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An explicit result of the sum of seven cubes ∗
, 2007
"... 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 nonnegative cubes. A proof was missing, as was fairly common at the time, the very notion of ..."
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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 nonnegative 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 nonnegative integers. Consequently, the integers we want to write as sums of cubes are assumed to be nonnegative. Maillet in [15] proved that twentyone cubes were enough to represent every (nonnegative) 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
"... 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 th ..."
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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
Sieve Methods
"... called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their i ..."
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called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their important applications in number theory. We apply sieves to study the distribution of squarefree numbers, smooth numbers, and prime numbers. The first chapter is a discussion of the basic sieve formulation of Legendre. We show that the distribution of squarefree numbers can be deduced using a squarefree sieve 1 . We give an account of improvements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun's Combinatorial sieve as presented in the modern language of [HR74]. We apply the general sieve to give a simpler