RELATIONS BETWEEN THE GAUSS-EISENSTEIN PRIME NUMBERS AND THEIR CORRELATION WITH SOPHIE GERMAIN PRIMES Pier Francesco Roggero, Michele Nardelli 1,2 , Francesco Di Noto 1 Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy Abstract In this paper we examine the relations between the Gauss prime numbers and the Eisenstein prime numbers and their correlation with Sophie Germain primes. Furthermore, we have described also various mathematical connections with some equations concerning the string theory. Versione 1.0 18/06/2013 Pagina 2 di 35 Index: 1. GAUSSIAN INTEGER ....................................................................................................................... 3 1.1 REMARKS ............................................................................................................................ 6 1.2 ON SOME EQUATIONS REGARDING LEMMAS AND THEOREMS OF GAUSSIAN PRIMES ………………………………………………………………………………………………...7 2. EISENSTEIN INTEGER ...................................................................................................................28 2.1 EISENSTEIN PRIME ......................................................................................................... 29 3. RELATIONS BEETWEEN GAUSSIAN PRIMES AND EISENSTEIN PRIMES .........................30 3.1 CORRELATION WITH SOPHIE GERMAIN PRIMES ....................................................32 3.2 PROOF THAT THERE ARE INFINITELY MANY SOPHIE GERMAIN PRIMES ........34 4. REFERENCES .........................................................................................................................………35 2 Versione 1.0 18/06/2013 Pagina 3 di 35 1. GAUSSIAN INTEGER A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. Definition Formally, Gaussian integers are the set Z[i] = {a + bi a,b ∈ Z}. Norm The norm of a Gaussian integer is the natural number defined as The norm is multiplicative: N(zw ) = N(z)N(w). The units of Z[i] are therefore precisely those elements with norm 1, i.e. the elements : 3 Versione 1.0 18/06/2013 Pagina 4 di 35 Gaussian primes in the complex plane are marked in red : A Gaussian integer is a Gaussian prime if and only if either: • a is prime number and b is zero, so we have a prime number of the form 4k+3 2 2 • both are nonzero and p = a + b = (a+bi) (a-bi) is a prime number (which will not be of the form 4k+3). The prime number 2 is a special case because is the only ramified prime in Z[i]). The integer 2 factors as 2 = (1+ i)(1− i) = i(1− i)2 as a Gaussian integer, the second factorization (in which i is a unit) showing that 2 is divisible by the square of a Gaussian prime; it is the unique prime number with this property. 4 Versione 1.0 18/06/2013 Pagina 5 di 35 Instead, the prime number 5 is a unique factorization 5 = (2+ i)(2−i). Gaussian primes are infinite, because they are infinite primes. 5 Versione 1.0 18/06/2013 Pagina 6 di 35 1.1 REMARKS The prime numbers that are congruent to 3 (mod.4) are also Gaussian primes. Since there are infinitely many rational primes of the form 4k +3, there are infinitely many necessarily Gaussian primes. The prime numbers congruent to 1 (mod.4) are the product of two distinct Gaussian primes. In fact, the primes of the form 4k +1 can always be written as the sum of two squares (Fermat's theorem on the sum of two squares) so we have: p = a 2 +b 2 = (a+bi)( a-bi) Numerical form. Since the form 4k + 3 corresponds to the form 6k '+1 with k' <k, for example, 19 = 4 * 4 +3 = 6 * 3 +1, with k > k ', in fact 4 > 3, the Gaussian primes are all of the form 6k '+ 1, and not of the form 6k' - 1, these being also of the form 4k +1, and then no-Gaussian primes. 