Yesterday, I read a blog about Fermat conjecture, reviewed its proof and find that this conjecture is not proved.Its proof did not prove anything at all.

Analyses:

Its proof idea is to try proving that

X ^ n + Y ^ n = Z ^ n

have at least one condition that satisfy this formula.

The contradiction reached does not imply anything. It can only imply that the proof works so far are not correct.

Following are the facts:

1. Elliptic curve is modular; actually there are infinitely many ways for its modularity

  calculations, so proofs of modularity is a trivial task that will not involve enormous

  works.

2. Modularity mapping may be divided into different categories, with each category’s works

  less a few days.

3. The Frey curve will not change its genus,that's determined by the nature of the curve

  structure.

4. By introducing Fermat Formula into Elliptic Curve, this make the difficult work even more

  difficult because you have the ambiguous modularity problem of the elliptic curves, and

  you add one more difficult problem of infinite polynomial equation to solve.

5. Its base conjecture, the epsilon conjecture, and or Ribet's theory have problems. Fermat

  Equation have infinitely many solutions that have nearly zero errors which should be all

  right for any mod calculation but the Frey curve do not have the ability to show these

  at all, and that's the obvious problem. Readers can check out the theses for these to

  verify.

The original proof not only contain many errors, but also logically disordered.

Fermat Conjecture is not proved at all. A problem went so long to solve,something must go wrong just like religions.

Now, let’s see why it was not proved.

Let

a ^ n + b ^ n = c ^ (n + d)

be always right, where a, b, c, n, are all positive integers and delta d is a real, and

Y ^ 2 = X (X - a ^ n) (X + b ^ n)

is elliptic curve, always modular, this is also a true Frey curve, always has genus of one, and this reasoning is always true because it is from identity equations.

Because a, b, c, d can take any values, and delta can be zero or not, it just do not care, this proved that Fermat Conjecture has nothing to do with Frey curve. Some theses claimed that Frey curve will not be modular or level reduction will make genus to become zero.This is not true except that wrong mod operations are performed.

So Frey curve and or Ribet's theory do not provide a way to Fermat Conjecture at all.

If you cannot see this, let’s further assume that

a ^ n + b ^ n = 0

This equation is always wrong, but no matter what values a, b, n are taken, following is always true.

The Frey curve is still elliptic, modular, it only change its locations for different a, b, and n. It is very clear that Frey curve cannot tell right or wrong.

Now,assume that

a ^ n + b ^ n = c ^ n

This is the Fermat Equation and the result is still the same.

The Frey curve is still elliptic, modular, it only change possible locations for different a, b, c, and n values if the equation need to be matched. Acyually, this time, there is no condition to match and the only possible condition is: Ribet's Theory is wrong.

The conclusion is that:

Frey curve only act as it is told what should be done, it do not have the ability to judge right or wrong. Fermat Conjecture has nothing to do with Frey curve and it need to be proved separately/

Actually, modularity theory can be defined by many different ways.

Proof:

The complete proof will use number theory, the theories of rational, real, complex numbers, modularity theory, the theories of modular form and modular curve, group theory, the theory of topology, and graph theory, etc. of course this will include arithmetic.

Because this will take some formal works and it is not necessary, I think knowing its analyses is enough.

So with so many stories behind, it must be true that history is written to think, not to believe.

The touching story is not so touching because almost everyone involved was wrong, and the story is actually very sad. Many life were lost, many families were broken, and many dreams that should continue were stopped.

We need only ask the question:

Do we have true experts? Let alone top experts.

There are more errors in mathematical world and science and in every works of life.

Expected to improve.