An explicit result of the sum of seven cubes

We prove that every integer ≥ exp(524) is a sum of seven non negative cubes. 1 History and statements In his 1770’s ”Meditationes Algebraicae”, E.Waring asserted that every positive integer is a sum of nine non-negative cubes. A proof was missing, as was fairly common at the time, the very notion of proof being not so clear. Notice that henceforth, we shall use cubes to denote cubes of non-negative integers. Consequently, the integers we want to write as sums of cubes are assumed to be non-negative. Maillet in [15] proved that twenty-one cubes were enough to represent every (non-negative) integer and later, Wieferich in [30] provided a proof to Waring’s statement (though his proof contained a mistake that was mended in [12]). The Göttingen school was in full bloom and Landau [13] showed that eight cubes suffice to represent every large enough integer. Dickson [7] improved on this statement by establishing that the only exceptions are 23 and 239. The reader will find a full history of the subject in chapter XXV of [8]. Finally, Linnik in [14] showed that every large enough integer is a sum of seven cubes. Since then, there has been no further improvements in terms of the number of cubes required. Notice that the circle method readily proves that almost all integers are sums of at most four cubes. From an experimental and heuristical viewpoint, computations and arguments developed in [2],[27], [16], [1], [6] tend to suggest that every integer ≥ 10 is a sum of four cubes. The argument in [6] even leads us to believe ∗MSC 2000 : primary 11P05, 11Y50 ; secondary 11B13