## Checking the odd Goldbach conjecture up to 10 20 (1998)

Venue: | Math. Comp |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Saouter98checkingthe,

author = {Yannick Saouter},

title = {Checking the odd Goldbach conjecture up to 10 20},

journal = {Math. Comp},

year = {1998},

pages = {863--866}

}

### OpenURL

### Abstract

Abstract. Vinogradov’s theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conjecture up to 10 20. 1.

### Citations

162 | Elliptic curves and primality proving
- Atkin, Morain
(Show Context)
Citation Context ...− pi < 4.1011 for all 0 ≤ i ≤ P − 1and pP >1020. The problem then is to have an efficient prime certificate. Indeed we need at least 250.106 prime numbers. If we use for instance Morain’s prover ECPP =-=[1]-=-, we see that numbers of 20 decimal digits are certified in approximately 1 second on Sun stations. Thus with forty machines (the number we used) the verification Received by the editor March 19, 1996... |

44 |
Representation of an odd number as a sum of three primes
- Vinogradov
- 1937
(Show Context)
Citation Context ...following property: every odd number greater than or equal to 7 is the sum of three prime numbers. This latter conjecture seems easier to deal with and it gives some results. For instance, Vinogradov =-=[8]-=- proved that it is true for all integer values greater than 3315. This bound was then reduced to 1043000 .Inthispaperwe investigate this conjecture numerically and prove it to be true for all integers... |

41 |
On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two
- Chen
- 1973
(Show Context)
Citation Context ... such that every integer is the sum of at most S primes [6], and (ii) every sufficiently large even integer may be written as the sum of a prime number and of the product of at most two prime numbers =-=[3]-=-. On the other hand, this conjecture has been numerically verified up to 4 × 1011 [7]. This conjecture, if true, would also imply the following property: every odd number greater than or equal to 7 is... |

9 |
New Primality Criteria and Factorizations of 2 w ±l
- Brillhart, Lehmer, et al.
(Show Context)
Citation Context ...ould have lasted more than two months. The next section describes the technique we used to avoid this problem. 3. Prime certificate The prime certificate we used was an implementation of Theorem 5 of =-=[2]-=-. Lemma 1. Let N = RF +1 be an odd integer where the entire factorization of F is known, F is even and gcd(R, F )=1. We suppose that there exists an integer a such that a N−1 ≡ 1 (mod N) and, for all ... |

9 |
Über additive Eigenschaften von Zahlen
- Schnirelmann
- 1933
(Show Context)
Citation Context ... problem is now known as the Goldbach 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 =-=[6]-=-, and (ii) every sufficiently large even integer may be written as the sum of a prime number and of the product of at most two prime numbers [3]. On the other hand, this conjecture has been numericall... |

4 |
The GNU multiple precision arithmetic library. -- Technical documentation
- Grandlung
- 1993
(Show Context)
Citation Context ...own the algorithm. The second difference 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 =-=[5]-=- and the computations took approximately four days. The first prime of the sequence was 138412033 and the last one was 100000000209366024193. Table 1 gives the first 100 values k, giving the prime num... |

4 |
On Vinogradov’s constant in Goldbach’s ternary problem
- Zinoviev
- 1997
(Show Context)
Citation Context ...e author wishes to especially thank the referee of the article whose advice on the first version was very helpful and who directed me to the reference [9]. 7. Late note During the year 1996, Zinoviev =-=[10]-=- proved under the assumption of the Generalized Riemann Hypothesis, that any odd number greater than 1020 is the sum of three prime numbers. Thus the current work fills the gap of the remaining cases.... |

3 |
Checking the goldbach conjecture up to 4:10
- Sinisalo
- 1993
(Show Context)
Citation Context ...tly large even integer may be written as the sum of a prime number and of the product of at most two prime numbers [3]. On the other hand, this conjecture has been numerically verified up to 4 × 1011 =-=[7]-=-. This conjecture, if true, would also imply the following property: every odd number greater than or equal to 7 is the sum of three prime numbers. This latter conjecture seems easier to deal with and... |

2 |
On odd Goldbach problem under general Riemann hypothesis
- Wang, Chen
- 1993
(Show Context)
Citation Context ...prime numbers. However, reaching the bound of 10 43000 encountered in Vinogradov’s theorem seems practically unfeasible. But under the assumption of generalized Riemann hypothesis, it has been proved =-=[9]-=- that this bound can be lowered to 3.2 × 10 49 . Such a bound is much more practicable and using the method described above, it should be possible to reach this bound for at most 7 prime numbers in qu... |

1 |
te Riele, and D.Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann Hypothesis
- Deshouillers, Effinger
- 1997
(Show Context)
Citation Context ...mann Hypothesis, that any odd number greater than 1020 is the sum of three prime numbers. Thus the current work fills the gap of the remaining cases. It has also to be quoted that Deshouillers et al. =-=[4]-=-, also performed a complete verification, by checking the binary Goldbach conjecture up to 1.615 × 1012 ,which allows to deduce the truth of the odd Goldbach conjecture up to 1020 by a theorem of Scho... |