

The Purpose Of Doubt In Knowledge Philosophy Essay
Info: 1284
words (5 pages) Essay
Published: 1st Jan 2015 in
Philosophy
The more I know about knowledge, then the more doubt will
develop into my mind. That is what I felt about doubt through
experiences in my life. We usually have doubt when we find it
hard to belief our accept something as the truth. Doubts exist
when our mind is in process of accessing something whether to
belief or not to believe. When we are 100% confident about
something that we think is true, then we will never
misunderstood the idea that had been brought in the first place.
Maybe without doubt, we would never come across the word
“evolution” because evolution is about some facts that are due
to the feeling of doubt. Key is something that can bring us
nearer to the truth and as generator to the knowledge that we
seek for. Doubt will lead us to find two form of knowledge.
First is the doubt in knowledge which can help us to get clearer
knowledge when we try to prove the claim made is true or not,
second is the doubt in knowledge will lead us to discover and
explore a new knowledge. Persian proverb says that “doubt is the
key to knowledge” because from doubt, we have the feeling to
know about something and perhaps will lead us to get clearer
knowledge or new knowledge. We need to fulfill certain criteria
before we can acknowledge something as a knowledge, such as it
must be justified by empirical evidence, logic, memory or public
authority. Without this criteria, a claimed knowledge made by
scientist or historian can be myth. History and Mathematics are
two different area of knowledge that can help us to find at what
extent doubt is used as an element to discover new knowledge.
History is a subjective parts of area of
knowledge and its talks about people and events of the past. It
is not just the compilation of the past that is being recorded
solely without any research and discussion, but after the facts
are being interpreted and analysed into something significant.
Lets take an example of a history about Piltdown man. For
instance, in 1912, a paleoantropologist, Charles dawson claimed
he have found human skull in England belong human ancestor aged
about 500 thousand years ago or known as Dawson’s downman.
According to him, it is a pioneer evidence of human evolution.
Most experts agreed that these specimens age about 500 thousand
years old and being displayed in museums for more than 40 years.
But out of sudden a few scientist doubt about the specimens was
from human ancestor because the skull fragment and the jawbone
obviously from two difference species. A few attempts were made
by scientists to prove this claim was wrong but they did not
succeed. Later, because of the advance in technology, a
paleonthologist, Kenneth Oakley has tried the technology of
flourine testing on Piltdown man in order to determine the
actual age of the fossils. He applied the technology on the
fossil and he found that the fossil is just a few thousand years
not 500 thousand years as claimed by Dawson. Further
investigation was made and it is found that the fossil is
combination of human skull and ape. A conclusion is made by
paleontologists that Piltdown man is a join together of a piece
of jigsaw puzzle with the right colour but wrong shape and was
immediately removed from the museum. Piltdown man is an example
that shows us that doubt can lead to get a clearer knowledge and
can bring us and nearer to the truth.
In order to agree the knowledge in history
can be gained because of doubt, we have to realize that it has
it own limitation and weakness. The weakness is doubts solely
cannot solve any puzzles that we are wonder about. It must be
accompanied by scientific method. The scientific method itself
still can be doubted because human who created method can still
make mistake. Maybe in the next few years someone will discover
that fluorine technique used to investigate the date of the
fossil is not accurate and valid enough to test for age of the
specimens. We have to be remembered that science always evolve
as time passes by and it is not impossible that more advance
techniques will be discovered until the actual date and time the
fossil belong to can be revealed. Henry Fairfield Osborn said
that “We have to be reminded over and over again that Nature is
full of paradoxes”. His statement is made to remind people that
those things that we see and expect in this world does not
necessary true as a lot more thing we don’t know about nature
and the most important thing is we did not exist at the time of
the history real event mentioned. Sometimes, historian only made
a speculation or lucky guess on the specimens they found as long
as it met their theoretical expectations. So it can be
understood that even an evolutionist will get the closest
interpretation on the fossil findings when they doubt and
willing to discuss among themselves.
