This paper presents both a result and method. The closed formula for the sum of first $$m+1,m \ge 0,$$ squares sequence $$F^{(k)}$$ where each member is previous k members with initial conditions $$k-1$$ zeroes followed by 1. generalized includes known sums Fibonacci numbers recent results Ohtsuka-Jakubczyk, Howard-Cooper, Schumacher, Prodinger-Selkirk cases $$k=2,3,4,5,6.$$ contributes coeffic...

Abstract Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory 175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in {Z})$ $x+3y\in \{4^k:k\in {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of tern...

cones of positive polynomials and their sums of squares relaxations

Let H2 denote the set of even integers n 6≡ 1 (mod 3). We prove that when H ≥ X, almost all integers n ∈ H2 ∩ (X,X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.