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PUBLICATION INTERNE No 977

VINOGRADOV’S THEOREM IS TRUE UP TO 1020

ISSN 1166-8687

YANNICK SAOUTER

IRISA CAMPUS UNIVERSITAIRE DE BEAULIEU - 35042 RENNES CEDEX - FRANCE

Te´l. : (33) 99 84 71 00 – Fax : (33) 99 84 71 71

Vinogradov's theorem is true up to 1020 Yannick Saoutery

Programme 1 | Architectures paralleles, bases de donnees, reseaux et systemes distribues Projet API Publication interne n977 | Decembre 1995 | 7 pages

Abstract: Vinogradov's theorem states that any suciently odd integer is the sum of three prime numbers. This theorem allows to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed to check this conjecture up to 1020. Key-words: Prime numbers; Goldbach's conjecture; Vinogradov's theorem

(Resume : tsvp) y

email: [email protected] CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

Centre National de la Recherche Scientifique (URA 227) Universite´ de Rennes 1 – Insa de Rennes

Institut National de Recherche en Informatique et en Automatique – unite´ de recherche de Rennes

Le theoreme de Vinogradov est vrai jusqu'a 1020

Resume : Le theoreme de Vinogradov demontre que tout entier impair susamment grand est la somme de trois nombres premiers. Ce theoreme permet de supposer la conjecture que cela soit vrai pour tout entier impair. Dans cet article, nous presentons la mise en uvre d'un algorithme qui a permis de veri er cette conjecture jusqu'a 1020. Mots-cle : Nombres premiers; Conjecture de Goldbach; Theoreme de Vinogradov

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1 INTRODUCTION

1 Introduction

Golbach stated in 1742 that every even integer greater than 2 is the sum of two prime numbers. This problem is noe known as the Goldbach's conjecture. This is still unsolved and the closest related results are that : (i) there exists an integer S such that every integer is the sum of at most S primes [1], and (ii) every suciently large even integer may be written as the sum of a prime number and of the product of at most two prime numbers[2]. On the other side, this conjecture has been numerically veri ed up to 1011[3]. This conjecture, if true, would also imply the following property: every odd number greater than 7 is the sum of three prime numbers. This latter conjecture seem easier to deal with and some results exist for it. For instance, Vinogradov [4] proved that it is true for all integer values greater than 3315 . This bound was reduced then to 1043000. In this paper we investigate numerically this conjecture and prove it to be true for all integers less than 1020.

2 Principle of the algorithm

Because of the huge size of the set of odd integers considered, systematic veri cation for all integers is impossible. But it is in fact possible to use partial results of the Goldbach's conjecture. Indeed if N is an odd integer, p a prime number and N ? p is the sum of two prime numbers then N is obviously the sum of three prime numbers. Then in virtue of the results of [3], if N is odd and if it exists a prime number p such that N ? p < 4:1011, then N is the sum of three prime numbers. So our algorithm just amounts to exhibit a sequence of increasing prime numbers pi, 0  i  P such that p0 < 4:1011, pi+1 ? pi < 4:1011 for all 0  i  P ? 1 and pP > 1020. The problem is then to have an ecient prime certi cate. Indeed we need at least 250:106 prime numbers. If we use for instance Morain's prover ECPP [5], we see that numbers of 20 decimal digits are certi ed in approximatively 1 second on Sun stations. Thus with forty machines (the number we use) the veri cation would had last more than two months. The next section describes the technique we used to avoid this problem.

3 Prime certi cate

The prime certi cate we used was an implementation of the lemma of [6]. Lemma 3.1 Let N = RF +1 an odd integer where the entire factorization of F is known and gcd(R; F) = 1. We suppose that there exists a such that aN ?1 = 1 [mod: N] and, for all prime factor pi of F , gcd(a(N ?1)=pi ?1; N) = 1. We pose then R = 2Fs + r with 0  r < 2F . We suppose N < 2F 3, then N is a prime number if and only if, either s = 0 or r2 ? 8s is not a perfect square. In practice if we use directly this criterion on any integer N possible we need to factorize N ? 1 to a sucient part of it. Although it is quite feasible for 20 digits numbers, it would slow down the algorithm much. So we decide to search for prime numbers of a special form: Theorem 3.1 Let N = 222:R + 1 with N < 1020 and R odd. Suppose that there exists a such that a(N ?1)=2 = ?1 [mod: N]. Then N is prime if and only if either s = 0 or r2 ? 8s is not a perfect square, r and s de ned as above.

