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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,448

An elliptic curve is of the form

y²=x³+ax²+bx+c where a,b, and c are integers.

When a=0, b=0 and c=-2, the equation becomes

y²=x³-2

Leonard Euler was the first to prove that this equation has only one solution, y=5 and x=3. Thus, he proved that 26 is the only number among the infinity of numbers which is jammed between a perfect square and a perfect cube.

The proof is lengthy and sophisticated even for the present day mathematicians.

The Prince of Mathematicians, Euler, who is belived to have contributed more to mathematics than any other individual was also the first to make amajore breakthrough in proving Fermat's Last theorem. He had proved that

a³+b³ = c³ has no solution where a,b, and c are whole numbers, the only exception being a=b=c=0.

Generations of mathematicians later proved Fermat's Last theorem for certain prime number values of n. It was only during the fag end of the twentieth century that Andrew Wiles finally succeeded in proving Fermat's Last Theorem.

Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Elliptic curves over the field of rationals are related to modular forms a result which was crucial to the proving of Fermats last theorem by Andrew Wiles.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,448

You're correct, JaneFairfax!

The relation between elliptic curves and modular forms was studied by Yukata Taniyama and Goro Shimura about 10 years after the second world war in Japan leading to the famous Shimura-Taniyama Conjecture on which Andrew Wiles' proof is based.

Go to this link.

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**100'****Member**- Registered: 2007-12-06
- Posts: 8

ganesh wrote:

Leonard Euler was the first to prove that this equation has only one solution, y=5 and x=3. Thus, he proved that 26 is the only number among the infinity of numbers which is jammed between a perfect square and a perfect cube.

The proof is lengthy and sophisticated even for the present day mathematicians.

The Prince of Mathematicians, Euler, who is belived to have contributed more to mathematics than any other individual was also the first to make amajore breakthrough in proving Fermat's Last theorem. He had proved that

a³+b³ = c³ has no solution where a,b, and c are whole numbers, the only exception being a=b=c=0.

Generations of mathematicians later proved Fermat's Last theorem for certain prime number values of n. It was only during the fag end of the twentieth century that Andrew Wiles finally succeeded in proving Fermat's Last Theorem.

It was Fermat who proved that 26 is the only number between a perfect square and cube.

Also, the proof that a³ + b³ = c³ has no solutions was only a tiny part of the solution. Find a piece of evidence in an infinity to support a hypothesis (regardless of how hard it was to find) is useless (this has been demonstrated by the Riemann Hypothesis in which millions of zeroes have been found upon Riemanns critical line, not furthering research particularly).

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