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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 positive density for sums of s pseudo sth powers when s ≥ 3. Moreover, our approach supports a conjecture of...

Sums of squares, cubes, and higher powers.

Jagy, William C., Kaplansky, Irving (1995)

Experimental Mathematics

Sums of squares in $ℤ [\sqrt{k}]$

Fernando Chamizo (1997)

Journal de théorie des nombres de Bordeaux

Sums of squares in rings of integers with 2 inverted

Gaël Collinet (2016)

Acta Arithmetica

We prove that in a ring of S-integers containing 1/2, any totally positive element is a sum of five squares. We also exhibit examples of such rings where some totally positive elements cannot be written as the sum of four squares.

Sums of three cubes, II

Trevor D. Wooley (2015)

Acta Arithmetica

Estimates are provided for sth moments of cubic smooth Weyl sums, when 4 ≤ s ≤ 8, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding X that are represented as the sum of three cubes of natural numbers.

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Sum-product estimates applied to Waring's problem mod $p$ .

Sums of 19 biquadrates : on the representation of large integers

Sums of 2m-th powers of rational functions in one variable over real closed fields.

Sums of cubes of polynomials

Sums of Cubes of Square-Free Numbers.

Sums of Cubes of Square-Free Numbers II.

Sums of distinct squares

Sums of fifth powers and related topics

Sums of Four Cubes.

Sums of k-th powers in the ring of polynomials with integer coefficients

Sums of powers: an arithmetic refinement to the probabilistic model of Erdős and Rényi

Sums of squares, cubes, and higher powers.

Sums of squares in $ℤ [\sqrt{k}]$

Sums of squares in rings of integers with 2 inverted

Sums of three cubes, II

Sums of three cubes in three linked three-progressions.

Sums of three squares in certain Cayley rings

Sums of two k th powers.

Sums of values of a rational function

Sur la décomposition d'un entier en une somme de puissances huitièmes d'entiers (Problème de Waring)

Currently displaying 21 – 40 of 45