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Watch carefully the first ten minutes of the documentary "The music of the primes"

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This is "The.Music.of.the.Primes.WS.PDTV.XviD-CCC" by bluelotus on Vimeo, the home for high quality videos and the people who love them.

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Morena,Alessia,Rosalina

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Federica;francesca e Caterins

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you have to upload your reserch on a word file

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Prime numbers hide your secrets                        Caterina,Francesca,Federica

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Prime numbers hide your secrets                        Caterina,Francesca,Federica

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The most notable practical use of prime numbers is in cryptography. Many popular algorithms used in public-key cryptography, which has numerous and extremely important security applications (your computer is probably using several of these algorithms at this very moment), are based on the fact that integer factorization is a "very hard" problem.

What this means is that the time required to factorize integers into their prime factors grows (roughly) exponentially with the number of bits in the integer. So if the encryption uses very large integers, it would take an unrealistic amount of time to "crack" it.

If (or when) quantum computers become a reality, they would have the potential to make all of these algorithms obsolete, since there are quantum algorithms (in particular Shor's algorithm) that can factor arbitrarily large integers much faster than any known classical algorithm. This has led to the very important field of post-quantum cryptography.

There are other, perhaps less notable, practical uses of prime numbers. Most of them are related to the fact that prime factorization is unique. A well-known example of this is Gödel numbering. This ingenious method allows you to encode any type of information of any length as a single integer, using products of powers of prime numbers. It has many important uses in mathematical proofs, most famously in the proofs of Gödel's incompleteness theore

Prime numbers are all the rage these days. I can tell something’s up when random people start asking me about the randomness of primes—without even knowing that I’m a mathematician! In the past couple of weeks we’ve heard about a beautiful result on the gaps between primes and about cicadas’ prime-numbered life cycles. Our current love affair with primes notwithstanding, many people have wondered whether this is all just abstract theoretical stuff or whether prime numbers have real-world applications.

In fact, they have applications to something as ubiquitous and mundane as making a purchase online. Every time you enter your credit card number on the Internet, prime numbers spring into action. Before your card number is sent over the wires, it must be encrypted for security, and once it’s received by the merchant, it must be decrypted. One of the most common encryption schemes, the RSA algorithm, is based on prime numbers. It uses a “public key,” information that is publicly available, and a “private key,” something that only the decoding party (merchant) has. Roughly speaking, the public key consists of a large number that is the product of two primes, and the private key consists of those two primes themselves. It’s very difficult to factor a given large number into primes. For example, it took researchers two years recently to factor a 232-digit number, even with hundreds of parallel computers. That’s why the RSA algorithm is so effective.

Introduction

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He is best known for Fermat's Last Theorem and Little Theorem.

Introduction to the theorem

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

ap≡a (mod. p)

For example, if a = 2 and p = 7, 27 = 128, and 128 − 2 = 7 × 18 is an integer multiple of 7.

If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols

ap-1≡1 (mod.p)

For example, if a = 2 and p = 7 then 26 = 64 and 64 − 1 = 63 is thus a multiple of 7.

Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem.

Dimostration

We can notice that we just need to prove that ap-1≡1 (mod.p)  for every whole number a who's prime with p. Multiplying both the numbers of the expression for a we obtain the exposed version at the beginning of the theorem's page. If a wouldn't be prime with p then ap≡0≡a (mod. p) and theorem would be true in every case.
Generalization

A little generalization of the theorem,who derives immediately for this,is the following:if p is prime and  m and n are positive whole numbers with m≡n (mod. p-1),the am≡an  (mod. p) for every whole number a. In this form,the theorem justifies the coding system of the public key RSA.

The little Fermat's theorem is generalized by the Euler's theorem: for every modulo n and every whole number a coprime to n,we have:

aᵠ(n)≡1 (mod. n)

Where ᵠ(n) stayes for the phi function of Euler,who counts the number of the wholes between 1 and n coprimes to n. We're talking about a generalization because if n=p  is a prime number,then ᵠ(p)= p-1
The RSA code was invented in 1977 by Ronald Rivest, Adi Shamir and Leonard Adleman.
It is used to encrypt information and messages.

It's an "asymmetric" system composed by "public keys".
You have two distinct keys for encrypt and decrypt.
If the first key is used to encrypt, the second is used to decrypt.
Despite the two keys depend on each other, it's not possible to go back from one to another to ensure the security.

KEY GENERATION:
They generate two keys, one direct and one reverse.
Only one key is made public so we create a list that includes all the direct keys while the revers keys are kept secret by users who use them to get the message.

To have a discrete security, we use keys bu 2048 bit.
512-bit keys and 1024-bit keys are too much used so less secure.
A 1024-bit key can be decrypted in one year.

FUNCTION/OPERATION:
Imagine that A has to send a secret message to B:
1) B chooses two large prime numbers and multiply them
2) B sends this number, that all can see, to A
3) A encrypts the message with this number
4) A sends the message to B, all can see the message but non the content
5) B uses prime numbers to decrypt the message
A and B take little time to encrypt and decrypt the message but the others would take a long time to discover the message.

KEY DISTRIBUTION:
We take two prime numbers: p=3, q=11
We multiply the two numbers: n=pq = 3x11 = 33 and  f(n) = (p-1)(q-1) = 2x10 = 20
We take a number e (not necessarily prime): e=7
We take a number d: d=3        (ed=7x3=21 tre lineette 1)

Public key: (n,e) --> (11,7)
Private key: (n,d) --> (11,3)

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Morena, Rosalina e Alessia

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io  amo lauuura

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CAterina,Federica,Francesca

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Modern Cryptography

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history of prime numbers

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The 7 great millennium problems

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lucia e letizia

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fede cate fra

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I)Prime numbers could be used in many shades of our daily life. Such as, they're used in Finance, during the second world war were used too, for computers' criptography.

II)A prime number is a natural number greater than one that has no positive divisors other than 1 and itself, instead a composite number is a positive integer that can be formed by multiplying two smaller positive integers.

III)The cicalas in North America have got a weird behaviour: in fact for hidden themslves from predators they go on lethargy for 17 years,so when they'll finish their cycle the predators are gone.

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Giulia Di Prinzio e Roberta Zuccarino