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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,478

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.

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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: 21,478

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.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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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One can define a "group law" on a cubic (and therefore an elliptic curve, by reducing the cubic to Weierstrass form).Suppose we have an elliptic curve , and denote the set of rational points on by . Poincaré saw that given three points on a line, one may form a group law by taking . That is, to add two points , we draw a line through and and take the third point of intersection to be . We then reflect in the x-axis to obtain a new point . We can then obtain a group law for the set of rational points and the group operation . The operation is closed, and it suffices to check that , that is, . We take the point at infinity to be the identity of the group. One can verify associativity in various ways -- one of which is by drawing lots of lines and verifying it geometrically -- and hence we have a group.Some authors write instead of . A nice photo from Wikipedia illustrates this:

*Last edited by zetafunc (2016-05-23 19:42:48)*

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With a little algebra, we can write a "doubling formula" for a point . We can obtain a new point whose co-ordinates are given by:And hence we can obtain explicit formulae for , and so on. (We'll see the importance of this later when we compute torsion subgroups of elliptic curves.)A natural question to ask is: given the algorithm for finding new rational points in the previous post, how many can we generate? Will we eventually return to our original point, or will this go on forever? It turns out that the answer to both of these questions is **yes**. If we do return to our original point, then we say that P has *finite order*, and write , to mean that P has order n. Such a point is called a *torsion point*.

If we do not return to our original point, then P is a point of infinite order.

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We denote the set of torsion points -- that is, the set of points of finite order on -- by . It turns out that this torsion subgroup is not only a subgroup of E(Q), but it is also finite. The point at infinity has order 1, by default. We have the following theorem:

**The Lutz-Nagell Theorem**

We can explicitly write the possibilities Tors(E(Q)) by Mazur's theorem.

**Mazur's Theorem**

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**Examples of Computing Tors(E(Q))**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

Hi;

Do you use the elliptic discriminant http://mathworld.wolfram.com/EllipticDiscriminant.html to get that

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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Hi,

I am using: for an elliptic curve . (in other words, without the multiplication by -16)

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 878

I am surprised to know about elliptic curves of rational numbers and corresponding modular form.

*Last edited by thickhead (2016-05-24 00:19:48)*

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**Another Example of Computing Tors(E(Q))**

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**Height and Elliptic Curves**

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**Néron-Tate Height**

Summing over k from m+1 to n, we get:

so that we have a Cauchy sequence (and therefore it converges). We call this limit the Néron-Tate height:

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**The Weierstrass Elliptic Function**

We define the Weierstrass elliptic function as follows:

.It can be shown that it converges absolutely, is periodic on the lattice and is even. Moreover, any meromorphic,

-periodic function is a rational function of and .We have an isomorphism of groups: . Hence, every lattice may be associated with an elliptic curve! (And the converse is true as well.)

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**Mordell's Theorem**

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**The j-invariant**

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