Hello friends ! In this chapter we're
going to see some properties and
curiosities about prime numbers. Since
Euclid proved that there are infinitely
many primes, they have a lot to offer.
There are infinitely many prime numbers.
Euclid proved this. There are always more
primes. No matter how high up we go
into the natural numbers, there will
always be more Primes. They never end.
The power of mathematical tools is
amazing. We know with certainty that
there are prime numbers in remote borders
that we will never explore, neither us
nor anyone else in the future. And now,
what do we see? Well, now we are assailed
by many doubts, which I will illustrate.
Let us tackle them.
What happens with the distribution of
prime numbers among natural numbers? This
will be discussed in detail in a later
chapter, because it's one of the
fundamental elements of this collection
of videos. But it is now interesting to
introduce some oddities. Ever since man
knows how to calculate, at first with a
pen a a paper, and already in the 20th
and 21st centuries, with computers, the
healthy sport of hunting prime numbers
has existed. There have always been
people who, through imaginative
procedures, have rivaled to obtain the
largest list of prime numbers, or the
largest prime number. That is, in this
area there are two battlefields: that of
those seeking the largest prime number
of all, regardless of which there may be
before, and that of those who seek the
largest list of consecutive primes.
Since there is no known practical
formula to tell if a number is a prime,
or to generate all prime numbers,
calculations must be made in the manner
of the sieve of eratosthenes. Not exactly,
as we have already said that such a
method is not speed efficient. Other
faster methods are used, although they
are mathematically very complex. There
are formulas that generate some prime
numbers, but here is not useful and magic
formula that generates all primes, and
only primes. Euclid already proved that
the existing number of primes is infinite,
so the task of searching them will never
end. Is it important to know large prime
numbers? Yes, it is, as many encryption
methods and other scientific
applications are based on this. The
largest known prime number. Let us look
at the very summarized history of the
largest known prime number. By 1588
Pietro Cataldi had verified that the
number two raised to the power of 17,
minus one,
that is, 131 071, was a prime number.
He also verified that 2 raised to the
power of 19, minus one, that is, five
hundred twenty four thousand two hundred
eighty-seven, was also a prime number.
These numbers have six digits.
Cataldi, like many others, thought that,
in most cases, number 2 raised to
the power of n, minus 1, when n was a
prime number, was also a prime. And this
is because up to 2 raised to 13, minus 1,
the formula gives mostly prime numbers
as result. Let us see: prime, prime, prime,
prime, non prime, prime. Later, in year 1640,
Pierre de Fermat worked with the case of
n equal to 23, showing that the result
was not a prime number.
Leonhard Euler searched and proved that,
in the case of n equal to 29
no prime number was obtained, either.
However, in 1772, Euler himself verified
that, in the case of n equal to 31, a
prime number was obtained. In 1867
Landry found a larger prime number: 2
raised to the power of 59, minus 1, over
one hundred seventy nine thousand nine
hundred fifty-one, with 13 digits. And, in
1876, Lucas verified that the number two
raised to the power of 127, minus 1, with
39 digits, was also a prime number. Until
1951 no greater prime number was found.
Ferrier was to find the next already
hugh forty four-digit number. Until Ferrier
no computers had been used. That
is, calculations were done by hand or, at
best in the 20th century, with mechanical
calculators. Of course, the sieve of
Eratosthenes method was not used, but
others based on theorems of Fermat and
Euler, and later, of Lucas, Lehmer, and many
others, which speeded up
calculations. Since then, the use of
computers caused an avalanche of
increasingly larger primes. Note that
we're not talking about lists of primes
up to a given maximum, but of the largest
prime number known at all times. To have
good information about the history of
calculation of prime numbers, and other
interesting aspects, I recommend you
study the website of Professor Chris
Caldwell, at the University of Tennessee,
at Martin. It is a very well documented
web. On the date of edition of this video
(May of 2018) the largest known prime
number is: this prime number, calculated
by Pace, Woltmann, Kurowsky, Blosser and
GIMPS, is known since December 26, 2017.
GIMPS is the acronym for Great Internet
Mersenne Prime Search: a collaborative
project of thousands of volunteers, who
use freely available software on their
computers to discover new very large
primes. This mega prime number has twenty
three million two hundred forty nine
thousand four hundred twenty five digits.
On one of the fullest pages of a common
book
there are about 2,500 characters. This
mean that, if we printed this number,
apart from the environmental crime
committed in electricity consumption, ink
and paper, we would use nine thousand
five hundred pages, the equivalent of
almost one meter-thick book. To get an
idea of the magnitude of this number, it
will suffice to know that the Universe
is estimated to have around ten to the
power of eighty atoms of matter. This is
only a one followed by 80 zeros, that is,
eighty-one digits. Less than pocket-money
compared with the super-special known prime
number with over 23 million digits. So
what? What does this mean? Well, it is not
really important, but if we think that this
number is a prime because has no
divisors,
none, then it might make us think a
little. Of all the possible ways there
are for a number to have divisors, and
such a large number even more so,
actually it does not. It has no divisors.
Are there larger prime numbers? Yes, there
are infinitely many larger prime
numbers. Euclid theorem says so. Prime
numbers hunters, even knowing that their
task is endless, engage seriously in
finding the next largest non prime. And
this is not just for the sake of finding
it, but also because of the great
technological device that is needed to
carried it out: very powerful computers, in
some cases, or clusters of thousands of
small computers, in other cases. We leave
things here for the moment. In the next
chapter we will see what the twin primes
numbers are, and the conjecture they gave
rise to. We will also see that there may
be deserts of prime numbers between
natural numbers. And, finally, we will know
another one of the problems about prime
numbers of very simple enunciation but,
nevertheless, without solution until
today: the Goldbach's conjecture.
Thank you for watching !
