Curious Squares

Continuants and some decompositions into squares

In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number $p$ as a sum of two squares, given a solution of $z^2+1\equiv0\pmod{p}$, and vice versa. In this paper, we extend the use of continuants to proper representations by sums of two squares in rings of polynomials on fields of characteristic different from 2. New deterministic algorithms for finding the corresponding proper representations are presented. Our approach will provide a new constructive proof of the four-square theorem …

Gaps in the fractional parts of square roots

Fractional parts of the first $N$ natural numbers fill the unit interval with asymptotically uniform density. However, the gaps around rational points shrink at an asymptotically lower rate $N^{-1/2}$, and their widths scale with the Thomae ("popcorn") function. This curious connection is derived and related geometrically to shadow pattern in the Euclid's orchard. Generalized cases of higher radicals and their convergence rates, are also investigated.

有限域中连续平方数或非平方数的三元组与四元组

Triples and quadruples of consecutive squares or non-squares in a finite field

本文研究有限域中连续平方数或非平方数的三元组与四元组计数，利用 Jacobsthal 和求值给出显式公式，推广了 Monzingo 的结果，并处理了 5 的幂阶情形。

序列一致划分与平方数划分

Sequentially congruent partitions and partitions into squares

本文通过构造双射证明序列一致划分与平方数划分计数相等，并推广至k次幂划分。