Welcome to this section, here you will find:
A cool quote
Euclides' proof of there being infinite prime numbers
How the cicada protects itself from parasites with prime numbers
Explanation of the two types of prime numbers (4n+/-1)
PGP (encrypting)
Most of the info here comes from the book "
Fermat's Enigma", which I have in Catalan. Therefore, the page number may be different.
If you have any doubt, want me to explain something,
or contribute (I'd put it with or without your name and e-mail address, as you want...), don't hesitate to contact me (carlesweb@pina.cat).
Cool quote...
Theoreticians of numbers consider
prime numbers to be the most important ones ,
because they are the atoms of mathematics. Prime
numbers are the bricks of numeric construction,
because all other numbers can be created by multiplying
combinations of prime numbers.
Bibliography: page 121 of " Fermat's Enigma"
Euclides' proof of there being infinite prime numbers...
The proof of there being an infinite number of prime numbers comes
from Euclides, and is a classical mathematical proof.
First, Euclides assumes that there is a finite list
of known prime numbers, and then proves that there must
exist an infinite number of additions to this list. Let the list have N prime numbers,
labelled P1, P2, P3, ... Pn.
We can then generate a new number Qa such that:
Qa=(P1 x P2 x P3 x ... x Pn) + 1
This new number, Qa, will either be a prime or not. If it is a prime, we have another number in our list, so it was not a complete list of all prime numbers. However, if Qa is not primo, then it can be divided evenly by some prime number. And this number cannot be one already in the list, because dividing Qa by any of them will give a remainder of 1. Therefore, there must be another prime, labelled Pn+1.
So either way we have found a new prime number that wasn't in the list. We can add it to the list and repeat the process forever. We will always find the same consequence. Therefore we can extend the list of known prime numbers forever, so there must be an infinite ammount of prime numbers.
Bibliography: page 122 of " Fermat's Enigma"
How the cicada protects itself from parasites with prime numbers.
Periodic cicadas, and specially the Magicicada
septendecim , have the longestlife cycles of all insects.
It starts underground, where the nymphs suck the sap of tree roots.
Then, after waiting for 17 years, adult cicadas go out
in great numbers and inivide the landscape. Some weeks afterwards,
they mate, lay eggs, and die.
The question that struck zoologists
was: Why is the life cycle so long? Why is it a prime number of years?
Another species, the Magicicada tredecim, appears every 13 years,
which indicates that life cycles with a prime number of years give some kind
of advantage for life conservation.
According to a theory, the cicada had a parasite with a life cycle, and it avoided it. If the parasite has a cycle of, lets say, 2 years, then the cicada wants to avoid a life cycle of a number of years that can be divided by 2, or they will coincide regularly. Likewise, if the parasite has life cycles of 3 years, they want to avoid one that can be divided by 3. At last, if they want to avoid them, the best tactic is to have one that is large and prime. Because nothing will divide 17, the Magicicada septendecim will rarely find its parasite. If the parasite's life cycle is 2 years, they will coincide every 34 years, and if it is of 16, only every 272 years!
If the parasite wants to fight back, it has only two cycles that increase the number of coincidences: the anual cycle and the same 17 year cycle of the cicada. However, it is unlikely that the parasite can appear and survive 17 years in a row, because during the first 16 there will be no cicadas to parasite. And if they want to get the 17 year cycle, they will have to evolve by first having a 16 year cycle. What this means is that during their evolution, they won't meet for 272 years, so they'll starve and die. Anyhow, the large prime number protects them.
This could explain why such a parasite has never been found! In its quest to meet the cicada, the parasite has likely elongated its life cycle so that it didn't coincide for so many years it became extint. The result is the cicada having a 17 year cycle which it no longer needs.
Bibliography: page 128 of " Fermat's Enigma"
4n+/-1 primes
Leonhard Euler, one of the greatest mathematicians
of the 18th century, tried to prove one of Fermat's observations, a theorem
dealing with prime numbers.
All prime numbers can be divided in two froups: the one formed by numbers
of the form 4n+1 and the one formed by primes of the form 4n-1, n being
some positive integer (except for 2, which is in none of the groups).
Thereore, 13 belongs to the first group (13=4x3+1) and 19 belongs to the
second (19=4x5-1). Fermat's prime numbers theorem said that the numbers of
the first group always equals the sum of two sqares (13=2²+3²)
and the numbers of the second group can never be expressed this way
(19=?²+?²). This property of prime numbers is beautiful and
simple, bux extremely difficult to prove. For Fermat, it was one of the
many proofs he decided to keep for himself. For Euler the challenge was to
rediscover Fermat's proof. Finally, in 1749, afterworking for 7 years and
almos 100 after the death of Fermat, Euler proved this theorem.
Bibliography: page 93 of " Fermat's Enigma"
PGP (encrypting)
Prime number theory is ne of the few areas of pure mathematics
with direct applications in the real world. One of those applications
is cryptography. Cryptography studies the methods of coding secret
messages so that they can only be decoded by the receiver, and not
by anyone else that may intercept them. The process of coding requires
a secret key. Usually, for decoding the message you just have to apply the
key in the opposite way of the one used for coding. With this process, the
key is the weakest element of the security chain. First, the sender and
the receiver must agree on the key used, and this information is a
risky process. If someone else intercepts the key while it is being
transmitted, he will be able to decode anything coded with that key. Also,
keys must be changed from time to time to ensure security and every time
that happens there is a new risk of the key being intercepted.
The problem with the key is that decoding the message is as easy as coding it because the same key is applied, only backwards.
During the 70s, Whitfield Diffie and Martin Hellman tried to find a
mathematical process taht would be easy to do in one direction but
extremely difficult in the other. Such a process would be a perfect key
for coded messages. For example, I could have a key divided in two parts
and publish the one corresponding to coding. Anyone could send me coded
messages, but I would be the only one to know the decoding part.
In 1977 Ronald Rivest, Adi Shamir and Leonard Adleman, a team of
mathematicians and computer scientists of the Massachusetts Institute of
Technology, realized that prime numbers were the ideal base for a process
of easy coding and difficult decoding.
When I wanted to have my own key, I would get two big prime numbers, up to 80 digits each, and multiply them to get an even bigger non-prime number. To code the message it would only be needed to know the big number; for decoding I would need the two prime factors. Now I can publish the big number and keep the two prime factors to myself. The difficulty in getting the factors from the product would be huge. If I published the number 589 as the coding key, a computer would spend less than a second finding out that the factors are 19 and 31, but we are talking about numbers of more than 100 digits, which would require years of calculations to the most potent computers in the world. Therefore, it would be enough to change the key once a year to be safe.
Bibliography: page 126 of " Fermat's Enigma"