A PROBLEM ON THE CONJECTURE CONCERNING THE DISTRIBUTION OF GENERALIZED FERMAT PRIME NUMBERS (A NEW METHOD FOR THE SEARCH FOR LARGE PRIMES)

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1 A PROBLEM ON THE CONJECTURE CONCERNING THE DISTRIBUTION OF GENERALIZED FERMAT PRIME NUMBERS A NEW METHOD FOR THE SEARCH FOR LARGE PRIMES) YVES GALLOT Abstract Is it possible to improve the convergence properties of the series for the computation of the C n involved in the distribution of the generalized Fermat prime numbers? If the answer to this question is yes, then the search for a large prime number P will be C logp ) times faster than today, where C Introduction In [2], based on Bateman and Horn conjecture [1][11] and on the distribution of the factors of the generalized Fermat numbers [3], Harvey Dubner and the author proposed the following conjecture: Conjecture 11 If E n B) is the number of primes of the form F b,n = b 2n + 1 for 2 b B, then E n B) C B n dt 2 n 2 log t where the constant C n is the infinite product 1 a np) p ) C n = 1 1 p ) = 1 a ) np) 1 p 1 and where p odd prime a n p) = p odd prime { 2 n if p 1 mod 2 n+1 ), 0 otherwise The actual distribution of generalized Fermat primes is in significant agreement with the values predicted by the conjecture for some polynomials of degree as large as 2 16 Today, we now that C 0 = 1 and in [9][10] Shans computed precisely C 1 and C 2 We indicate in this paper a method for the computation of the first C n ; however the method becomes unpractical for n > 20 Today, no relation is nown for a fast computation of other C n and we have to search for the smallest primes of the form 2 n to estimate them What will be the consequences, if a formula, involving only some functions and series and not the primes of the form 2 n+1 + 1, exists to compute precisely C n? Date: November 18, 2001; revised March 17, Mathematics Subject Classification Primary 11Y11; Secondary 11A41 Key words and phrases prime numbers, prime distribution, generalized Fermat numbers 1

2 2 YVES GALLOT The search for a large prime of the form 2 n + 1 will be about n/200 times faster than it is today For n 10 7, the search will be times faster! The chance for a number of the form N = b n ± 1 to be prime depends on its size but is virtually independent of, b, n as long as N passed a trial division test up to the bound logn) Today, to find a prime, we have to chec the primality of all the numbers that passed a trial division test In practice, we have to test about 002 logp ) numbers to find a prime P If C n can be computed quicly and precisely, the minima of the sequence C n will indicate to us where the primes of the form 2 n +1, with small, are For example, the estimates for {C 27, C 28, C 29, C 30 } are {195, 192, 178, 220}: it indicates that it is probably faster to search for a prime of the form 2 n + 1 for n = rather than for the other values And we find that the smallest primes of each form are , , and If C n can be used as an indicator, then we will just have to test two or three numbers to find a prime of the form 3 2 n + 1 or 5 2 n + 1 rather than, today, about 002 log2)n numbers: the search for a large prime number P will be about 001 logp ) times faster if the indicator C n can be evaluated quicly and if we test the form 2 n + 1 The following sections indicate a formula for C n but that is quicly unpractical for large n Can you improve this formula and speed up the prime number search? and Let the infinite products C n s) = P n s) = 2 Definitions p odd prime p 12 n+1 ) 1 a n p)p s 1 p s 1 p s 1 + p s ) 2 n 1 Let χ be a Dirichlet character The L-series attached to χ is defined by χn) Ls, χ) = n s = 1 χp)p s ) 1, Res) > 1 p n=1 Let Xm) be the character group Z/2 m Z) For m = 1, only the trivial character χ 0 is defined and Ls, χ 0 ) is the Dirichlet lambda function λs) = n=0 2n + 1) s = 1 2 s )ζs) For m = 2, there are two characters: the trivial one and the character χ 4 defined by χ 4 n) = 4 n ), where a n ) is the Kronecer symbol with the definition a 2 ) = 0 for a even Ls, χ 4 ) = n=0 1)n 2n + 1) s is the Dirichlet beta function βs) For m 3, the group Z/2 m Z) is generated by 1 and 5 Thus a character is uniquely determined by its value in 1 and 5 The order of 1 is 2 and the order of 5 is ϕ2 m )/2 For every a {0, 1} and b {1,, ϕ2 m )/2}, we can associate the character χ a,b uniquely determined by χ a,b 1) = 1) a and χ a,b 5) = exp2πib/2 m 2 ) Note that for m = 3, the two primitive characters are real There L-series are Ls, χ 0,1 ) = n=1 2 n )n s and Ls, χ 1,1 ) = n=1 2 n )n s