In other area of knowledge such as
Mathematics doubt is still exist but in different view and
perspective. This is because Mathematics differs from history in
a way that it is more to objective, less ambiguous, less open to
interpretation and more clearly defined. Math is more to numbers
and can be differentiate very clearly which one is wrong and
which one is right or in other word the answer is not
subjective. Mathematics knowledge arises from problems that we
face everyday and the solution in Mathematics is free from both
observation and induction. In mathematics, people not always
doubt whether the knowledge claim by mathematician is true or
not because mathematics have the ability of selfproving and its
objectivity, but people do doubt whether the claim made by
mathematician about certain theorem can be modified in order to
form a more perfect theorem. The simplest example is prime
number formula. The definition of prime number is the number
which can divide by 1 and itself. As a student, I myself also
have doubt why number 1 did not count as prime number while
number 1 can be divided by 1 and itself. But for mathematicians,
they doubt whether prime number have a formula so that it can
cover all prime numbers until infinity. After fail to solve this
problem, Mathematicians claim that this problem is unsolvable.
This unsolvable problem remains for 2300 years. Although this
problem remains unsolved for so long, that doest mean
mathematician had give up. The doubt grows deeper until in year
2003, Professor.S.M.R Hashemi Moosavi found the formula for
prime number. This shows that from doubt that prime number
formula is unsolved, we now can enjoy the fruitful of the doubt
although it takes decades or centuries.
However, no ones would know that claim
made by Professor S.M.R Hashemi Moosavi is the most perfect one
to suit the formula for prime number because at this time no one
can prove his theorem is wrong. Unlike the previous
Mathematician, they also once had claimed that they have found
this formula, but it only last for a certain period because
someone had proven it only applicable up to certain numbers, not
included until the infinity. We also can doubt what the last
number of prime number is since the ending of infinity itself is
unknown.
Having discussed doubt is the key to knowledge from the
perspective of two areas of knowledge namely history and
mathematics makes me realized that doubt do exist in the process
of searching for knowledge. Although the process obtaining
knowledge though the feeling of doubt is differ in History and
Mathematics due to its subjectivity, objectivity and its nature,
the final ending will bring us to only one destination; gaining
knowledge. Therefore, doubt is the key to knowledge is agreed in
these two areas of knowledge.
Reference : UKEssays.
(November 2018). The Purpose Of Doubt In Knowledge Philosophy
Essay. Retrieved from
https://www.ukessays.com/essays/philosophy/thepurposeofdoubtinknowledgephilosophyessay.php?vref=1
Breaking the code  The
Moosavi Formula
I have found the formula to
determine the prime at the
kth level. This formula is
named as Moosavi formula
formulated by Prof. SMR
Hashemi Moosavi. I would say
that this is the final
formula that unveils the
mystery of primes. You will
no longer need a computer to
determine the prime at the
kth level if you can do it
by hands. Here's the list:
Reference:
www.primenumbersformula.com
Issue: Breaking the Code
Part 2 Which is which? The
Sieve Algorithms or The
Primality Test Equations
Yes, I agree with Mr. Jay
that the modern sieve of
Atkin is more complicated,
but faster when properly
optimized.[1] Atkin's
algorithm is actually more
efficient than the sieve of
eratosthenes when run in
computers. This is somehow
due to the fact the
algorithm takes less memory
of the computer than its
predecessor the sieve of
eratosthenes. This can be
proven by the statement from
Wikipedia that the Atkin
sieve computes primes up to
N using O(N/log log N)
operations with only N1/2 +
o(1) bits of memory. That is
a little better than
the sieve of Eratosthenes
which uses O(N) operations
and O(N1/2(log log N)/log N)
bits of memory.[2] However,
this does not amplify so
much for there is only minor
advantage when both are
compared. But, what I am
quite unsure of, is the
findings of my pal that the
sieve of Eratosthenes is
useful only for relatively
small primes.[3]
Before we further our
discussion, friends, may I
know your opinions on the
following questions:
1. "Which is
better?"  Which do
you think is better in
finding the large or very
large primes: The algorithm
method? or Is it the use of
primality test equations?
2. "How large is
your large?"  How
large is your large prime
that you think this is the
largest prime?
3. "How fast are
you?"  Which do
you is more faster, the
manual method or the
computer generated method of
locating prime?
If you have comments about
my reaction on this topic.
Please feel free to write me
or post a comment in this
site. Thanks.
References:
Jay08_20, [usepmat_group]primes,
1/19/2010 [1], [3]
http://en.wikipedia.org/wiki/Sieve_of_Atkin,
(Date accessed: 01/19/2010)
[2]
http://stackoverflow.com/questions/622/mostefficientcodeforthefirst10000primenumbers
Issue: How
can we determine large and
very large primes?
