April 1999

URSUS serie particulars on the same same pipe

When I'll back to Italy after the Chicago convention I'll be more rigth on this Goldbach conjecture...

Goldbach's conjecture (1742, letter from Christian Goldbach 1690-1764 to Leonhard Euler, 1707-1783) states that every positive even integral greater than 2 is the sum of two prime numbers (or that every positive even integral greater than 4 is the sum of two odd prime numbers). Before C. Goldbach, E. Waring (1734-1798) formulated an equivalent statement, later published in his book "Meditationes algebraicae" (a part of this work was published in 1760; it was printed in 1762 as "Miscellanea analytica" and, finally, in 1782). Until today (1999) the problem is open: the conjecture is not proved so it cannot be considered a real "theorem". In 20th century, V. Brun (1919 and 1920), G.H. Hardy and J.E. Littlewood (1923), J.G. van der Corput (1937), T. Estermann (1938), I.M. Vinogradov (1937), N.G. Cudakov (1938 and 1947), J.R. Chen (1966 and 1973-1978), R.C. Vaughan (1972), D. Zwillinger (1979) published some important works about Goldbach's conjecture; in 1965, M.L. Stein and P.R. Stein verified the conjecture up to 100 millions (moreover a check up to 400000 millions is made by Sinisalo, 1993).