3 A PROBLEM ON THE DISTRIBUTION OF GF PRIMES 3 3 Shans formula In [9], Daniel Shans developed a method to compute with accuracy and efficiently C 1 and in [10], he used a similar method to compute C 2 Lemma 31 If a is a positive even integer and if x < 1 a, then ) 1 x n ban) 1 ax = 1 + x n where n=1 31) b a n) = 1 2n µd)a n/d d n d odd Proof We expand both sides in Maclaurin series and identify the corresponding coefficients This yields the condition 2 n ) n b a d d = an d n d odd Now applying the Möbius inversion formula we obtain 31) For s > 1, C n s) = λs) p 12 n+1 ) = λs)p n s) ) 1 2n p 12 n+1 ) p s ) 1 2n 1 + p s p s 1 p s ) 2 n 1 By lemma 31 and since b 2 n1) = 2 n 1, we obtain the relation ) 1 p s b2 n ) C n s) = λs)p n s) 1 + p s It can be rewritten as Formula 32 p 12 n+1 ) C n s) = λs)p n s) P n s) b 2 n ) 2 n 1 It is easy to verify that We have from 32) 4 Computation of the constant C 1 P 1 s) = λ2s) C 1 s) = λs) λs)βs) λ2s) λs)βs) ) b2 ) λ2s) λs)βs)

4 4 YVES GALLOT The product converges for s = 1, λ2) = π2 8 and β1) = π 4 then 41) C 1 = π ) b2) λ2) 2 λ)β) It is easy to verify that 5 Computation of the constant C 2 P 2 s) = λ2s) 2 λs)βs)ls, χ 0,1 )Ls, χ 1,1 ) We have from 32) λ2s) 2 C 2 s) = λs) λs)βs)ls, χ 0,1 )Ls, χ 1,1 ) λ2s) 2 λs)βs)ls, χ 0,1 )Ls, χ 1,1 ) The product converges for s = 1, L1, χ 1,1 ) = π 2 and L1, χ 2 0,1) = log1+ 2 π 2 51) C 2 = 4 log1 + λ2) 2 ) b 4 ) 2 2) λ)β)l, χ 0,1 )L, χ 1,1 ) 6 Computation of C n Pieter Moree indicated to the author [7] a generalization of Shans formula: Theorem 61 P n s) = M n2s) 2 L n s) where L n s) = Ls, χ) and χ Xn+1) M 1 s) = λs) 2) ) b 4 ) 2 then and for n 2 M n s) = χ Xn) χ 1)=1 Ls, χ) Proof An elementary proof that requires no algebraic number theory) is indicated here Let Gs, χ) = p =1 χp ) p s Gs, χ) provides an unambiguous definition for log Ls, χ) [5, p 256] For n 2, log M n2s) 2 = 2 G2s, χ) Gs, χ) L n s) = p χ Xn) χ 1)=1 2 =1 χ Xn) χ 1)=1 χ Xn+1) χp ) p 2s χ Xn+1) ) χp ) p s

