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Note :
You are just permitted to use these
information with mentioning the reference
www.primenumbersformula.com
Note :
This 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 and
 functions are discovered by discoverer
(Prof.S.M.R.Hashemi Moosavi). The software of this function is produced.
Note :
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 prof.S.M.R.Hashemi Moosavi.
Note :
The 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.
New
click here to
see the cover and table of
contents of the book" The discovery of prime numbers formula and its
results"
Short introduction
:
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
unsolvable
problems related to them.
Discovery results
:
1.
The distinction of the prime numbers.
2.The
generating formula of the prime numbers.
3.The
definition of the prime numbers set by using the
generating
function of prime numbers.
4.The
generating formula of the Mersenne prime numbers.
5.The
determining of the k-th prime number.
6.Solving
Riemann zeta equation by using the determining of
the number of the prime numbers less than or equal arbitrary
number (N) exactly.
7.Study of the guesses of Goldbuch and Hardy.
8.The
proof of being infinity of the prime twin couples.
Note:
the results of this great discovery “the formula of generating
prime numbers discovery” has been sent to most of research centers
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,
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 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 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 by-products 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 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 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 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.
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 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 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.
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