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
Last Theorem ():
Special case is of an elliptic curve (Non – Singular Cubic
It is enough in the below general elliptic curve:
Or elliptic curve HM (*) we assume:
after replacing in (*) or (**):
Elliptic curve (*) is Non – Singular.
First Fermat's equation is multiplied
We assume .
We know that Proofed
an elliptic curve.
New Proof of Fermat's Last Theorem by HM Final – Main
Elliptic curve HM (*) is non – Singular, because:
(Three different roots)
 S. M. R. Hashemi Moosavi, The Discovery of Prime Numbers
Formula & It's Results (2003).
 S. M. R. Hashemi Moosavi, Generalization of Fermat's Last
Theorem and Solution of Beal's Equation by HM Theorems (2016).
 S. M. R. Hashemi Moosavi, 31 Methods for Solving Cubic
Equations and Applications (2016).
Introduction to Elliptic Curves
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
integers (There is a reason for the strange choice of indices on
but we won't go into it here). we want to consider the set of
satisfy this equation.
To make things easier, let us focus on the special case in which
the equation is of the form
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
we know, from elementary analysis, that an equation
a smooth curve exactly when there are no points on the curve at
which both partial derivatives of
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
other words, a point will be "bad" exactly when its
coordinate is Zero and its
coordinate is a double root of the polynomial
of degree 3, this gives us only three possibilities:
no multiple roots, and the equation defines an elliptic curve
(Three distinct roots),
example, elliptic curve
three distinct roots).
a double root (curve has a node), (For example, curve
a triple root (curve has a cusp), (For example, curve
the roots of the polynomial
the discriminant for the equation
out to be
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
actually a polynomial in the coefficents 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
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
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.
Elliptic curve of the form
is a elliptic curve of Non – Singular if
not a double root or
a triple root. In fact below equation
has three distinct roots, if
no multiple roots, and
Non –Singular cubic elliptic curve.
I am Dr.S.M.R.Hasheimi Moosavi.
I have discovered the formula of the prime numbers after 20
years of research and injury. I solved the
problems related to them.
The distinction of the prime numbers.
generating formula of the prime numbers.
definition of the prime numbers set by using
the generating function of
generating formula of the Mersenne prime numbers.
determining of the k-th prime number.
Riemann zeta equation by using the determining of
number of the prime numbers less than or equal arbitrary number (N) exactly.
of the guesses of Goldbuch and Hardy.
proof of being infinity of the prime twin couples.
the results of this great discovery "the formula of generating
prime numbers discovery" has been sent to most of research
and universities of the world.
What is a prime number?
A prime number is
a positive integer that has exactly two positive integer
factors, 1 and itself. For example, if we list the
factors of 28, we have 1, 2, 4, 7, 14, and 28. That's
six factors. If we list the factors of 29, we only have
1 and 29. That's 2. So we say that 29 is a prime number,
but 28 isn't.
Another way of
saying this is that a prime number is a whole number
that is not the product of two smaller numbers.
Note that the
definition of a prime number doesn't allow 1 to be a
prime number: 1 only has one factor, namely 1. Prime
numbers have exactly two factors, not "at most
two" or anything like that. When a number has more than
two factors it is called a composite number.
Here are the first
few prime numbers:
2, 3, 5, 7, 11,
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199,
The Largest Known Primes
on the Web
What are Mersenne
primes and why do we search for them?
A Mersenne prime is a prime
of the form 2P-1. The first Mersenne primes are 3, 7,
31, 127, etc. There are only 41 known Mersenne primes.
GIMPS, the Great Internet
Mersenne Prime Search, was formed in January 1996 to discover
new world-record-size Mersenne primes. GIMPS harnesses the power
of thousands of small computers like yours to search for these
"needles in a haystack".
Most GIMPS members join the
search for the thrill of possibly discovering a record-setting,
rare, and historic new Mersenne prime. Of course, there are many
more information about Mersenne primes , History ,Theorems and
click here .
Why do people find these primes?
"Why?" we are often asked,
"why would anyone want to find a prime that big?"" I often now
answer with "did you ever collect anything?"" or "did you ever
try to win a competition?"" Much of the answer for why we
collect large primes is the same as why we might collect other
rare items. Below I will present a more complete answer divided
into several parts.
For the by-products of the
People collect rare and beautiful items
For the glory!
To test the hardware
To learn more about their distribution
This does not exhaust the
list of reasons, for example some might be motivated by primary
research or a need for publication. Many others just hate to see
a good machine wasting cycles (sitting idle or running an inane
Perhaps these arguments
will not convince you. If not, just recall that the eye may not
see what the ear hears, but that does not reduce the value of
sound. There are always melodies beyond our grasp.
