Computational Number Theory & Enumeration

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Abstractions about divisibility, polynomial arithmetic, special integer sequences, combinatorial transversals, computability, and formulas for numerical constants.

13 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bailey–Borwein–Plouffe Formula — A base-16 rational series for π whose radix alignment lets modular exponentiation recover hexadecimal or binary digits at a distant position without first generating the intervening expansion.
• Condensed Detachment — An inference rule that unifies an implication’s antecedent with a minor premise and detaches the most general resulting consequent.
• Division Algorithm — A terminating integer-arithmetic procedure that returns quotient and remainder satisfying a declared Euclidean division contract.
• Divisor Function — The multiplicative arithmetic-function family \(\sigma_z(n)=\sum_{d\mid n}d^z\), including divisor count and divisor sum as distinguished cases.
• Giuga Number — A composite integer n for which every prime divisor p satisfies p dividing n/p minus 1, a restrictive factorwise congruence linked to Giuga's primality conjecture.
• Multiplicative Digital Root — The terminal single base-b digit reached by repeatedly replacing a nonnegative integer with the product of its digits, paired with multiplicative persistence as the number of iterations required to reach that fixed point.
• Polynomial Ring — Adjoin one or more algebraically free commuting indeterminates to a coefficient ring, with finite coefficient support and the universal substitution property.
• Primefree Sequence — A nontrivial Fibonacci-type integer sequence begun from coprime composite seeds and proved to contain only composite terms, typically by a finite cover of periodic modular divisibility classes.
• Refactorable number — Classify a positive integer as refactorable when the number of its positive divisors divides the integer itself.
• S-m-n Theorem — Effectively specialize a program index by fixing some inputs, producing a new index whose remaining-input behavior exactly matches the original program with those parameters embedded.
• Sparsely Totient Number — A natural number n whose Euler totient is a strict suffix minimum: every larger integer m has φ(m) greater than φ(n).
• Transversal (Combinatorics) — A collision-free assignment that chooses one member from each indexed set, equivalently a matching that saturates the family side of its incidence graph.
• Untouchable Number — A positive integer lying outside the image of the aliquot-sum function, classified by the nonexistence of any positive integer whose proper divisors sum to it.