effective strategy for showing that x^p + y^p = z^p had no solutions for any given odd prime p. By 1993, computing technology was sufficiently advanced to prove that x^p + y^p = z^p had no solutions for p < 4*(10^6) (and that there were no solutions in the first case for p < 7*(10^14)). But these methods had no hope of providing a proof of the theorem for all p.

THE NEW SHIFT TOWARDS THE PROOF- In the January of 1954, a talented young mathematician at the University of Tokyo paid a routine visit to his departmental library. Goro Shimura was in search of a copy of Mathematica Annalan, VOL-24. In particular he was after a paper by Deuring on his algebraic theory of complex multiplication, which he needed in order to help him with a particularly awkward and esoteric calculation.

To his surprise, the volume was already borrowed by Yutaka Taniyama who lived on the other side of the campus. These two went on to change mathematicians' perception towards FERMAT'S LAST THEOREM forever(though indirectly).

THE TANIYAMA-SHIMURA CONJECTURE- According to the TANIYAMA-SHIMURA CONJECTURE, every elliptic equation has a modular form linked with it, or more mathematically “ Elliptic curves over the field of rational numbers are related to modular forms.”

THE UNEXPECTED- During the autumn of 1984, a selected group of number theorists gathered for a symposium in Oberwolfach, a small town in Germany's Black Forest. Here, a mathematician by the name of Gerhard Frey, firmly established a link between the TANIYAMA-SHIMURA CONJECTURE and FERMAT’S LAST THEOREM. He stated that if someone proved the TANIYAMA-SHIMURA CONJECTURE to be correct he will automatically prove the FERMAT’S LAST THEOREM.

FREY’S CALCULATION AND HIS FINAL CONCLUSION- According to Frey, if FERMAT’S LAST THEOREM is false, i.e if it has at least one solution(let that be (A,B,C) for n= N), then

A^N + B^N = C^N. Now, Frey rearranged this equation with a rigorous mathematical procedure, and transformed it into an elliptic equation. The equation is:-

y^2 = x^3 + (A^N - B^N)x^2 - (AB)^N

Now Frey’s elliptic equation is only a phantom equation as its existence is conditional on the fact that FERMAT’S LAST THEOREM is false.

The final conclusions of Frey are as follows:-

• Iff Fermat’s Last Theorem is wrong, then Frey’s elliptic equation exists.

• Frey’s elliptic equation is so weird that it can never be modular.

• The Taniyama-Shimura Conjecture says that every elliptic equation is related to a modular form.

• Therefore the Taniyama-Shimura Conjecture is wrong.

Alternatively,

• Iff the Taniyama-Shimura conjecture is true, then every elliptic equation has a modular form.

• If every elliptic equation has a modular form, then Frey’s elliptic equation is forbidden to exist.

• If so, then there can be no solution to Fermat’s Last Theorem.

• Thus Fermat’s Last Theorem is true.

KEN RIBET had successfully proved this link established by Frey.

THE WAR ENDS- After years of hard work, there was finally a tool to solve the Fermat’s Last theorem, and ANDREW WILES in utter secrecy solved the theorem by proving the TANIYAMA-SHIMURA conjecture in the year 1993. But the proof had some mistakes and he published the correct proof in the year 1995, thus bringing THE MATHEMATICAL WAR to an end.