Euclid may have been the
first to define primality in his Elements approximately 300 BC.
His goal was to characterize the even perfect numbers (numbers
like 6 and 28 who are equal to the sum of their aliquot
divisors: 6 = 1+2+3, 28=1+2+4+7+14). He realized that the even
perfect numbers (no odd perfect numbers are known) are all
closely related to the primes of the form 2p-1
for some prime p (now called Mersennes). So the quest for
these jewels began near 300 BC.
Large primes (especially of
this form) were then studied (in chronological order) by Cataldi,
Descartes, Fermat, Mersenne, Frenicle, Leibniz, Euler, Landry,
Lucas, Catalan, Sylvester, Cunningham, Pepin, Putnam and Lehmer
(to name a few). How can we resist joining such an illustrious
Much of elementary number
theory was developed while deciding how to handle large numbers,
how to characterize their factors and discover those which are
prime. In short, the tradition of seeking large primes
(especially the Mersennes) has been long and fruitful It is a
tradition well worth continuing.
2. For the
by-products of the quest
Being the first to put a
man on the moon had great political value for the United States
of America, but what was perhaps of the most lasting value to
the society was the by-products of the race. By-products such
as the new technologies and materials that were developed for
the race that are now common everyday items, and the
improvements to education's infrastructure that led many man and
women into productive lives as scientists and engineers.
The same is true for the
quest for record primes. In the tradition section above I
listed some of the giants who were in the search (such as
Euclid, Euler and Fermat). They left in their wake some of the
greatest theorems of elementary number theory (such as Fermat''s
little theorem and quadratic reciprocity).
More recently, the search
has demanded new and faster ways of multiplying large integers.
In 1968 Strassen discovered how to multiply quickly using
Fast Fourier Transforms.
He and Schönhage refined and published the method in 1971. GIMPS
now uses an improved version of their algorithm developed by the
long time Mersenne searcher Richard Crandall .
The Mersenne search is also
used by school teachers to involve their students in
mathematical research, and perhaps to excite them into careers
in science or engineering. And these are just a few of the
by-products of the search.
collect rare and beautiful items
Mersenne primes, which are
usually the largest known primes, are both rare and beautiful.
Since Euclid initiated the search for and study of Mersennes
approximately 300 BC, very few have been found. Just 37 in all
of human history--that is rare!
But they are also
beautiful. Mathematics, like all fields of study, has a
definite notion of beauty. What qualities are perceived as
beautiful in mathematics? We look for proofs that are short,
concise, clear, and if possible that combine previous disparate
concepts or teach you something new. Mersennes have one of the
simplest possible forms for primes, 2n-1. The
proof of their primality has an elegant simplicity. Mersennes
are beautiful and have some surprising applications.
4. For the
Why do athletes try to run
faster than anyone else, jump higher, throw a javelin further?
Is it because they use the skills of javelin throwing in their
jobs? Not likely. More probably it is the desire to compete
(and to win!)
This desire to compete is
not always directed against other humans. Rock climbers may see
a cliff as a challenge. Mountain climbers can not resist certain
Look at the incredible size
of these giant primes! Those who found them are like the
athletes in that they outran their competition. They are like
the mountain climbers in that they have scaled to new heights.
Their greatest contribution to mankind is not merely pragmatic,
it is to the curiosity and spirit of man. If we lose the desire
to do better, will we still be complete?
5. To test
Since the dawn of
electronic computing, programs for finding primes have been used
as a test of the hardware. For example, software routines from
the GIMPS project were used by Intel to test Pentium II and
Pentium Pro chips before they were shipped. So a great many of
the readers of this page have directly benefited from the
search for Mersennes.
Slowinski, who has help
find more Mersennes than any other, works for Cray Research and
they use his program as a hardware test. The infamous Pentium
bug was found in a related effort as
was calculating the twin prime constant.
Why are prime programs used
this way? They are intensely CPU and bus bound. They are
relatively short, give an easily checked answer (when run on a
known prime they should output true after their billions of
calculations). They can easily be run in the background while
other "more important" tasks run, and they are usually easy to
stop and restart.
learn more about their distribution
Though mathematics is not
an experimental science, we often look for examples to test
conjectures (which we hope to then prove). As the number of
examples increase, so does (in a sense) our understanding of the
prime numbers formula software. Open source code of
Seyyed Alireza Hashemi Moosavi's program is in follow
that is based on H.M formula of prime numbers that was
discovered in the year of 2003 Prof. Seyyed Mohammad
Reza Hashemi Moosavi.