5 A PROBLEM ON THE DISTRIBUTION OF GF PRIMES 5 χ Xn+1) χp ) = ϕ2 n+1 )δ 2 n+11, p ), where δ m a, b) = 1 if a b mod m) and δ m a, b) = 0 otherwise 2 χ Xn) χp ) = χ Xn) χ p ) + χp )) = χ 1)=1 ϕ2 n )δ 2 n 1, p ) + δ 2 n1, p )) = ϕ2 n )δ 2 n+11, p 2 ) Thus, log M n2s) 2 = ) 2 n δ L n s) 2 n+11, p 2 ) p 2s 2 n δ 2 2 n+11, p ) p s p =1 = 2 n odd p 12 n+1 ) p s Let ω be the order of p modulo 2 n+1 ω divides ϕ2 n+1 ) = 2 n and if p 1 mod 2 n+1 ), ω divides odd then ω = 1 and p 1 mod 2 n+1 ) It follows that log M n2s) 2 = 2 n p s L n s) p 12 n+1 ) odd ) 1 p = 2 n 1 s log 1 + p s = log P n s) p 12 n+1 ) The product converges for s = 1, then we have from 32) Formula 62 C n = M n2) 2 Mn 2) 2 ) b 2 n ) 2 n 1, L n 1) L n ) where L n 1) is the residue of the Dedeind zeta function of Qζ 2 n+1) at s = 1 7 Estimates of C n For the computation of L n ), we use the relation L n s) = L n 1 s) L 0 s) = χ Xn+1) χ primitive { 1 if = 1, λ) otherwise and for the computation of M n ), n 2 M n s) = M n 1 s) M 2 s) = λ) χ Xn), χ 1)=1 χ primitive Ls, χ) Ls, χ) For Res) > 1, Ls, χ) can be evaluated quicly by the formula f Ls, χ) = f s χn)ζ s, n ), f n=1 where f is the conductor of χ and ζs, a) = n=0 a + n) s is the Hurwitz zeta function L1, χ) can also be evaluated as a sum of f terms by Theorem 49 of [12]:

6 6 YVES GALLOT Theorem 71 Let χ be a non trivial Dirichlet character of conductor f and τχ) = f a=1 χa)e2πia/f be a Gauss sum Then L1, χ) = { πi τχ) f f 2 a=1 τχ) f f a=1 χa)a if χ 1) = 1, χa) log 2 sinπa/f) if χ 1) = 1 τχ) fi δ Note that, by remaring that χ = 1, we can exclude the Gauss sums associated to f of the computation Because of the cancellation of the series, the usage of high precision is required We used Pari/GP calculator [8] and GNU MP [4] for the computation Table 1 Results n 1/L n 1) C n The results lead us to propose 8 Future Studies Conjecture 81 lim L n1)c n = 1 n and to use the local maxima of L n 1) as indicators for the primes of the form 2 n But if theorem 71 is used for the computation, we cannot estimate L n 1) or C n in a reasonable amount of time for n 100 and today the only probable prime that the computation indicated is = What is interesting in the method is that no Euclidean division was required for the computation, except some modulo 2 m for the estimates of the characters However, to become practical, a fast method for the computation of the residue of ζ Qζ2 n+1 )s) at s = 1 is necessary

7 A PROBLEM ON THE DISTRIBUTION OF GF PRIMES 7 Now, if we consider that the primes of the form 2 n are uniformly distributed with a density function defined by the theorem of de la Vallée-Poussin, we have C n n + 1) log 2 [6] This result can be used to normalize our indicator Table 2 Estimates of 1/L n 1) n 1/L n 1) n + 1) log 2 L n 1)n + 1) log 2 first prime 2 n Accorting to results of Table 2, the average behaviour of L n 1) is [n+1) log 2] 1 and its behaviour depends mainly on the first prime of the form 2 n We propose Conjecture 82 Let R n be the residue of the Dedeind zeta function ζ Qζ2 n )s) at s = 1 We have lim m 1 m m R n log 2 n = 1 n=1 There exist a constant 0 < c < 1 and a constant C > 1 such that for all n c R n log 2 n C References 1 P T Bateman and R A Horn, A Heuristic Asymptotic Formula Concerning the Distribution of Prime Numbers, Math Comp ), H Dubner and Y Gallot, Distribution of generalized Fermat prime numbers, Math Comp ), H Dubner and W Keller, Factors of generalized Fermat numbers, Math Comp ), The GMP library, The GNU MP web pages, 5 K Ireland and M Rosen, A Classical Introduction to Modern Number Theory, 2nd ed, Springer-Verlag, New Yor, A Kulsha, communication to PrimeNumber egroup, 24 december 2001

8 8 YVES GALLOT 7 P Moree, private communication, The PARI Group, PARI / GP, 9 D Shans, On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n 2 + a, Math Comp ), D Shans, On Numbers of the Form n 4 + 1, Math Comp ), A Schinzel and W Sierpińsi, Sur certaines hypothèses concernant les nombres premiers, Acta Arith ), , Erratum ), L C Washington, Introduction to Cyclotomic Fields, 2nd ed, Springer-Verlag, New Yor, address: galloty@wanadoofr