Engineering Mathematics

Shanks's SQUFOF Factoring Method

The SQUFOF algorithm: continued fraction expansion of the square root, square forms, the reverse cycle, multipliers, and why it excels in a narrow but important range.

Skew-free Algebras

Skew congruences in direct products, skew-free algebras and varieties, the relationship to congruence permutability and distributivity, and the role in Boolean representations.

Solving Polynomial Equations Modulo p

Root-finding modulo a prime: the gcd with x^p − x, equal-degree splitting by random shifts, and the special low-degree cases that admit closed forms.

Square Roots Modulo a Prime

Computing modular square roots: the p ≡ 3 (mod 4) shortcut, the Tonelli–Shanks algorithm and its 2-adic structure, and Cornacchia's algorithm for x² + dy² = p.

Engineering Mathematics13 Aug 2026 · 6 min read

Steiner Triple Systems, Squags and Sloops

Steiner triple systems recast as algebras: squags (Steiner quasigroups) and sloops (Steiner loops), the equational axioms for each, and what the variety structure delivers that …

Engineering Mathematics13 Aug 2026 · 6 min read

Stone Duality and Boolean Spaces

Stone duality: the Stone space of ultrafilters, Boolean spaces as compact Hausdorff totally disconnected spaces, the dual equivalence of categories, and how it seeds the Boolean…

Subdirect Products and Birkhoff's Subdirect Representation Theorem

Subdirect products, subdirect irreducibility, the monolith, and Birkhoff's subdirect representation theorem with its Zorn's lemma proof.

Subexponential Factoring and Index Calculus

Smooth number densities, the index calculus method for discrete logarithms, the quadratic sieve and number field sieve for factoring, practical improvements, and what record com…

Subgroups: Restriction, Corestriction and Transfer

Restriction and corestriction (transfer) maps, the index formula, consequences for torsion, and detection on Sylow subgroups.

Subuniverses, Subalgebras and the Generation Operator

Subuniverses, subalgebras, the generation operator Sg(X), and the two equivalent descriptions of generation as intersection from above and as term-closure from below.

Syntactic Monoids and Kleene's Theorem

The syntactic congruence and syntactic monoid, Kleene's characterisation of recognisable languages as the regular ones, Schützenberger's star-free theorem, and Eilenberg's varie…

Tame Congruence Theory

Tame congruence theory: minimal sets, the five local types, type sets of varieties, and how the classification refines and largely supersedes the 1981 congruence-condition hiera…

Terms, Term Algebras and Term Operations

Terms as syntactic objects, the term algebra T(X) and its absolute freeness, the distinction between a term and the operation it induces, and why the term algebra is the referen…

The Algebraic Approach to Constraint Satisfaction

The constraint satisfaction problem, polymorphisms as an algebraic invariant, the Feder–Vardi dichotomy conjecture, the algebraic reformulation, and the 2017 resolution.

The Boolean Prime Ideal Theorem

The Boolean Prime Ideal Theorem, its equivalent formulations, its strength relative to the axiom of choice, and which results in this collection depend on it.

The Center of an Algebra and Affine Representation

The centre of an arbitrary algebra, its first-order definition, the characterisation of algebras with Z(A) = ∇ as polynomially equivalent to modules, and the commutator programm…

The Chevalley–Eilenberg Resolution

The Chevalley-Eilenberg resolution of the trivial module, its differential, the resulting cochain complex, and the bound on cohomological dimension.

The Cohen–Lenstra Heuristics

The Cohen-Lenstra heuristics for class groups of quadratic fields, the weighting principle, predicted divisibility frequencies, and their role in validating computations.

The Compactness Theorem and its Consequences

The compactness theorem, its proof by ultraproducts, the standard consequences including non-standard models, and the systematic list of properties first-order logic cannot expr…

The Congruence Lattice Con A

The congruence lattice: completeness, algebraicity, its position inside Eq(A), simple and subdirectly irreducible algebras read off from it, and the classification of varieties …

The Continued Fraction Factoring Method

CFRAC: generating small residues from the continued fraction expansion of the square root, smoothness testing, the linear algebra step, and its historical role as precursor to t…

The Distribution of Primes

Chebyshev bounds, Bertrand's postulate, Mertens' theorems, the prime number theorem and primes in arithmetic progressions — the density results that make random prime generation…

The Elliptic Curve Method (ECM)

Lenstra's elliptic curve factoring method: curves modulo a composite, the failed inversion that reveals a factor, stage one and stage two, curve parameterisations, and the role …

Engineering Mathematics13 Aug 2026 · 11 min read

The Euclidean Algorithm and GCD Computation

Euclid's algorithm and its worst case, the binary GCD, Lehmer's method for multiprecision operands, and how to choose between them by operand size.

