Engineering

Bertrand's Postulate

Bertrand's postulate that a prime always lies between n and 2n, its elementary proof, and its use in algorithm analysis.

Better Smoothness Density Estimates

Refined estimates for smooth number density and how they determine optimal sieve parameters.

Binary Powering and Exponentiation Chains

Square-and-multiply exponentiation, left-to-right and right-to-left variants, windowing, and why exponentiation cost drives primality testing.

Binary Quadratic Forms and the Ideal Correspondence

The dictionary between binary quadratic forms and ideals of a quadratic order, and why the form language is computationally preferable.

Brute-Force Discrete Logarithm Search

Exhaustive search for discrete logarithms, its cost, and its role as the baseline against which other methods are measured.

Buchmann's Sub-exponential Algorithm: Overview

Buchmann's algorithm for class groups and units of arbitrary number fields, its structure, and where its cost concentrates.

Cantor-Zassenhaus Equal Degree Splitting

Splitting a product of irreducibles of equal degree by random elements, the probability analysis, and the characteristic two variant.

Engineering Mathematics13 Aug 2026 · 22 min read

Casting Out Vectors: Reducing a Spanning Set

Keep the vectors indexed by pivot columns and discard the rest: the basis-of-a-span theorem turns any generating set into a linearly independent one with the same span.

Engineering Mathematics13 Aug 2026 · 24 min read

Characterization of Vector Spaces

Every vector space of dimension n is isomorphic to complex n-space, and two finite-dimensional spaces are isomorphic exactly when their dimensions agree.

Chebyshev's Theorem on the Density of Primes

Chebyshev's elementary bounds on the prime counting function, the binomial coefficient argument, and what they establish short of the prime number theorem.

Checking the Condition C_p

The central congruence condition of the Jacobi sum test, what it asserts, and how it is verified in practice.

Chinese Remainder Theorem Algorithms

Reconstructing an integer from residues, Garner's incremental method, and CRT as a strategy for controlling coefficient growth.

Chinese Remaindering and Polynomial Interpolation

The Chinese remainder theorem for polynomials, and Lagrange interpolation as its special case.

Choosing an Algorithm: Decision Guide

A consolidated decision guide across the main computational tasks in this collection.

Class Group and Unit Computation: the Computational Problem

Why class group and unit computation are a single problem, what the combined algorithm produces, and how the results are verified.

Class Numbers from Analytic Class Number Formulas

Using L-functions and the analytic class number formula to compute or verify class numbers and regulators.

Coefficient Explosion in Hermite Normal Form Computation

Why intermediate entries in normal form computation grow so violently, how to recognise it, and the three standard mitigations.

Engineering Mathematics13 Aug 2026 · 25 min read

Column Space of a Nonsingular Matrix

A square matrix is nonsingular exactly when its column space is all of C^n: proof, the right-inverse construction and the cost of a deficient column space.

Column Space Spanned by Original Columns

Row-reduce, note the pivot columns, then take those columns of the original matrix: a basis for the column space built from genuine columns rather than reduced ones.

Engineering Mathematics13 Aug 2026 · 24 min read

Column Spaces and Systems of Equations

The column space of a matrix is the span of its columns: exactly the set of right-hand sides for which a linear system is consistent, plus the membership test.

Column Vectors: Equality, Addition and Scalar Multiplication

The vector space of column vectors, with entrywise definitions of equality, addition and scalar multiplication, and how one vector equation replaces a system.

Engineering Mathematics13 Aug 2026 · 19 min read

Complex Arithmetic and Vectors

Conjugation of a column vector entry by entry, and the theorems showing it distributes over vector addition and over scalar multiplication with the scalar conjugated.

Complex Multiplication and Class Numbers

The link between curves with complex multiplication and class groups of imaginary quadratic orders, and the Hilbert class polynomial.

Complex Number Arithmetic

Addition, subtraction, multiplication and division of complex numbers, the imaginary unit, reciprocals, and why linear algebra takes its scalars from C.