6 Versione 1.0 18/06/2013 Pagina 7 di 35 1.2 On some equations regarding Lemmas and Theorems of Gaussian integers, mod-gaussian convergence of a sum over primes and Gaussian primes . Now we see some Lemmas and Theorems concerning the ring of Gaussian integers Z[i], N(a + bi ) = a2 + b2 . We note that the letter p denote the Gaussian prime. Lemma 1 ( ) s = ( ) s = res Fk s x / s Ck x , res Fk∗ s x / s Ck∗x , (1.1) s=1 s=1 where π ∞ τ ( )−τ ( − ) π ∞ ( )− ( − ) k a k a 1 tk a tk a 1 C = ∏1+ , (1.2) C ∗ = ∏1+ , (1.3) k ∑ a () k ∑ a () 4 p a=2 N p 4 p a=2 N p As a consequence of the following representation, ( ) = ( ) k −1( ) (k −k 2 ) 2/ ( ) (−k 3 +6k 2 −5k ) 6/ ( )× (k 3 −4k 2 +3k ) 2/ ( ) (3k 4 −26 k 3 +57 k 2 −34 k ) 24/ ( ) ( ) Fk s Z s Z 2s Z 5s Z 6s Z 7s Z 8s fk s (1.4) we have ( ) ∞ τ ( ) ∞ τ ( )−τ ( − ) F s a −1 a a 1 k = ∏1+ k ()1− p = ∏1+ k k (1.5) () ∑ as () ∑ as () Z s p a=1 N p p a=2 N p ( ) ( ) = and so function Fk s / Z s is regular in the neighbourhood of s 1. At the same time we have 7 Versione 1.0 18/06/2013 Pagina 8 di 35 π ()()()= χ ζ = res Z s L ,1 4 res s , (1.6) s=1 s=1 4 which implies (1.2). Numerical values of Ck and Ck∗ can be, for example: ≈ ≈ C2 ,1156101 , C2∗ ,1 524172 . We note that the value 1,156101 is very near to the value 1,15522604 and that the value 1,524172 is very near to the value 1,52313377 ( ,11561 ≈ ,11552 ; ,1 5241 ≈ ,1 5231 ). Values concerning the new universal musical system based on fractional powers of Phi and Pigreco. Theorem 1 ( ) = + ( 2/1 3+4(k −1) 3/ ) ( ) = + ( 2/1 3+2(k 2 +k −2) 3/ ) M k x Ck x O x log x , (1.7) M k∗ x Ck∗x O x log x , (1.8) where Ck and Ck∗ were defined in (1.2) and (1.3). By Perron formula and by the following expressions, τ (e)( ) << ε (e)(α ) << ε (α ) (e)(α ) << ε (α ) k∗ n n , tk N , tk∗ N , (1.9) for c = 1+ /1 log x , log T ≈ log x we have s 1+ε c+iT () = 1 () x + x M k x ∫ Fk s ds O , (1.10) 2πi c−iT s T that, for the eq. (1.7), can be rewritten also as follows: s 1+ε c+iT ( ) = + ( 2/1 3+4(k −1) 3/ ) = 1 () x + x M k x Ck x O x log x ∫ Fk s ds O . (1.10b) 2πi c−iT s T 8 Versione 1.0 18/06/2013 Pagina 9 di 35 Suppose d = 2/1 − /1 log x . Let us shift the interval of integration to [d − iT ,d + iT ]. To do this consider an integral about a closed rectangle path with vertexes in d − iT , d + iT , c + iT and c − iT . There are two poles in s = 1 and s = 2/1 inside the contour. The residue at s = 1 was calculated in (1.1). The residue at s = 2/1 is equal to Dx 2/1 , D = const and will be absorbed by error term (see below). Identity (1.4) implies ( ) = ( ) k −1( ) ( ) Fk s Z s Z 2s fk s , (1.11) ( ) > ε > where fk s is regular for Rs 3/1 , so for each 0 it is uniformly bounded for Rs > 3/1 + ε . Let us estimate the error term. The error term absorbs values of integrals about three sides of the integration’s rectangle. We take into ( ) = ζ ( ) ( χ ) account Z s s L s, 4 . On the horizontal segments we have + s c iT − x − σ − ∫ Z()()s Z k 1 2s ds << max Z()()σ + iT Z k 1 2σ + 2iT x T 1 << d +iT s σ ∈[]d ,c 2/1 2θ −1 4(k −1) 3/ −1 3/4 << x T log T + xT log T . (1.12) − It is well-known that ζ (s) ≈ (s −1) 1 in the neighbourhood of s = 1. So on [d,d + i] we get s d +i 1 1 ()()k −1 x << 2/1 ζ k −1()+ << 2/1 dt << 2/1 k −1 ∫Z s Z 2s ds x ∫2d 2it dt x ∫ k −1 x log x , (1.13) d s 0 0 it − /1 log x and for the rest of the vertical segment we have s 4/1 2/1 d +iT TT T ()− ()()k −1 x << ζ ()+ 4 dt ()+ χ 4 dt ()+ 2 k 1 dt << ∫ Z s Z 2s ds ∫2/1 it ∫ L 2/1 it , 4 ∫ Z 1 2it d +i s 1t 1 t 1 t ()− + 2/1 + ()− << x 2/1 (log 5 T ⋅ log 8 k 1 3/ 1 T ) << x 2/1 log 3 4 k 1 3/ T . (1.14) The choice T = x 2/1 +ε finishes the proof of (1.7). 9 Versione 1.0 18/06/2013 Pagina 10 di 35 In conclusion, we note that multiplying both the sides of (1.14) for 8, we obtain the equivalent expression: s 4/1 2/1 d +iT TT T ()− ()()k −1 x << ζ ()+ 4 dt ()+ χ 4 dt ()+ 2 k 1 dt << 8∫ Z s Z 2s ds 8 ∫2/1 it ∫ L 2/1 it , 4 ∫ Z 1 2it d +i s 1t 1 t 1 t ()− + /1 2 + ()− << 8x 2/1 (log 5 T ⋅ log 8 k 1 3/ 1 T ) << 8x 2/1 log 3 4 k 1 3/ T , (1.14b) that we can connect to the modes corresponding to the physical vibrations of the superstrings by the following Ramanujan modular equation : ∞ cos πtxw ' −π 2 e x w'dx ∫0 π 142 4 anti log cosh x ⋅ πt 2 2 − w' t w' 4 φ () 1 e w' itw ' 8 = .