The Ext Functor

Ext defined via extensions, via projective resolutions and via injective resolutions, the agreement of the three definitions, and the basic vanishing criteria.

The Extended Euclidean Algorithm and Modular Inverses

The extended Euclidean algorithm, its loop invariants, half-extended variants, modular inversion, and rational reconstruction from a partial remainder sequence.

The Finite Basis Problem after Tarski

The resolution of Tarski's finite basis problem, the positive finite basis theorems obtained since Baker's, and what the negative answer means for how the question is now approa…

The Five-Term Exact Sequence

The five-term exact sequence for a normal subgroup, inflation and restriction maps, transgression, and its derivation from the Lyndon-Hochschild-Serre spectral sequence.

The Grothendieck Spectral Sequence

The Grothendieck spectral sequence for a composite of functors, the acyclicity hypothesis, the five-term sequence and edge maps, and the standard specialisations.

The Group Ring and the Augmentation Ideal

The group ring, modules over it as representations, the augmentation map and its kernel, and the relation between the augmentation ideal and the abelianisation.

The Hermite Normal Form

Definition and uniqueness of the Hermite normal form, algorithms for computing it, entry explosion and its remedies, and its role as the representation of choice for modules and…

The Hom Functor and Left Exactness

The Hom functor in both variables, its covariant and contravariant forms, left exactness, and the precise sense in which it fails to be exact.

The Irredundant Basis Theorem

Irredundant bases, the failure of the exchange property outside vector spaces, and the theorem that the set of irredundant basis sizes of a finitely generated algebra has no gaps.

The Jacobi Sum Primality Test

The Adleman-Pomerance-Rumely test and its Cohen-Lenstra refinement: cyclotomic extensions, characters and Jacobi sums, the basic test, and the final trial division stage.

The Künneth Formula

The Kunneth theorem for complexes over a PID, the short exact sequence with the Tor correction term, splitting, and the hypotheses that are genuinely needed.

The Ladder of an Exact Couple and Rees Systems

The ladder diagram of an exact couple, Rees systems, and the systematic treatment of filtered chain complexes and their limits.

The LLL Lattice Reduction Algorithm

The LLL algorithm: reduction conditions, the swap-and-reduce loop, the potential function that proves polynomial termination, quality guarantees, and the deep-insertion and floa…

The Long Exact Homology Sequence

The long exact sequence associated to a short exact sequence of complexes, construction of the connecting homomorphism, naturality, and the standard applications.

The Long Exact Sequences of Derived Functors

Derivation of the long exact sequences of derived functors via the horseshoe lemma, their naturality, dimension shifting and the standard computational patterns.

The Lyndon–Hochschild–Serre Spectral Sequence

The LHS spectral sequence for a normal subgroup, its E2 page, the action of the quotient, the five-term sequence, and worked applications.

The Number Field Sieve

The number field sieve: polynomial selection, sieving over two sides, the algebraic factor base and character columns, the square root step in a number field, and the special nu…

The Quadratic Sieve and MPQS

The quadratic sieve, sieving by roots of a polynomial, the multiple polynomial variation, large prime variations, and the linear algebra stage.

Engineering Mathematics13 Aug 2026 · 7 min read

The RSA Cryptosystem

RSA key generation, encryption and signing, the correctness proof from Euler's theorem, why textbook RSA is insecure, padding schemes, known attacks including small exponent and…

Engineering Mathematics13 Aug 2026 · 11 min read

The Seventeen Open Problems: Status Then and Now

The source's seventeen open problems listed in full, grouped by section, with what can be said about their status and an explicit account of where certainty ends.

The Smith Normal Form and Its Applications

The Smith normal form, elementary divisors, the structure theorem for finitely generated abelian groups, and the application to class group structure determination.

The Stein–Serre Theorem for Abelian Groups

The Stein-Serre theorem characterising countable torsion-free abelian groups that are free, via vanishing of Ext, and related Whitehead-type problems.

The Subalgebra Lattice as an Algebraic Lattice

The subuniverse lattice as an algebraic lattice, the Birkhoff–Frink representation theorem, and why an exact characterisation closes the question for arbitrary algebras while le…

The Subresultant Algorithm, Resultants and Discriminants

The subresultant polynomial remainder sequence, the resultant as a determinant and as a product over roots, the discriminant, and their use in elimination and in number field ar…