This document gives the formal proofs of the following results about the sums of two and four squares: 1. Any prime number p ≡ 1 mod 4 can be written as the sum of two squares. 2. (Lagrange) Any natural number can be written as the sum of four

A magic square is an n × n array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for n = 4, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the n = 3 case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane a...

This paper builds a bridge between research on regulatory focus in motivation and classification learning. It tests the hypothesis that a fit between the situational regulatory focus and the reward structure of the task leads to greater cognitive flexibility than a mismatch between situational focus and the reward structure and that the fit between the regulatory focusinduced processing charact...

Article history: Received 17 March 2014 Received in revised form 5 May 2014 Accepted 8 May 2014 Available online xxxx In this invited response, we first note that contrary to the claims of Haier and colleagues, they were not part of the review process for the article Fractionating Human Intelligence. We then expand on the rationale underlying the primary objective of the study, to determine whe...

Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized in two directions. The one is a horizontal generalization which is known as polygonal number theorem, and the other is a higher dimensional generalization which is known as Hilber-Waring problem. In this paper, we shall generalize Lagrange’s sum of four s...

The goal of this article is to prove the Sum Squares Conjecture for real polynomials $$r(z,\bar{z})$$ on $$\mathbb {C}^3$$ with diagonal coefficient matrix. This conjecture describes possible values rank $$r(z,\bar{z}) \left\Vert {z} \right\Vert ^2$$ under hypothesis that $$r(z,\bar{z})\left\Vert ^2=\left\Vert {h(z)} some holomorphic polynomial mapping h. Our approach connect problem degree est...