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.
Core Idea¶
A primefree sequence, in its established nontrivial number-theoretic sense, is a Fibonacci-type sequence of positive integers defined by A_n = A_(n-1) + A_(n-2) whose two initial terms are coprime composites and whose every later term is also composite. Coprimality excludes the easy construction in which one shared prime divisor propagates forever. What makes the object interesting is a global proof: finitely many primes, each dividing terms in periodic residue classes of indices, collectively cover all indices.
Ronald Graham's 1964 paper established the construction method. John Nicol later summarized the Graham, Knuth, and Wilf developments, corrected and generalized the covering approach, and produced the smaller initial pair 407389224418, 76343678551. The abstraction is therefore not one famous list of huge integers.
Scope of Application¶
The concept belongs to elementary and computational number theory, modular recurrence sequences, covering congruences, and the design of explicit all-composite constructions. Graham used ranks of apparition of primes in the Fibonacci sequence to produce index classes on which divisibility repeats. Nicol refined the construction and gave a smaller proven pair, showing that the identity survives changes in seeds and cover details.
The scope includes searches for smaller initial pairs, verification of proposed covering sets, and comparisons among constructions. It excludes merely empirical claims based on a computed prefix and broad uses of “prime-free” for sets, intervals, or unrelated sequences.
Clarity¶
The abstraction makes three obligations explicit. First, seed order matters because the recurrence is directional; swapping the pair generally produces a different sequence. Second, pairwise coprimality matters because it rules out a common factor propagating from the start. Third, a finite modular cover matters because infinitely many composite claims cannot be certified by finite enumeration.
Manages Complexity¶
Without modular structure, the claim “every term is composite” presents infinitely many separate factorization problems and the terms grow exponentially. A covering proof compresses those obligations into finitely many triples: prime divisor, period, and covered index residues. Once the initial modular states and coverage are verified, induction and periodicity extend the result to all terms.
Abstract Reasoning¶
The recurrence yields the identity A_(m+n) = F_(n-1) A_m + F_n A_(m+1), with the usual indexing conventions. If p divides the relevant Fibonacci number, this identity explains why a zero term modulo p reappears in an arithmetic progression of indices. The finite cover then permits a universal inference: for any index, choose a covering congruence class and its assigned prime; that prime divides the indexed term.
Knowledge Transfer¶
Within recurrence number theory, the method transfers literally: analyze the finite state evolution modulo each prime, identify periodic zero positions, and solve a covering problem over index classes. Other linear recurrences may support analogous constructions, but they require their own state transitions and existence proofs.
The named “primefree sequence” does not transfer as a prime abstraction outside mathematics. Its portable parents are Periodicity, Recurrence, and Covering Set. A scheduling system can use periodic residue covers, but calling the schedule a primefree sequence would be metaphor rather than recognition of this object.
Relationships to Other Abstractions¶
Current abstraction Primefree Sequence Domain-specific
Parents (1) — more general patterns this builds on
-
Primefree Sequence presupposes Recurrence Prime
The candidate directly presupposes
prime:recurrence: its identity fixes a second-order additive rule that generates every term from the ordered seeds.
Hierarchy path (1) — routes to 1 parentless root
- Primefree Sequence → Recurrence
Neighborhood in Abstraction Space¶
Primefree Sequence sits in a sparse region of the domain-specific corpus (69th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Computational Number Theory & Enumeration (13 abstractions)
Nearest neighbors
- Giuga Number — 0.88
- Covering Set — 0.86
- Divisor Function — 0.85
- Fermat number — 0.84
- Elliptic Divisibility Sequence — 0.84
Computed from structural-signature embeddings · 2026-09-08