Introduction to Elliptic Curves


Seyyed Mohammad Reza Hashemi Moosavi


Elliptic curves are a special kind of algebraic curves which have a very rich arithmetical structure.

There are several fancy ways of defining them. but for our purposes we can just define them as the set of points satisfying a polynomial equation of a certain form. To be

 specific, consider an equation of the form

where the are integers (There is a reason for the strange choice of indices on the , but we won't go into it here). we want to consider the set of points which satisfy

 this equation.

To make things easier, let us focus on the special case in which the equation is of the form

with a cubic polynomial (in other words, we're assuming ). In this case (*), it's very easy to determine when there can be singular points, and even what

 sort of singular points they will be. If we put

Then we have


we know, the curve will be smooth if there are no common solutions of the equations

Attention. we know, from elementary analysis, that an equation  defines a smooth curve exactly when there are no points on the curve at which both partial

derivatives of  vanish.

in other  words, the curve will be smooth if there are no common solutions of the equations (**).

And the condition for a point to be "bad" be comes

which boils down to  In other words, a point will be "bad" exactly when its - coordinate is Zero and its - coordinate is a double root of the

 polynomial . since is of degree 3, this gives us only three possibilities:

·         has no multiple roots, and the equation defines an elliptic curve (Three distinct roots),  (For example, elliptic curve  has three distinct  roots).

·          has a double root (curve has a node), (For example, curve  has a node).

·         has a triple root (curve has a cusp), (For example, curve  has a cusp).

Attention. If and are the roots of the polynomial , the discriminant for the equation turns out to be

where is a constant.

This does just what we want:

If two of the roots are equal, it is Zero, and if not, not. Further more, it is not too hard to see that is actually a polynomial in the coefficients of  , which is what we

claimed. In other words, all that the discriminant is doing for us is giving a direct algebraic procedure for determining whether there are singular points.

while this analysis applies specifically to curves of the form  it actually extends to all equations of the sort we are considering there is at most one singular point

 and it is either a node or a cusp.

Attention. with some examples in hand, we can proceed to deeper waters. In order to understand the connection we are going to establish between elliptic curves and Fermat's

 Last Theorem, we need to review quite a large portion of what is known about the rich arithmetic structure of these curves.

Conclusion. Elliptic curve of the form

is a elliptic curve of Non – Singular if  has not a double root or

a triple root. In fact below equation

has three distinct roots, if

Attention. If then has no multiple roots, and is a

Non –Singular cubic elliptic curve.


horizontal rulehorizontal rule

3 New Proof of Fermat's last Theorem by HM Final Main Theorem


Seyyed Mohammad Reza Hashemi Moosavi

·       HM Final – Main Theorem

For every odd number Fermat's Last Theorem ():

Special case is of an elliptic curve (Non – Singular Cubic Curve) HM:

·       Proof:

It is enough in the below general elliptic curve:



Or elliptic curve HM (*) we assume:




then after replacing in (*) or (**):

(odd numbers)


§  Elliptic curve (*) is Non – Singular.

§  First Fermat's equation is multiplied .

§  We assume .

§  We know that Proofed is an elliptic curve.


New Proof of Fermat's Last Theorem by HM Final – Main Theorem

Because:  ;






    (odd numbers);






§  Elliptic curve HM (*) is non – Singular, because:


    (Three different roots)

·        References

[1] S. M. R. Hashemi Moosavi, The Discovery of Prime Numbers Formula & It's Results (2003). (ISBN: 978-600-94467-9-7).

[2] S. M. R. Hashemi Moosavi, Generalization of Fermat's Last Theorem and Solution of Beal's Equation by HM Theorems (2016). (ISBN: 978-600-94467-3-5).

[3] S. M. R. Hashemi Moosavi, 31 Methods for Solving Cubic Equations and Applications (2016). (ISBN: 978-600-94467-7-3).