You are just permitted to
use the subject with mentioning the reference address of
formula is one of the on-to generating functions for the
prime numbers that for every natural "m" it generates
all the prime numbers (3,5,7,2,11,13,2,17,19, ...)
The presented "H.M" functions by
Discoverer (Prof. Seyyed Mohammad Reza Hashemi Moosavi)
are 6 numbers that 4 functions are by Wilson's theorem
and one of them by Euler's function(
and one of them by
functions are discovered by discoverer (Prof.S.M.R.Hashemi
Moosavi). The software of this function is produced.
In Year of 2007 (AAAS)
National Association Of Academies
Of Science " (USA) Awarded an A++ = Excellent
grade to prime numbers formula and its results by
discovery of prime numbers formula and its results has
been published under an article in journal of "Roshd
of Borhan " ,
associated with Ministry of Education.
More explain about
discovery of prime numbers formula
proof of prime numbers was propound 300 years before the
Christian era by Euclid and since that time great
mathematician like Euler try to discover a formula for
production of prime numbers.
Euler could define
a quadratic function which give prime numbers for forty
prime number which are uninterrupted and also Fermat
presented a formula to obtain prime numbers and later,
it breached by Euler for n = 5.
Many of other
mathematicians achieved to violating and especially
formula and finally they found that discovery of prime
numbers formula is impossible and this problem will be
In fact this
discovery means that one of complicated and unsolvable
mathematics problem was solved and this discovery give
this fact to man that earthy human can solve other
unsolvable problems with research and effort.
I have worked
about this problem around 20 years and I found this fact
that I can't comeback from this path which I came and I
promised that search about this un solvable problem till
end of my life even if I couldn't achieve the final
Of course my
research had result with patronage under god and trust
in god and I discovered the formula of prime numbers.
How is the usage of this formula
in mathematics and other sciences?
A result for
discovery of prime numbers formula is the solution of
Riemann Zeta equation which is on of seven universal
unsolvable problems in mathematics millennium and it's
solution need to gain the number determination equation
of prime number for any desirable number n carefully
(with prime numbers formula).
Another result is
determination of Kth desirable prime number and other
usages are definition of prime number set, proof of
infinity of prime twin pairs, considering of the guesses
of Goldbuch and Hardy, gaining the generator formula of
numbers and also very unknown and big prime numbers and
other problems related to prime numbers.
But the basic and
cardinal usage of this formula is in coding and decoding
that usually use from very big prime numbers for this
and before it is necessary to gain them with complicated
mathematics methods. But with presenting of this
formula, definition of coding and decoding system became
easy and I invent a system for coding with this formula
that I presented this system inventions registration
Euclid’s theory about infinite prime numbers in 300 B.C
Most of the mathematicians and other researchers have
been curious to find a formula which could generates
prime numbers. After many years later, some
mathematicians like Euler and Fermat presented some
formulas to generate prime numbers limitedly.
mathematicians like Hardy and Courant and many other
researchers officially announced that such a formula
can’t be found and in follow to prove their wrong idea
they started to publish some Algebraic theorems in their
Niven and Mills in relation to prime numbers function
proved the above theorem. But their parameters have
never been determined.
determining the number of prime numbers was very
important problem. So Gauss and other mathematicians
started to set some tables for them.
We knew that
there is no exact formula to determine the number of
prime numbers exactly. This problem is known as Zeta
Riemann equation which was one of the seven known
unsolvable problems of the world that after my discovery
on 5th August 2003, one of them is no more
unsolvable with the prime numbers formula accurately you
can absolutely generate all prime numbers to the nth
consequent generate of prime numbers formula resulted in
defining the set of prime numbers and so many other
unbelievable results until now like breaking the code of
RSA and AES by the use of prime numbers formula and
other sets like Mersenne prime, perfect numbers and so
many important sets and results just related to the
field of number theory and basic sciences.
prime numbers formula by Prof. Seyyed Mohammadreza
Hashemi Moosavi caused so many results in basic sciences
that we will mention a little part in follow:
Distinction of prime numbers.
Defining a formula for generating prime numbers.
Definition of prime numbers set by using the
generating function of prime numbers.
Defining a formula to generate the Mersenne prime
Determination of Nth prime number.
Solving Riemann Zeta equation by using the
determination of the number of prime number less than or
equal to arbitrary number N exactly.
The proof of guesses of Goldbuch and Hardy.
The proof of infinity of the prime twin couples.
Determining a general series of answer for
This formula has so many unknown applications in
Cryptography, generating Titan Mersenne prime numbers
and other sciences like solving NP.
Click the link to download or open the PDF file of
New Proof of Fermat's
by Final - Main HM Theorem