This
question on how to determine
large and very large primes
are most likely asked by
most mathematicians,
mathematics enthusiasts and
students. Even long long
time ago, the search for
large primes was a focus of
study and research.
Fortunately, the people in
the "gone but not forgotten"
era came up with theorems,
algorithms and formulas in
getting these primes to
satisfy the need. Among the
contributors were Euclid who
developed the Euclidian
algorithm, Father Mersenne
with his famous Mersenne
primes, Eratosthenes with
his Sieves of Eratosthenes,
Fermat with his
Factorization Method and
Little Theorem and others
like, Gauss, Reimann and
Euler who belong to the
antique generation. Further
studies were also been
conducted by enthusiasts in
the middle ages and they
came up to determine large
primes without the help of a
computer and mostly, done
with their bare hands. To
name few of them, they are
Anton Felkel, J.P. Kulik,
EDF Meissel, BK Parady, J.R.
Smith and S. Zarantonello,
they are the ones who used
to publish tables of primes
in large numbers as we say
it. Most of their work were
based on the theorems and
correlations from the
prototypes of antique
generation.
So, what
is the formula to determine
exact large or very large
primes? The
answer is that based on the
study, the accurate, exact
and realistic method of
determining large primes is
by the use of Sieves
of Erastosthenes. From
the name itself you can
think that numbers in a
range are being sieved or
"filtered" and that you can
come up with real and exact
primes. Here is an example
of primes determined by the
method in a range less than
100 or 10^2:
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....
Of
course, the only problem
with this method is that
this is very tedious when
done by hands especially
when expanding in the range
to 10^3, 10^4, or 10^n,
where: n may be any number
greater than 2. However,
with the advent of
computers, large primes may
be generated with the Sieve
of Eratosthenes faster than
any publishable table may be
read (Anderson and Bell,
1997, p.116). This is done
by writing a program usually
in high level computer
language like Turbo Pascal
or Turbo C using syntax such
as arrays and loops that
would command the computer
to generate large or very
large primes to a certain
range. (Note: I am sorry
that I could not discuss the
programming technique here
but I could give the key
points in making a flowchart
for this program.)
The second question one
might ask is: What
is the use of other theorems
or formulas made by
Mersenne, Fermat or Gauss in
determining large primes? The
answer to that is  the
theorems or formulas made by
these mathematicians were
simple assumptions or
estimations to get large
primes and may or may not be
representing exact single
large primes. Let us take
the definition of Mersenne
which states that a Mersenne
number is an integer of the
form M(n) = 2^n  1. If a
Mersenne number is prime, it
is called a Mersenne prime
or can be stated this way
"If M(n) is prime, then n
is a prime." Let us
therefore prove if this is
absolute.
Proof:
n
2^n  1 = M (n)
Remarks
1
2^1  1 = 1
Empty
2 2^2 
1 = 3
2 is prime;
M (n) = 3 prime
3
2^3  1
= 7
3 is prime; M (n)
= 7 prime
4
2^4  1 = 15 = 3 x 5
4 is not
prime; M (n) = 15 composite
5
2^5  1
= 31
5 is prime; M (n)
= 31 prime
6
2^6  1 = 63 = 3^2 x 7
6 is not
prime; M (n) = 63 composite
7
2^7  1 = 127
7
is prime; so M (n) = 127
prime
8
2^8  1 = 255 = 3 x 5 x
17 8 is not prime;
M (n) = 255 composite
9
2^9  1 = 511 = 7 x 73
9 is not
prime; M (n) = 511 composite
10
2^10  1 = 1023 = 3 x 11
x 31 10 is not prime; M (n)
=1023 composite
11 2^11 
1 = 2047 = 23 x 89
11 is prime; M (n) = 2047
composite ?
As you can see in the table
above, when n = 11; M (n) =
2047 = 23 x 89 which does
not produce a single prime
but rather composite.
Therefore, we can deduce
that Mersenne formula is
good only for assumption for
predicting primes but does
not determine exact single
large prime. The same thing
also happens when we use
Fermat number which is
defined as a Fermat number
is an integer of the form
F (n) = 2^2n + 1. If a Fermat
number is prime, it is
called a prime.
Unfortunately, the Fermat
number, F(5) = 4294967297 is
divisible by 641 and
therefore, not a prime
number and so Fermat number
is also not absolute.
