The usual division algorithms on ${\mathbb {Z}}$ and ${\mathbb {Z}}[i]$ measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi _R$. To the author’s knowledge, the integers, ${\mathbb {Z}}$ and the Gaussians, ${\mathbb {Z}}[i]$ are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi _{{\mathbb {Z}}}$ and $\phi _{{\mathbb {Z}}[i]}$. This article presents the first division algorithm (that the author knows of) for ${\mathbb {Z}}[i]$ relative to $\phi _{{\mathbb {Z}}[i]}$, empowering readers to perform the Euclidean algorithm on ${\mathbb {Z}}[i]$ using its minimal Euclidean function.

We devise schemes for producing, in the least possible time, p identical objects with n agents that work at differing speeds. This involves halting the process to transfer production across agent types. For the case of two types of agent, we construct schemes based on the Euclidean algorithm that seeks to minimize the number of pauses in production.

A modified form of Euclid’s algorithm has gained popularity among musical composers following Toussaint’s 2005 survey of so-called Euclidean rhythms in world music. We offer a method to easily calculate Euclid’s algorithm by hand as a modification of Bresenham’s line-drawing algorithm. Notably, this modified algorithm is a nonrecursive matrix construction, using only modular arithmetic and combinatorics. This construction does not outperform the traditional divide-with-remainder method; it is presented for combinatorial interest and ease of hand computation.

We analyse the behaviour of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field $\mathbb{F}_{q}$ of q elements when the highest degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f that are relatively prime with g and for the average degree of $\gcd(g,f)$. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (g,f) as above.

It is shown that a real Hankel matrix admits an approximate block diagonalization inwhich the successive transformation matrices are upper triangular Toeplitz matrices. Thestructure of this factorization was first fully discussed in [1]. This approach isextended to obtain the quotients and the remainders appearing in the Euclidean algorithmapplied to two polynomials u(x) andv(x) of degree n andm, respectively, whith m <n