Proof Application of the previous lemma. We have indeed 2:(2 ) > 10 . Moreover if N is prime then a N ? = = J(a; N) [mod: N], where N is the Jacobi symbol. There always exist a in this case such that J(a; N) = ?1. We have then aN ? = 1 [mod: N] and gcd(a N ? = ? 1; N) = gcd(2; N) = 1 since N is odd and so the lemma applies. Reciprocally if there exists a such that a N ? = = ?1 [mod: N], then aN ? = 1 [mod: N] and gcd(a N ? = ? 22 3

(

20

1) 2

1

(

(

1) 2

1; N) = 1. Thus the lemma also applies in this case.

1) 2

1

(

1) 2

2

Now we have:

Lemma 3.2 Let N be a prime number of the form 10:k + 3. Then J(5; N) = ?1. Proof Since N is odd and relatively prime with 5 the reciprocity quadratic law applies and we have: J(5; N) = J(N; 5):(?1) 5?2 1 : N?2 1 . But 5?2 1 : N2?1 = N ? 1 = 10:k + 2 and thus J(5; N) = J(N; 5) = J(3; 5). But 3 is not a quadratic residue modulo 5 and so J(5; N) = ?1. 2 We have then trivially:

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Yannick Saouter

Theorem 3.2 Let N = 2 :R + 1 with N < 10 and R odd. Suppose that 5 N ? = = ?1 [mod: N]. Then N is prime if and only if either s = 0 or r ? 8s is not a perfect square, r and s de ned as above. 22

20

(

1) 2

2

The least prime number of the forms 222:R + 1, with R odd, and 10:k + 3 at the same time is equal to 138412033 = 33  222 + 1. You may note that if you increment or decrement any such number with a multiple of 10:222 then this number will be still of the desired form (made exception of the fact that it is necessarily a prime number). Then our algorithm was the following: 1. Let p0 = 138412033. 2. If pi is a prime number according 3.2, increment i by one and set pi+1 to pi plus 95360:222 and go on with step 2. 3. If pi is not a prime number decrement pi by 10:222 and go on step 2. 4. Repeat while pi < 1020. The litteral value 95360:222 is in fact the largest multiple of 10:222 smaller than 4:1011. Then if the later algorithm ends, the theorem of Vinogradov is true up to 1020. You may note that the converse is false.

4 Implementations and results

The algorithm was not in fact implemented exactly this way. At rst, before to apply the theorem 3.2, a partial sieving was done to discard number having small divisors. The sieve used the ten rst prime numbers and was very ecient: a great part of the remaining integers revealed to be prime and thus it is quite clear that a larger sieve might have slowed down the algorithm. The second dierence is that the research area was split into 40 subparts and distributed in parallel on 40 Sun stations. The code was written using the GMP multiprecision library[7] and the computations took approximatively four days. The rst prime of the sequence was 138412033 and the last one was 100000000387414228993.

5 Conclusion

With this method it is quite hard to gain yet orders of magnitude unless great progress is done on Goldbach's conjecture. However these kinds of computations enable to have better minoration of Schnirelmann's constant. But with a bound of 1043000 for Vinogradov's theorem the upper bound we obtain with this study is 3:1042980. Goldbach's conjecture and Vinogradov's theorem if true would imply that this constant is in fact equal to 3.

6 Addendum

In article [8], Bob Silvermann claimed that Vinogradov's theorem has been established for all odd integers greater than 1020. This result if true, would then establish the entire Vinogradov theorem. It would then also imply that every even integer greater than 2 is the sum of at most 4 prime numbers.

References

[1] L. Schnirelmann. { U ber additive Eigenschaften von Zahlen. { Math. Ann., (107):649{660, 1933. [2] J.R. Chen. { On the representation of a large even integer as the sum of a prime and the product of at most two primes. { Kexue Tongbao, (17):385{386, 1966. [3] M.K. Sinisalo. { Checking the goldbach conjecture up to 4:1011. { Math. Comp., 61(204):931{934, 1993. [4] I.M. Vinogradov. { Representation of an odd number as the sum of three primes. { Dokl. Akad. Nauk SSSR, (15):169{172, 1937. [5] A.O.L. Atkin and F. Morain. { Elliptic curves and primality proving. { Math. Comp., 61(203), 1993.

REFERENCES

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[6] J. Brillhart, D.H. Lehmer, and J.L. Selfridge. { New primality criteria and factorizations of 2m  1. { Math. Comp., 29(130):620{647, 1975. [7] T. Grandlung. { The GNU multiple precision arithmetic library. { Technical documentation, 1993. [8] B. Silvermann. { Goldbach conjecture. { Article of sci.math, 1995.