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The variance associated with the distribution of sums of two unlike powers in arithmetic progressions is evaluated asymptotically.

Asympototic formulae are obtained for the variance associated with the distribution in arithmetic progressions of the numbers that are the sum of two positive cubes of integers.

AbstractPaucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

In this paper, we continue our investigation of the distribution of the values of 1x 3 1 + 2x 3 2+sx 3 s at integer points when s = 7 and 8. Under suitable conditions on the real algebraic numbers s it is shown that for any real the inequality j 1x 3 1 +sx 3 s - j <(max j xj j )-has infinitely many integral solutions provided that 23 90 when s = 8; and 1 360 when s = 7

We show that a sum of four non-degenerate binary cubic forms with integral coefficients necessarily possesses a non-trivial rational zero. When each of these binary cubic forms has non-zero discriminant, we are able to obtain bounds on the number, N(P), of integral zeros of the sum inside a box of size P of the shape P5−N(P)P5. Finally, given two binary cubic forms with non-zero discriminant, we show that almost all integers, lying in those congruence classes permitted by local solubility conditions, are represented as the sum of the aforementioned forms.

An upper bound for the numberE(N) of numbers not exceedingN and not being the sum of four cubes is given; namelyE(N)131/147+e.