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]

Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum

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