The third question one might
ask is: Can we
assume for a large prime
number using the definitions
or theorems made by antique
mathematicians? My
answer is yes. We
could use the ancient
formulas to predict or test
a certain large number if
its a prime or composite.
The method is simple you
just assume a large or very
large number and test it for
primality. This is done
using the theorem that if
the positive integer n is a
composite integer, then n
has a prime factor p such
that p^2 < n.This is
also similar with the
suggestion made by Dr.
Wilkinson in his
conversation with Mr. Ulric
Cajuste
([usepmat_group]primes,1/16/10).
Let us take a proof on this:
ex. determine whether n =
521 is prime?
Proof:
We only
need to consider primes p
that are less than or equal
to 2 for 22^2 = 484 which is
less than 521 according to
the theorem p^2 < n.
If we are to use 23^2 = 529
this would be greater than
521 and therefore, does not
conform to the theorem.
Hence, we have to use 22.
The primes less than 22 or
equal to 22 are 2, 3, 5, 7,
11, 13, 17, and 19. Trying
these primes to be divided
to 521, we find that none of
the primes divides 521.
Therefore, we can say the
521 is a prime.
The only disadvantage with
this method is that we just
have to keep on guessing
numbers until we find
numbers to be prime.
Is there another
formula aside from Sieves of
Eratosthenes that could
generate large primes? Yes,
I have found one but this is
subject yet to scrutiny. The
equation looks like this:
P (n) =
(2n + 1  p) d N (n) + p = p
(2n + 1 / p)^ d N (n)
where: N
if N = 2n + 1 is prime
p if N = 2n + 1 is
composite (p: arbitrary
prime number)
This equation was formulated
by Prof. S.M.R. Hashemi
Moosavi who claimed that he
solved the problem on prime
number generation. To prove
this formula would be for
our next discussion.
What is the rationale
behind why computer
companies offer prices to
those who can get larger
primes? I
believe that the rationale
behind offering prices by
computer companies is that
1.) as a sort of
advertisement 2.) to test
their product, say computer.
I am convinced that the
greater weight falls to the
second reason since computer
companies tend to test their
hardware such as processors
or chipsets if it can
perform long and complex
mathematical operations,
including also is the speed
and time to process
instructions.
If you have comments about
my reaction on this topic.
Please feel free to write me
or post a comment. Thanks.
References:
James A.
Anderson and James M Bell,
Number Theory with
Applications, 1997, Prentice
Hall Publishing
Prof.
S.M.R. Hashemi Moosavi, The
Discovery of onto
generating function of the
prime numbers and its
results,
www.primenumbersformula.com,
Date accessed: January 17,
2010
Ultima_syd, [usepmat_group]primes,
1/16/2010

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,
etc.

you can
read about
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 2^{P}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 worldrecordsize 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 recordsetting,
rare, and historic new Mersenne prime. Of course, there are many
other reasons.
for
more information about Mersenne primes , History ,Theorems and
lists
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.
1.
Tradition!
2.
For the byproducts of the
quest
3.
People collect rare and beautiful items
4.
For the glory!
5.
To test the hardware
6.
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
screen saver).
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.
1.
Tradition!
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 2^{p}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
group?
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
byproducts 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 byproducts of the race. Byproducts 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
byproducts of the search.
3. People
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 historythat 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, 2^{n}1. The
proof of their primality has an elegant simplicity. Mersennes
are beautiful and have some surprising applications.
4. For the
glory!
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
mountains.
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
the hardware
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
Thomas Nicely
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.
6. To
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
distribution.
by
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
and
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 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
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.
3 New Proof of Fermat's last Theorem by
HM Final Main Theorem
by
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)
Attention:
§
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);
;
;
;
Attention:
§
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: 9786009446797).
[2] S. M. R. Hashemi Moosavi, Generalization of Fermat's Last
Theorem and Solution of Beal's Equation by HM Theorems (2016).
(ISBN: 9786009446735).
[3] S. M. R. Hashemi Moosavi, 31 Methods for Solving Cubic
Equations and Applications (2016). (ISBN: 9786009446773).
Magic squares
By Seyyed Alireza Hashemi Moosavi
date: 2021/13/07
Click the link to download or open the PDF file of
New Proof of Fermat's
last Theorem
by Final  Main HM Theorem 