A new approach of the concept of prime number Jamel Ghannouchi To cite this version: Jamel Ghannouchi. A new approach of the concept of prime number. 4 pages. 24. <hal-3943> HAL Id: hal-3943 https://hal.archives-ouvertes.fr/hal-3943 Submitted on Jun 24 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
A new approach of the concept of prime number Jamel Ghanouchi jamel.ghanouchi@live.com Ecole Superieure de Sciences et Techniques de Tunis Abstract (MSC=D4)In this document, we deal with the concept of prime number and define the real primes together with an application to cryptology. Keywords : Primes ; Cryptology. Defintion The prime numbers are called primes because they are the bricks of the numbers : Each number n can be written as n = p n i when p j are primes and n j are integers. This writing is called the decomposition in prime factors of the numbern. In fact, this definition is a very particular case of another one much more general. Effectively, ifn j are rationals, everything changes. Let the decomposition in prime factors of an integer n wghen n j are rationalsn = p n i. In this writhing, the p j have no reason to be the same than before, they become a convention. For example, if we decide that 6 is conventionally prime, we have2 = 6 4 and each number can be written in function of 6 and a rational exponent of it instead of 2. If we decide conventionally that F n = 2 2n + is prime n, and it is possible by the fact that GCD(F n,f m ) = when n m, then each new prime (new primes=bricks with rational exponents in the writing) replaces anther one in the list of the old primes (old primes=bricks with integral exponents in the writing). Example : If by convention, F 5 = 2 25 + = 4294967297 = 64.6747 is prime, we can decide that it replaces 64 = F 5.6747 which becomes compound and 6747 is prime or 64 is prime and 67447 = F 5.64 is compound. In all cases, the advantage is that we have a formula which gives for each n a prime. And we can see the the primes are infinite.
Let us generalize to the reals! The following numbers can be called bricks or elements, because they constitute the bricks of the numbers. They exist, of course, and we prefer to call them prime numbers because they really generalize the concept of primes. Let us see this : A real number is compound if it is equal to ±p n...p n i i where p j are prime numbers and n j are rationals. We define other real prime numbers which can not be expressed like this :π,e,ln(2). Thus q p = p q is compound, for example. We can remark that such numbers really generalize the concept of prime, as they can be written only asp=p. Also n p + is conventionally prime, with p prime, hence p = (p )( p+) is compound! And 2 i p = (p )( 2 i p+) ( 2i p+)...( p+) And, conventionally, π and e are primes instead of π n and e m with (n )(m ) which are compound. We define the GCD of two numbers : Ifp andp 2 are prime real numbers p p 2 GCD(p,p 2 ) = n n 2 < GCD(p n ) = n n 2 > ;n > GCd(p n ) = p min(n,n 2 ) n n 2 > ;n < GCd(p n ) = p max(n,n 2 ) n i,p n p2 2 n...pj j ) = GCD(p n pn 2 2...pn i i,j (GCD(p n i i,p n j j )) And ifx = p n i andy = p m l l...p m l j l i theny divisesxifgcd(x,y). Thus 3 does not divide the prime 3, for example. 2 Theorem Ifpis prime then a R, k ;a p = a+kp Proof of the theorem We have k,k a p = a = (a m. u m ), a m N (a p m. u m )+kp = ((a m +k p). u m )+kp = a+k p 2
Cryptology Leu us build real primes P and Q. We have p a prime and u n a sequence. We know that p n = + u n pn is prime, for n [,N], with N enough great P = p N. Also with another prime q and another sequence v n, we have another real prime with M enough great, Q = q M. As P + is prime and Q + is prime, let n = P + Q. Let e = α + u P + v Q and let d = kn e. If we have n and e public keys, the message is M by M = C +e+kn and the cipher is C = M e+k n = M +d+k n. α Another possibility is to taken = PQ ande = thennandeare (P ) u (Q ) v the public keys andm = C e +kn thenc = M d +k n withd = (P )u (Q ) v. α Example : p = 79, q = 83 p = 5 79+ = 3.3962299 q = 7 83+ = 2.879983394 P = p + = 2.8428878 Q = q + = 2.6975392 n = PQ = 7.62946543 e = 9378.2 (P ) 7 (Q ) 6 = 5.73775223 = 2.6345566 ((P )(Q )) M = 79.836 M e = 79.836 2.6345566 = 4.6268 (n) Then d = (P )7 (Q ) 6 9378.2 =.379628286 3445.7.62946543 + 4.6268 = 258.2768 AndC = 258.2768.379628286 = 79.8637364 C = 79.864 The probabilities What is the probability that a real x between x and x + dx is prime? If p(x [x,x+dx]) is that probability Effectively, as p(x [x,x+dx]) = d(log(x)) x = dx x 2 log(+ dx x ) = log(x+dx ) = log(x+dx) log(x) = dx x x = d(log(x)) Thus p(x [x,x+dx]) = p(x [,x+dx]) p(x [,x]) = log(x+dx) log(x) x+dx x = log(x+dx) log(x) = log(x+dx) log(x) = d(log(x)) x x x x How many primes are there betweenxandx+dx? there are dx π(x) = d(log(x)) = 3
Conclusion We did not present it like this, but we have given a new method of cryptology which can be used with efficiency in several fields like internet security. Références [] Alan Baker, Transcendental number theory Cambridge University Press, (975). 4