Erdős–Woods number¶

A positive integer k for which some interval of k+1 consecutive integers has every interior member sharing a nontrivial common divisor with at least one endpoint.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4410
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Elementary and Computational Number Theory → Mathematics

Core Idea¶

The property is existential in the interval start, connects covering systems of prime divisors to Diophantine constraints and has a finite computable initial sequence under the standard convention. Prime divisors of the two endpoints cover each interior offset: for every intermediate integer, a gcd with the left or right endpoint exceeds one, while the endpoint distance remains k. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

Buddy-Number Bridges

Think of a row of counting numbers with a start number and an end number. Some numbers are 'buddies' because they are in the same times table, like 6 and 9, which are both in the 3 times table. A number k is special if you can find a row k steps long where every number in the middle is a buddy of the start or of the end.

Ends That Cover the Middle

Pick a starting number a and look at the stretch from a to a+k. Now check every number strictly between the two ends. For each one, it must share a common factor bigger than 1 with the left end or with the right end. If you can find any starting point a where this works, then k is called an Erdős–Woods number. You only need one starting point that works, and you can check it by testing every middle number.

Endpoint-Factor Covering Distances

A positive integer k is an Erdős–Woods number if there exists some integer a such that every integer strictly between a and a+k satisfies gcd(m, a) > 1 or gcd(m, a+k) > 1. In other words, the prime factors of the two endpoints together 'cover' every interior number. The definition is existential: k qualifies if at least one witness start a exists. Checking a proposed witness just means testing each interior offset. Finding witnesses, or showing none exist, means solving congruence conditions on a, which links the idea to covering systems. Conventions matter, so you have to state whether the endpoints are included and how published sequences are indexed.

An Erdős–Woods number is a distance k for which some interval [a, a+k] has every interior integer sharing a nontrivial factor with one of its endpoints. Formally, for each i with 0 < i < k, either gcd(a+i, a) > 1 or gcd(a+i, a+k) > 1. Since gcd(a+i, a) = gcd(i, a) and gcd(a+i, a+k) = gcd(k−i, a+k), the problem becomes a covering problem. You assign each interior offset to a prime dividing a or a+k and then solve the resulting system of congruences, for example with the Chinese remainder theorem. The definition is existential in a. A witness can be verified directly, but proving that a given k is not an Erdős–Woods number requires ruling out every start. Under a fixed convention on endpoints, the initial terms of the sequence can be computed. Claims about minimal witnesses, or about sequence indices, depend on these conventions being stated explicitly.

Scope of Application¶

Erdős–Woods number belongs to elementary and computational number theory and is useful where the analyst can specify the typed elementary and computational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences are explicit. The scope is broad within that domain but bounded by the need for the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Erdős–Woods number. Erdős–Woods number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed elementary and computational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of elementary and computational number theory because they reuse the typed elementary and computational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Prime divisors of the two endpoints cover each interior offset: for every intermediate integer, a gcd with the left or right endpoint exceeds one, while the endpoint distance remains k., and type the carrier, state every parameter and convention in the definition, test that the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Erdős–Woods number Domain-specific

Parents (1) — outgoing typed relations

Erdős–Woods number is a kind of Coverage / Reachability Prime

It remains its own entry because its identity is fixed by the positive distance k, witness interval start, endpoint inclusion convention, every interior offset, gcd condition and nontrivial threshold, witness verification, minimality if claimed, and indexing of published sequences.

Hierarchy paths (2) — routes to 2 parentless roots

Erdős–Woods number → is a kind of Coverage / Reachability → is a kind of Completeness

Show alternative path (1)

Neighborhood in Abstraction Space¶

Erdős–Woods number sits in a crowded region of the domain-specific corpus (9^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer & Divisor Number Theory (45 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25