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Prime numbers are extremely important because they are the basis for many daily applications. In fact, thanks to the primes, we can today count on cryptosystems, that allow us to have home banking service, email or online shopping (among other things). Prime numbers have been studied for thousands of years and still remain one of the “mysteries” of mathematics …

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Although Euclid contributed to study of prime numbers – he came to the conclusion that they are infinite in number – and Fermat and Mersenne discovered that numbers that can be written under a certain form are primes, there are still a multitude of these numbers that are not known and on which research is still being done.

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Pierre De Fermat was a lawyer by profession and an amateur mathematician. He rarely published his mathematical discoveries. He was one the inventors of analytic geometry and came up with some of the fundamental ideas of calculus.

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He is probably most famous for a problem that went unsolved until 1994; that the equation x n + y n = z n has no non-trivial solution when n>2. One of Fermat’s books contained a handwritten note in the margin declaring that he had a proof for this equation, but it would not fit in the margin. He never published his proof, nor was it found after his death. In 1994 Andrew Wiles worked out a proof of this equation using advanced modern techniques.

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In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a non-negative integer.

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Examples…. When n = 0; therefore F = 2 1 + 1 = 2 + 1 = 3 When n = 1; therefore F 1 = 2 2 + 1 = 4 + 1 = 5 When n = 2; therefore F 2 = 2 4 + 1 = 16 + 1 = 17 When n = 3; therefore F 3 = 2 8 + 1 = 256 + 1 = 257 When n = 4; therefore F 4 = 2 16 + 1 = 65536 + 1 = 65537

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LESS COMMON The less common is a number of the form obtained by setting in a Fermat polynomial, the first few of which are 3, 5, 9, 17, 33,... These are given by the binomial number of the form The first few for, 1, 2,... are 3, 5, 17, 257, 65537, 4294967297,... SPECIAL CASE

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Fermat first conjectured that all the numbers in the form, are primes. However, in 1732, Leonhard Euler refuted this claim by showing that F5 = 2^32+ 1 = 4,294,967,297 = 641 x 6,700,417 is a composite. Primes in this form are called Fermat primes. Up-to-date there are only five known Fermat primes.

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A Fermat number Fn=2^2^n+ 1 (for n ≥ 1) can be thought of as a square whose side length is 2^2^(n-1 ) plus a unit square (see figure 1 ). Figure 1. F2 = 42 + 1 = 17

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Hence, determining whether a (Fermat) number is a composite or not is equivalent to determining whether we can rearrange the unit-square blocks to form a rectangle (see figure2). Moreover, determining whether an integer d divides a (Fermat) number is the same as deciding whether we can reorganize the blocks to form a rectangle with base d. Figure2. F2 = 17 is not a composite because no matter how you rearrange the blocks, you cannot get a rectangle. Figure3. F2 = 17 is not divisible by 3.

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The below table only shows the factoring status of Fermat numbers up to n = 100. Prime Composite with no known factors Composite with complete factorization Composite with incomplete factorization Unknown

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12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100 0 Total number of Fn primes known 5 Largest Fn prime known F4 = 65537 Total number of Fn composites known 243 Largest ten Fn composites known F476624, F495728, F567233, F585042, F617813, F67205, F960897, F2145351, F2167797, F2478792

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The Fermat numbers form a sequence in the form F n = 2 2n + 1, n = 0, 1, 2,... Clearly all the Fermat numbers are odd. Moreover, as we'll see shortly, any two are mutually prime. In other words, each has a prime factor not shared by any other. Hence, the number of primes cannot be finite.

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All Fermat’s number are odd due to the addition of one to an even number. No Fermat number can be expressed as the sum of two primes, with the exception of F 1 = 2 + 3. No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime. With the exception of F 0 and F 1, the last digit of a Fermat number is 7. The sum of the reciprocals of all the Fermat numbers is irrational.

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Generation of pseudo-random numbers. But what are pseudo random numbers???? Well, these are computer-generated random number. The prefix pseudo- is used to distinguish this type of number from a "truly" random number generated by a random physical process such as radioactive decay. These numbers appear to be random, but are not truly so. They represent a statistical randomness.

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How do we use Fermat’s prime to generate Pseudorandom Number Generation Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1 … N, where N is a power of 2.

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Euler Totient Function: (n) (n) = how many numbers there are between 1 and n-1 that are relatively prime to n. (4) = 2 (1, 3 are relatively prime to 4) (5) = 4 (1, 2, 3, 4 are relatively prime to 5) (6) = 2 (1, 5 are relatively prime to 6) (7) = 6 (1, 2, 3, 4, 5, 6 are relatively prime to 7)

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As we can observe from (5) and (7), (n) will be n-1 whenever n is a prime number. This implies that (n) will be easy to calculate when n has exactly two different prime factors: (P * Q) = (P-1)*(Q-1), if P and Q are prime.

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Let’s test the theorem: If a = 5 and p = 6 Then (6) = (2-1) * (3-1) = 2 So, 5 (6) = 25 and 25 = 24+1 = 6*4+1 => 25 = 1(mod 6) OR 25 % 6 = 1 It also follows that a (p)+1 a(mod p) so that p does not necessarily need to be relatively prime to a.

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Euler’s theorem uses modulus arithmetic which helps to lay the foundation for RSA encryption. To construct a personal cipher key we need an appropriate value, we will call it variable R. So, we select two very large prime numbers U and V and multiply them. => (R) = (U-1)*(V-1). This makes R difficult to factor, since the fewer factors a number has, the longer it takes to find them.

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We also define the variables P and Q. P is an arbitrary number that is relatively prime to (R). Q is the calculated inverse of P in (mod (R)). We use P and R to create a public key, and Q and R to create a private key. This yields P*Q 1(mod (R) ). The result is that too much information is lost in the encryption due to the modulus arithmetic to decipher a privately encrypted RSA message without the use of the public key. Unless the would- be decipherer had enough time and processing power to attempt a brute-force factorization. But, the larger the primes, the longer it takes to factor their product.

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Marin Mersenne, Marin Mersennus or le Père Mersenne (8 September 1588 – 1 September 1648) was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics”. Mersenne was "the center of the world of science and mathematics during the first half of the 1600s.””

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A Mersenne (also spelled Marsenne) prime is a specific type of prime number. It must be reducible to the form 2 n - 1, where n is a prime number. The first few known values of n that produce Mersenne primes are where n = 2, 3, 5, 7, 13, 17, 19, 31, 61, and 89.

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In August 2008, Edson Smith, a system administrator, found the largest prime number known till date. The number (which is a Mersenne prime) is 12,978,189 digits long. It would take nearly two-and-a-half months to write out and, if printed, would stretch out for 30 miles.

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In the last January, a new prime number was discovered, 257,885,161-1, bringing a new enthusiasm to mathematicians because of the potential importance of the discovery. The mystery surrounding prime numbers is that we do not know a single pattern to describe them all, a property that makes them so precious and useful!

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The above picture is a matrix of 64 rows and 64 columns of altogether 4096 black or white squares. The squares are numbered (in row major order) from 0 to 4095. The n-th square is black if the number 2n + 1 is prime. For example, the black square in the right upper corner represents the prime 127, and the black square in the right lower corner represents the prime 8191. You can observe, besides many twin primes, the two kinds of triple primes (patterns xx_x resp. x_xx ) and some quadruple primes (pattern xx_xx ). Prime numbers < 8192

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RITIKA KUMARI SHUBHANGI AGARWAL RAHUL CHAUBEY HIMANSHU YADAV GROUP 6

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