Checking the Odd Goldbach Conjecture Up to 1020

Checking the Odd Goldbach Conjecture Up to 1020

Starr, Norton

Checking the Odd Goldbach Conjecture Up to 10^sup 20^, Yannick Saouter. Mathematics of Computation 67:222 (1998) 863-866.

This paper ends with a considerably stronger statement than indicated by the title. The Goldbach conjecture implies the odd Goldbach conjecture, namely, every odd number greater than 6 is the sum of three primes. A last-minute addition to this article reports that if the Generalized Riemann Hypothesis (GRH) is assumed, then the odd Goldbach conjecture also is true. This is because Dmitrii Zinoviev of Ohio State recently showed that if the GRH is assumed, then any odd number greater than 1020 is the sum of three primes. [“On Vinogradov’s Constant in Goldbach’s Ternary Problem,” D. Zinoviev, J. Number Theory 65 (1997) 334-358. Also see “A Complete Vinogradov 3-primes Theorem under the Riemann Hypothesis,” J.-M. Deshouillers, G. Effinger, H. te Riele, D. Zinoviev. Elec. Res. Ann. Amer. Math. Soc. 3 (1997) 99-104.] In his Journal of Number Theory article, Zinoviev remarks that if one does not assume the GRH, the odd Goldbach conjecture is valid for odd numbers larger than “about 1043’000.” Readers may wish to ponder the effort needed to check the resulting huge initial interval. NS

Copyright Mathematical Association Of America Mar 1999

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