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.

Version: v1 · 2026-08-30 · History
Domain-specific #: 2528
Origin domain: number theory
Subdomain: recurrence sequences

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.^[1]^[2]

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.^[2] The abstraction is therefore not one famous list of huge integers. It is a reusable object class and proof architecture: coprime composite seeds, second-order additive recurrence, periodic modular divisibility, complete index cover, and all-composite conclusion.

Structural Signature¶

Recognition roles:

ordered seed pair: positive composite integers A_0 and A_1;
nontriviality condition: gcd(A_0,A_1)=1;
Fibonacci recurrence: every later term is the sum of the preceding two;
index-to-prime assignment: each index lies in a residue class assigned a prime divisor;
modular periodicity: divisibility by each selected prime repeats at a proved period;
finite covering family: the residue classes cover every nonnegative index; and
proper-divisor check: assigned divisors are smaller than the corresponding positive terms, so divisibility proves compositeness.

The operational test has two levels. A sequence qualifies extensionally only if no term is prime and the seed pair is coprime. A reference-grade construction also needs a proof covering infinitely many indices; checking a long prefix is evidence, not certification.

What It Is Not¶

It is not any sequence lacking primes. The constant sequence 4,4,4,..., a list of even numbers, or an arbitrary enumeration of composites does not have the Fibonacci recurrence and coprime-seed constraint. It is not the ordinary Fibonacci sequence with a finite run of composite terms. It is not a sequence whose initial values share a factor: if p divides both seeds, induction immediately makes p divide every term, which is precisely the triviality excluded.

It is also not a “prime Fibonacci sequence,” a Fibonacci-prime list, or Conway's subprime recurrence. Those change the object or filter its terms. Here the recurrence stays exactly additive; only the initial conditions are engineered.

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.^[1] Nicol refined the construction and gave a smaller proven pair, showing that the identity survives changes in seeds and cover details.^[2]

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. The title's specialist meaning is narrower than its ordinary compositional meaning.

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.

The core proof step is transparent. If a prime p divides A_r and the sequence modulo p returns to the same relevant state after period m, then p divides all terms with indices congruent to r mod m. A collection of such congruence classes proves primefreeness only if their union contains every index. Missing even one residue class leaves infinitely many potentially uncovered positions.

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.

This compression preserves important information: the recurrence, seed order, gcd, cover completeness, and divisor witnesses. It deliberately discards full integer factorizations, which are unnecessary. A term needs one proper divisor, not its entire prime decomposition.

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.^[1] 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.

The gcd invariant also transfers down the sequence: consecutive terms have the same gcd as the seed pair because gcd(A_n,A_(n+1)) = gcd(A_n,A_(n+1)-A_n). Thus coprime seeds make every consecutive pair coprime. That does not by itself prove each term composite; it only blocks a shared-factor shortcut.

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.

Examples¶

Nicol's pair. Start with A_0=407389224418 and A_1=76343678551. Their sum is A_2=483732902969; subsequent terms are 560076581520, 1043809484489, and 1603886066009. Direct checks give divisors 109 of A_2, 2 of A_3, 107 of A_4, and 17 of A_5. Nicol proves that a finite modular cover continues this divisor assignment for every index, not merely these examples.^[2]

Trivial rejected construction. Start 6,10. The gcd is 2, and every later sum is even. The resulting sequence contains only composites once values exceed 2, but it fails the coprime-seed recognition condition.

Finite-prefix non-example. A reversed or guessed seed pair may produce hundreds of composite terms before a prime occurs. Unless a complete cover or another infinite proof is supplied, the observation does not establish a primefree sequence.

Structural Tensions¶

Existential property versus proof certificate. “No term is prime” defines the outcome, but infinite verification needs structure. Diagnostic: require a complete modular cover or an equivalently global proof rather than a search cutoff.
Coprime seeds versus trivial shared divisor. Removing the gcd condition makes construction effortless but destroys the intended problem. Diagnostic: compute the initial gcd before analyzing any later term.
Seed order versus unordered pair. The same two integers in reverse order generate a different trajectory. Diagnostic: calculate the third and several modular states in the declared order before reusing a certificate.
Smaller seeds versus manageable cover. Reducing initial size is attractive, but a smaller pair may require a harder or nonexistent finite cover. Diagnostic: verify coverage and divisibility independently of digit count.
Autonomy versus reduction. Recurrence, Periodicity, and Covering Set explain the machinery, yet their conjunction does not require coprime composite seeds and all-composite terms. Diagnostic: remove the primefree invariant; if the remaining construction admits prime terms, the candidate retains an autonomous residual.

Structural–Framed Character¶

The abstraction is strongly structural: ordered integers, gcd, linear recurrence, congruences, periods, and coverings determine membership. Historical names and computational searches frame which examples are prominent, but they do not change the proof conditions.

Its specialist convention is also real. In ordinary English, any prime-avoiding sequence could be “prime free.” The literature's Fibonacci-like, coprime-seed usage supplies a narrower domain frame that the dossier preserves.

Structural Core vs. Domain Accent¶

The portable core is finite certification of an infinite property through periodic classes that cover all indices. The domain accent supplies integer primality, Fibonacci recurrence, ranks of apparition, coprime composite seeds, and proper prime divisors. Remove those terms and the recognizable object disappears.

The candidate is therefore domain-specific rather than prime. Periodic covering recurs elsewhere, but “primefree sequence” itself does not retain literal vocabulary and recognition conditions across three unrelated domains.

The candidate directly presupposes prime:recurrence: its identity fixes a second-order additive rule that generates every term from the ordered seeds. prime:periodicity explains why modular divisibility witnesses can repeat. A domain_specific:covering_set is a powerful and typical certificate that every generated term is composite, but it is a proof object rather than the resulting sequence and is not constitutive of every conceivable proof of the all-composite invariant.

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 (69^th 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

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Covering Set: the proof instrument assigning a divisor to every index class, not the sequence itself.
Fibonacci prime: a prime term of the ordinary Fibonacci sequence, the opposite outcome.
Composite sequence: may lack the recurrence and coprimality conditions.
Shared-factor recurrence: all terms inherit one divisor from the seeds and fail nontriviality.
Lucas sequence: a broader family of second-order recurrences; only specially seeded all-composite instances meet this identity.
Elliptic divisibility sequence: governed by elliptic-curve arithmetic and divisibility laws, not the Fibonacci sum recurrence.
Primefree prefix: finite computation cannot certify the infinite property.
Subprime Fibonacci sequence: changes each recurrence step by dividing composite sums, so it is a different dynamical rule.

References¶

[1] R. L. Graham, “A Fibonacci-Like Sequence of Composite Numbers,” Mathematics Magazine 37, no. 5 (1964): 322–324, https://doi.org/10.1080/0025570X.1964.11975551; author-hosted scan, https://fanchung.ucsd.edu/ron/papers/64_06_fibonacci.pdf. registry ↩a ↩b ↩c

[2] John W. Nicol, “A Fibonacci-like Sequence of Composite Numbers,” Electronic Journal of Combinatorics 6, no. 1 (1999), R44, https://doi.org/10.37236/1476. registry ↩a ↩b ↩c ↩d