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 001 1 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+1 + 1 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, 2003 2000 Mathematics Subject Classification Primary 11Y11; Secondary 11A41 Key words and phrases prime numbers, prime distribution, generalized Fermat numbers 1
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 50000 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 = 29 + 1 rather than for the other values And we find that the smallest primes of each form are 12 2 28 + 1, 6 2 29 + 1, 3 2 30 + 1 and 35 2 31 + 1 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
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 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
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 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 1 12732395447351626862 13728134628182460091 2 18393323355189883003 26789638797482848822 3 21525897547289665031 20927941299213300766 4 35915460044718845396 36714321229370805404 5 36517070262282297544 36129244862406263646 6 41255743008723022645 39427412953667399869 7 38076566382722473439 31089645815159960954 8 74360874409142208222 74348059978748568639 9 75184624012206212999 74890662797425630491 10 80721025282979537844 80193434982306030483 11 73647294084873125710 72245969049003170901 12 85063380378154203965 84253498784241795333 13 85931795231960285064 84678857199473387694 14 83718452818332958280 80096845351535704233 15 70545211775956337581 58026588347082479139 16 11263974068691738207 11195714229391949615 17 11189718898237277808 11004300588768807590 18 13040977439195566699 19 13129323890520994181 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+1 + 1 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 2 15+1 + 1 = 65537 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
A PROBLEM ON THE DISTRIBUTION OF GF PRIMES 7 Now, if we consider that the primes of the form 2 n+1 + 1 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+1 + 1 1 1273240 1386294 1088793 1 2 2 + 1 2 1839332 2079442 1130542 2 2 3 + 1 3 2152590 2772589 1288025 1 2 4 + 1 4 3591546 3465736 0964970 3 2 5 + 1 5 3651707 4158883 1138887 3 2 6 + 1 6 4125574 4852030 1176086 2 2 7 + 1 7 3807657 5545177 1456323 1 2 8 + 1 8 7436087 6238325 0838926 15 2 9 + 1 9 7518462 6931472 0921927 12 2 10 + 1 10 8072103 7624619 0944564 6 2 11 + 1 11 7364729 8317766 1129406 3 2 12 + 1 12 8506338 9010913 1059318 5 2 13 + 1 13 8593180 9704061 1129275 4 2 14 + 1 14 8371845 10397208 1241925 2 2 15 + 1 15 7054521 11090355 1572092 1 2 16 + 1 16 11263974 11783502 1046123 6 2 17 + 1 17 11189719 12476649 1115010 3 2 18 + 1 18 13040977 13169796 1009878 11 2 19 + 1 19 13129324 13862944 1055876 13 2 20 + 1 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+1 + 1 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 16 1962), 363 367 2 H Dubner and Y Gallot, Distribution of generalized Fermat prime numbers, Math Comp 71 2002), 825 832 3 H Dubner and W Keller, Factors of generalized Fermat numbers, Math Comp 64 1995), 397 405 4 The GMP library, The GNU MP web pages, http://wwwswoxcom/gmp/ 5 K Ireland and M Rosen, A Classical Introduction to Modern Number Theory, 2nd ed, Springer-Verlag, New Yor, 1990 6 A Kulsha, communication to PrimeNumber egroup, 24 december 2001
8 YVES GALLOT 7 P Moree, private communication, 2002 8 The PARI Group, PARI / GP, http://wwwparigp-homede 9 D Shans, On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n 2 + a, Math Comp 14 1960), 321 332 10 D Shans, On Numbers of the Form n 4 + 1, Math Comp 15 1961), 186 189 11 A Schinzel and W Sierpińsi, Sur certaines hypothèses concernant les nombres premiers, Acta Arith 4 1958), 185 208, Erratum 5 1959), 259 12 L C Washington, Introduction to Cyclotomic Fields, 2nd ed, Springer-Verlag, New Yor, 1997 E-mail address: galloty@wanadoofr