Hello friends ! Welcome to a new chapter:
chapter 5 !  I'm going to introduce you to
two 17th century mathematicians, who made
us advance in the knowledge of prime
numbers:
the Frenchmen Pierre de Fermat and Marin
Mersenne, contemporaries and friends. Both
mathematicians gave rise to the so
called Fermat's and Mersenne's numbers, two
sets of numbers that promised to have
very interesting properties. For its
usefulness in the creation of algorithms
to find prime numbers, Mersenne numbers
became very famous, and important, and are
used a lot today. But this doesn't mean
that Fermat's work was inferior. Pierre de
Fermat was an excellent mathematician,
author of the very famous Fermat's Last
Theorem, which was proved in 1995 by the
British mathematician Andrew Wiles. We
will meet him again in future chapters.
Pierre de Fermat, an excellent
mathematician and lawyer, in a wish that
was not satisfied in previous centuries
about how to generate prime numbers,
devised what he believed satisfied
this desire: a formula that will give
prime numbers. Not all of them, of course,
although only primes. These were the
so-called Fermat's prime numbers. The non
proven conjecture was that the following
sequence of numbers: 2 raised to 2
raised to n, plus 1, only generated prime
numbers. That is, making n equal to 1, 2, 3,
4, etc., the result of the operation of
raising 2 to the power of the result of
2 raised to n, plus 1, would give a prime
number. Let us see what happens with the
sequence of Fermat numbers: for n equal
to 1 the result is 2 raised to 2 raised
to 1, plus 1, that is, 5, which is a prime;
for n equal to 2 the result is 2 raised
to 2 raised to 2, plus 1, which is
equivalent to 2 raised to 4, plus 1, that
is, 17, which is a prime; for n equal to 3
the result is 2 raised to 8, plus 1, that
is 257, which is also a prime. This looks
promising.
Here is where Fermat began to get inspired.
For n equal to 4 the result is 2
raised to 16, plus 1, which is equal to
65 537, and
it is also a prime. Fermat, filled with
joy, went into a tailspin. What if n is equal
to 5? This is what would happen: 2 raised
to 2 raised to 5, plus 1, is equal to 2
raised to 32, plus 1. The fifth Fermat
number has 10 digits, and its value is
4 294 957 297.
 
 
Is this number a prime?
Here Fermat has no
answer, because the mathematical tools
available at the time prevented him from
knowing so. But, possessed by an
astonishing arrogance, and knowing that
no one would contradict him, at least
before his death, he conjectured that it
was also a prime. And not only the fifth
number, but, for that matter, all the rest
should also be primes. The sixth Fermat
number already has twenty digits, and the
eighth Fermat number is already greater
than the number of atoms in the Universe.
Fermat numbers grow extremely fast as
exponent n grows. Fermat conjectured
that all numbers generated by his
fantastic formula had to be prime, and so
they became known as Fermat's prime
numbers, and no one would dare
question it. But, dear friends, one
century later came the great Swiss
mathematicians Leonhard Euler, and
nothing escaped him. He realized that
the fifth Fermat number was divisible by
641, so the following could be written:
so, it wasn't a prime. How did Euler do it?
Well, out of curiosity, with skill and
patience. Thus, he proved, with a
counterexample, that the conjecture was
false. But, by then,
Fermat's prime numbers had already been
engraved with Golden letters in the
history of mathematics. In fact, only the
first four Fermat prime numbers are known.
There may be more, sure there are more. But
they are not known. Despite the
mathematical talent of Fermat, who went
down in history mainly because of the
enunciation of his "last theorem", here he
failed.
Moved by the same interests, the french
monk, and mathematician, Marine Mersenne,
contemporary and friend of Fermat, conjectured
another idea also based on playing with
the powers of 2. He established that
the sequence of numbers 2 raised to n,
minus 1, where n was a prime number
should generate prime numbers. This
sequence, for the first value of n
Prime, 2, 3, 5, 7, 11, ...  is as follows: the
first four are prime numbers, but 2047 is
not a prime, as it is a product of 23 and
89, so that not all values of n causes the
formula to provide a prime number. This
was, of course, annoying to Mersenne, as
it broke the beauty of conjecture. Aware
of this, Mersenne stubbornly dared to say
that his formula would always give a
prime number for the following values of
n: 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. And,
here is the astonishing part, because
Mersenne did not prove it! He could not do it.
Besides, how could he boldly say
that 2 raised to 257,
minus 1, was a prime number? This number
has 78 digits, and Mersenne did not have a
clue if it was a prime or not. Later,
already in the 20th century, it could be
found that Mersenne was wrong when n was
equal to 67 and 257. In addition, he had
left along the path to his formula the
primes values 61, 89 and 107, which did
give prime numbers as result, and that
he did not consider. That is, Mersenne was
another daring person when it came to
making conjectures. What happens is that,
unlike with Fermat's Primes,
those of Mersenne have been very useful. We
do not know how many Mersenne primes
numbers exist, but they are abundant, and
it turns out, and I will not go into
detail, that there is a relatively fast
method to know whether a Mersenne
number is prime or not. Relatively fast
means working with computers today, but
still fast compared to other potential
prime numbers. Thus, it is not surprising
for prime number hunters, who want to
undertake a new challenge, to work with
potential Mersenne prime numbers. We saw
in last chapter that number: is a prime
number, the largest known prime number at
the time I make this video. This is a
Mersenne prime number, were n is equal to
Dear viewer, the last chapters have been a
presentation of the prime numbers. We
have started by knowing their features,
and their seemingly chaotic behavior. We have
why, from the beginning of mathematics,
they have aroused so much curiosity. A
race of research began centuries ago,
reaching our times, and for now there are
no signs of it coming to an end. There
are many unanswered questions, many paths
to knowledge have been open during
hundreds of years. We will follow several
of them through all the chapters, and
they will lead us to the most beautiful
landscapes. The best way to start with
the walk is to learn about the life and
work of one of the best mathematicians
of all times, who made significant
progress in the understanding of prime
numbers: the great
Leonhard Euler. But this we leave for
the next chapter.
Thank you for watching !
