Engineering Mathematics

The Dirichlet Unit Theorem, Computationally

The structure of the unit group, its rank from the signature, and what computing units actually requires.

The Elliptic Curve Method: Stage One

ECM stage one: multiplying a point by a highly smooth scalar to reach the identity in one component.

The Error Term in the Prime Number Theorem

How closely li(x) approximates pi(x), the connection to zeta zeros, and what the Riemann hypothesis would give.

The Euclidean Algorithm: Classical and Binary Variants

The classical Euclidean algorithm, the binary variant that replaces division with shifts, and how to choose between them.

The Existence of Finite Fields

Construction of a field of any prime power order, and the proof that one exists for every such order.

The Extended Euclidean Algorithm

Computing Bezout coefficients alongside the gcd, the recurrence for the coefficient sequences, and the size bounds that make it practical.

The Extended Euclidean Algorithm and Bezout Coefficients

Computing Bezout coefficients alongside the GCD, modular inversion as its principal application, and controlling coefficient growth.

The Fermat Test and Carmichael Numbers

The Fermat primality test, pseudoprimes, and the Carmichael numbers that defeat it for every base.

The Field of Fractions of an Integral Domain

Constructing the field of fractions of an integral domain, its universal property, and the standard examples.

Engineering Mathematics13 Aug 2026 · 9 min read

The Four Core Computational Tasks of Number Fields

The four fundamental computational problems for a number field, their dependencies, and what counts as a complete answer to each.

Engineering Mathematics13 Aug 2026 · 24 min read

The Four Fundamental Subsets of a Matrix

The null space, row space, column space and left null space of a matrix, all obtained from the two submatrices C and L of one extended echelon form reduction.

The Frobenius Map

The Frobenius endomorphism, its fixed field, its order, and the Galois structure of finite field extensions.

The Fundamental Unit of a Real Quadratic Field

Computing the fundamental unit of a real quadratic field by continued fractions, its connection to the Pell equation, and its size.

The Galois Group Computation Problem

What computing a Galois group means, why the splitting field cannot be constructed, and the transitive group classification that makes the problem finite.

Engineering Mathematics13 Aug 2026 · 24 min read

The Goldilocks Theorem: Size, Independence and Spanning

With the dimension known, a candidate set that is too large is dependent, one too small cannot span, and one of the right size needs only a single check.

The Goldwasser-Kilian Primality Test

Primality proving by elliptic curves with known point counts, its recursive certificate, and the point-counting bottleneck.

The Gram-Schmidt Orthogonalisation Procedure

Gram-Schmidt orthogonalisation, the coefficients that drive lattice reduction, and why the orthogonal vectors themselves are not lattice vectors.

Engineering Mathematics13 Aug 2026 · 24 min read

The Gram-Schmidt Procedure

Gram-Schmidt converts a linearly independent set into an orthogonal set with the same span by successive projections, and it yields the QR factorisation.

The Group Law on an Elliptic Curve

The chord-and-tangent addition law, its explicit formulas, and the coordinate systems that avoid inversion.

The Hermite Normal Form Algorithm

The Hermite normal form, the classical column-reduction algorithm, and the modular variant that bounds entry growth.

The Ideal Class Group

The class group as the obstruction to unique factorisation, its finiteness, and what a complete computation must produce.

Engineering Mathematics13 Aug 2026 · 22 min read

The Inverse of a Matrix

Definition of the matrix inverse: the two-sided condition AB = BA = I, why only square matrices qualify, uniqueness, and how the inverse solves Ax = b.

The Inverse of a Matrix

Matrix inversion over a field, its computation by elimination, and why explicit inversion is usually the wrong operation.

The Jacobi Symbol

The Jacobi symbol as the multiplicative extension of the Legendre symbol to odd composite moduli, and what it does and does not tell you.

The Law of Quadratic Reciprocity

The law of quadratic reciprocity, its statement, and why it makes symbol evaluation efficient.

Engineering Mathematics13 Aug 2026 · 23 min read

The Left Null Space

The left null space of a matrix: definition as the null space of the transpose, its interpretation as recorded row operations, dimension m minus rank, and use in redundancy anal…

The Legendre Symbol

The Legendre symbol as a multiplicative character, its evaluation by Euler's criterion, and the supplementary laws.

Engineering Mathematics7 Aug 2026 · 3 min read

The LLL Lattice Basis Reduction Algorithm

The LLL algorithm: size reduction interleaved with swaps under the Lovasz condition, and the guarantees it provides in polynomial time.

The LLL Lattice Basis Reduction Algorithm

The LLL algorithm: size reduction interleaved with swaps under the Lovasz condition, and the guarantees it provides in polynomial time.

The Logarithmic Embedding and the Unit Lattice

Mapping units into a real vector space by logarithms of conjugate absolute values, and the lattice this produces.

The Matrix (Regular) Representation of Algebraic Numbers

Representing a field element by its multiplication matrix, and reading trace, norm and characteristic polynomial off that matrix.

Engineering Mathematics13 Aug 2026 · 22 min read

The Matrix-Vector Product

The matrix-vector product defined as a linear combination of the columns of a matrix, the equation Ax = b for a linear system, and when equal products force equal matrices.

The Maximal Order Problem

The maximal order problem, its reduction to a prime-by-prime question, and the factoring obstruction that limits it.

The Miller-Rabin Primality Test

The Miller-Rabin test, the witness structure, the one-quarter bound, and why it is the practical standard.

The Modern Primality Testing Landscape

The methods available for proving primality, their complexities, and which to use for a given size of candidate.

The Multiple Polynomial Quadratic Sieve

MPQS: using many polynomials with short intervals to keep values small, and the self-initialising variant.

Engineering Mathematics13 Aug 2026 · 21 min read

The Norm of a Vector

The norm of a vector: definition from the moduli of its entries, its identity with the inner product, positive definiteness, unit vectors and stable computation.

The Number Field Sieve and Factoring Records

The number field sieve, the current state of factoring records, and how key size recommendations follow from them.

The Order of a Group Element

The order of an element, its relationship to the group order, and the computational cost of determining it.

The Pocklington-Lehmer N-1 Primality Test

Proving primality from a partial factorisation of one less than the candidate, and the certificate this produces.

The Pohst-Zassenhaus Theorem

The theorem underlying maximal order computation: enlarging an order by the ring of multipliers of its radical.

The Pollard p-1 Method

Pollard's p-1 method, its dependence on the smoothness of the group order, and why that dependence is its fatal limitation.

The Pollard Rho Factoring Method

Pollard's rho method: cycle detection in a pseudorandom sequence, the birthday bound, and Brent's improvement.

The Polynomial Euclidean Algorithm over a Field

The Euclidean and extended Euclidean algorithms for polynomials over a field, and their role in inversion and interpolation.

The Polynomial Reduction Algorithm

Finding a small defining polynomial for a number field using lattice reduction on the maximal order, and why this pays for itself.

The Prime Number Theorem

The prime number theorem, its equivalent formulations, and the logarithmic integral as the superior approximation.

The Quadratic Residuosity Assumption

The quadratic residuosity assumption, its use in probabilistic encryption, and its relationship to factoring.

The Quadratic Sieve Algorithm

The quadratic sieve: candidate generation near the square root, sieving for smoothness, and its practical range.