On sums which are powers
نویسندگان
چکیده
منابع مشابه
Sums of powers : an arithmetic refinement to the probabilistic model of Erdős and Rényi
Erdős and Rényi proposed in 1960 a probabilistic model for sums of s integral sth powers. Their model leads almost surely to a positive density for sums of s pseudo sth powers, which does not reflect the case of sums of two squares. We refine their model by adding arithmetical considerations and show that our model is in accordance with a zero density for sums of two pseudo-squares and a positi...
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We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinit...
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1. Introduction. Many of the most perplexing problems in number theory arise from the interplay of addition and multiplication. One important class of such problems is those in which we ask which numbers can be expressed as sums of some numbers which are defined multiplicatively. Such classes of numbers include nth powers (in this paper, for any fixed n; in a more general treatment, n could als...
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In [3] Berg, Christensen and Ressel prove that the closure of the cone of sums of squares ∑R[X]2 in the polynomial ring R[X] := R[X1, . . . , Xn] in the topology induced by the `1-norm is equal to Pos([−1, 1]n), the cone consisting of all polynomials which are non-negative on the hypercube [−1, 1]n. The result is deduced as a corollary of a general result, also established in [3], which is vali...
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1 Background The problem of cubes that are sums of consecutive cubes goes back to Euler ([10] art. 249) who noted the remarkable relation 33 + 43 + 53 = 63. Similar problems were considered by several mathematicians during the nineteenth and early twentieth century as surveyed in Dickson’sHistory of the Theory of Numbers ([7] p. 582–588). These questions are still of interest today. For example...
متن کاملA q-Analogue of Faulhaber's Formula for Sums of Powers
Generalizing the formulas of Warnaar and Schlosser, we prove that Schlosser’s qanalogue of the sums of powers has a similar formula, which can be considered as a q-analogue of Faulhaber’s formula. We also show that there is an analogous formula for alternating sums of a q-analogue of powers. MR Subject Classifications: Primary 05A30; Secondary 05A15;
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تاریخ انتشار 2005