A new universal Waring theorem for eighth powers

On the Waring-Goldbach problem for eighth and higher powers

On a Waring-Goldbach type problem for fourth powers

In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers. MSC: 11P05, 11P32, 11P55.

Theorem on Fourth Powers 319 Extensions of Waring ' S T H E O R E M on Fourth Powers

is said to be of order n and weight &1+&2+ • • • + a n . Since ax* = x*+ • • • +^ 4 , to ^ terms, a form of weight w is equal to a sum of w biquadrates. But 79 is not a sum of fewer than nineteen biquadrates. Hence 19 is the minimum weight of a form (1) which represents all positive integers. Let ƒ be a form (1) which represents p, and let ai = r+s. The form g = (r, s, a2, • • • , an) shall be ...

Waring's Theorem for Binary Powers

A natural number is a binary k’th power if its binary representation consists of k consecutive identical blocks. We prove an analogue of Waring’s theorem for sums of binary k’th powers. More precisely, we show that for each integer k ≥ 2, there exists a Supported by NSF Award CCF-1553288 (CAREER) and a Sloan Research Fellowship. Supported by NSERC Discovery Grant #105829/2013.

A remark on a theorem of the Goldbach - Waring type