Composition of Binary Quadratic Forms

Gauss composition as the group law on form classes, its practical computation, and reduction between compositions.

Engineering Mathematics13 Aug 2026 · 9 min read

Computational Algebraic Number Theory: Field Overview

What computational algebraic number theory actually computes, why the problems are hard, and how the subject's algorithms fit together.

Engineering Mathematics13 Aug 2026 · 9 min read

Computational Number Theory and Algebra: Field Overview

How number theory, abstract algebra and algorithm analysis combine into a single computational discipline, and how the KEVOS Mathematics library is organised.

Computational Number Theory: Tools and Libraries

Software for computational number theory and algebra, what each is suited to, and how to choose.

Computing Column Spaces

Four methods for describing the column space of a matrix: raw columns, pivot columns, the transpose-and-reduce route, and expressing the column space as a null space via extende…

Engineering Mathematics13 Aug 2026 · 22 min read

Computing Determinants

Minors, cofactors and expansion about any row or column: how to evaluate a determinant along the sparsest line, and why triangular matrices collapse to a diagonal product.

Computing Eigenvalues and Eigenvectors

Computing eigenvalues as roots of the characteristic polynomial and eigenvectors as null spaces: the determinant test, eigenspaces and basis extraction.

Engineering Mathematics13 Aug 2026 · 24 min read

Computing Eigenvectors of Linear Transformations

How to compute eigenvalues and eigenvectors of a linear transformation on an abstract vector space: build a matrix representation, solve, then un-coordinatise.

Computing in the Integers Modulo n

Implementing arithmetic in Z_n: representative choice, reduction after each operation, inversion, and the cost of each primitive.

Computing Minimal Polynomials in Quotient Algebras

Finding the minimal polynomial of an element of a polynomial quotient algebra by linear algebra on its powers.

Computing Minimal Polynomials of Sequences

The Berlekamp-Massey algorithm and its Euclidean equivalent for finding the shortest linear recurrence.

Computing Minimal Polynomials over Finite Fields

Computing the minimal polynomial of a finite field element using conjugates or linear algebra.

Computing Modular Square Roots: Composite Modulus

Square roots modulo a composite, the four roots for a semiprime, and the equivalence with factoring.

Computing Modular Square Roots: Prime Modulus

Extracting square roots modulo a prime, the easy case for p congruent to 3 mod 4, and the randomised algorithm in general.

Computing Modular Square Roots: Prime Power Modulus

Lifting a square root from a prime to a prime power by Hensel's method, and the special handling powers of two require.

Computing Rank, Kernel and Image

Extracting rank, kernel basis and image basis from an echelon form, and the applications to polynomial factorisation.

Engineering Mathematics13 Aug 2026 · 21 min read

Computing the Inverse of a Matrix

How to compute a matrix inverse: the two-by-two determinant formula, the column-by-column view, and Gauss-Jordan reduction of the augmented array [A | I].

Computing the Jacobi Symbol

The Euclid-style algorithm for evaluating a Jacobi symbol in quadratic time without factoring either argument.

Computing the Structure of Residue Rings

The structure of the unit group of a residue ring of the maximal order, and its computation by decomposition and lifting.

Computing with Elliptic Curves over C

Practical computation with elliptic curves over the complex numbers: periods, the torus map, and heights.

Conditional Probability and Independence

Conditional probability, Bayes' theorem, independence and the distinction between pairwise and mutual independence.

Congruences and Modular Arithmetic

Congruence as an equivalence relation compatible with arithmetic, and the basic manipulation rules including where cancellation fails.

Conjugates of Complex Numbers

The complex conjugate: definition, the theorems that it respects addition and multiplication, involution, conjugate root pairs and its role in adjoints and inner products.

Conjugates, Norms and Traces

Conjugates of a finite field element, the norm and trace maps, and their surjectivity onto the subfield.