factoring

Engineering Mathematics14 Aug 2026 · 5 min read

Factoring Polynomials: GCF, Difference of Squares and Trinomials

Factor polynomials using greatest common factors, difference of two squares and basic trinomial patterns, with multiplication checks.

Engineering Mathematics14 Aug 2026 · 13 min read

Quadratic Equations by Factoring

Handbook guide to quadratic equations by factoring, with rules, worked examples, verification, common errors and practical engineering applications.

Engineering Mathematics14 Aug 2026 · 5 min read

Solving Quadratic Equations by Factoring

Solve quadratic equations by arranging them equal to zero, factoring completely and applying the zero-product property.

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.

Primality Versus Factoring: Framing the Problems

Why proving compositeness is easy, proving primality is harder, and factoring is harder still — and what this asymmetry means in practice.

Shanks's Class Group Factoring Method

Shanks's method factoring an integer by finding an ambiguous form in the class group of the corresponding discriminant.

Shanks's Square Forms Factorisation (SQUFOF)

SQUFOF: factoring by finding a square form in the cycle of an indefinite quadratic form, and why it excels for small inputs.

Engineering Mathematics13 Aug 2026 · 7 min read

The Elliptic Curve Method: Stage One

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

Engineering Mathematics13 Aug 2026 · 7 min read

The Pollard Rho Factoring Method

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

The Schnorr-Lenstra Class Group Factoring Method

Factoring via class groups of quadratic orders, and its place as the conceptual bridge to the elliptic curve method.

Engineering Mathematics13 Aug 2026 · 7 min read

Trial Division and Lehman's Method

Trial division as the first factoring step, its cost, and Lehman's improvement on Fermat's method.

Engineering Mathematics13 Aug 2026 · 10 min read

Classical Factoring: Trial Division, Fermat and Lehman

Trial division and wheel factorisation, Fermat's difference of squares method, Lehman's improvement, and the role of these methods as a preprocessing stage.

Pollard's p−1 Method and Its Relatives

The p−1 method with its two stages, smoothness assumptions, the p+1 method using Lucas sequences, and the implications for choosing cryptographic primes.

Primality Testing versus Factoring

The separation between primality testing and factoring, compositeness tests versus primality proofs, probable primes, pseudoprimes, and how to select a testing strategy.

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.

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…

Engineering Mathematics13 Aug 2026 · 10 min read

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 · 10 min read

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.