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.)]]>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:

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

]]>**The Lutz-Nagell Theorem**

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

**Mazur's Theorem**

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

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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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