Stoneham number¶

In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12292
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Stoneham number is treated here as the recurring number theory identity summarized by this source-grounded definition: In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.

In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. For coprime numbers b, c > 1, the Stoneham number α b,c is defined as. \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}.

It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 . In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α b,c . In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.

For Stoneham number, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in number theory, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 .
Constitutive relation — In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
Operating condition — For coprime numbers b, c > 1, the Stoneham number α b,c is defined as.
Recognition evidence — \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}.
Admissible variation — In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α b,c .
Characteristic consequence — It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 .
Failure boundary — In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.

What It Is Not¶

Not the whole field of number theory. The node requires the specific identity stated by In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
Not an over-broad reading. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
Not an over-broad reading. For coprime numbers b, c > 1, the Stoneham number α b,c is defined as.
Not an over-broad reading. \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}.
Not automatically Dudeney number. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Stoneham number applies literally inside number theory wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
Documented setting. For coprime numbers b, c > 1, the Stoneham number α b,c is defined as.
Documented setting. \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}.
Documented setting. It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 .
Documented setting. In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α b,c .
Documented setting. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.

Outside number theory, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Classification or should be marked as analogy.

Clarity¶

A clear use of Stoneham number names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. The strongest recognition evidence in the frozen account is: \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Stoneham number compresses multiple number theory details into a stable diagnostic relation. The source shows both the central mechanism—in mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.—and the practical consequence—it was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 . This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the number theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
Check operation and conditions. For coprime numbers b, c > 1, the Stoneham number α b,c is defined as.
Demand recognition evidence. \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}.
Test variation. Change an implementation or setting while preserving in 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α b,c .
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Classification.

Knowledge Transfer¶

Within the home domain. Knowledge about Stoneham number transfers literally when a new case preserves the same carrier type, relation, and recognition test. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. For coprime numbers b, c > 1, the Stoneham number α b,c is defined as.

Beyond the home domain. No canonical parent is asserted for Stoneham number. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G; recognition evidence → \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}

Applied / In Practice¶

For coprime numbers b, c > 1, the Stoneham number α b,c is defined as. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → the applied context; invariant → In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G; boundary → the case exits the class when in mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G

Structural Tensions¶

T1 — Stable identity versus admissible variation. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. For coprime numbers b, c > 1, the Stoneham number α b,c is defined as. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. \alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{ck}c^k}. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Stoneham number literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Stoneham number distinguish that the broader parent Classification leaves together?

Structural–Framed Character¶

Stoneham number is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Its framed side is the number theory vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: For coprime numbers b, c > 1, the Stoneham number α b,c is defined as. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: It was shown by Stoneham in 1973 that α b,c is b-normal whenever c is an odd prime and b is a primitive root of c 2 . In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. It further constrains recognition and variation through: For coprime numbers b, c > 1, the Stoneham number α b,c is defined as. \alpha{b,c} = \sum{n=c^k>1} \frac{1}{b^nn} = \sum{k=1}^\infty \frac{1}{b^{ck}c^k}.

What is domain-bound. number theory supplies the operative entities, technical vocabulary, warrants, and exceptions that make Stoneham number literal. Its documented scope includes the condition that In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Another bounded application condition is that For coprime numbers b, c > 1, the Stoneham number α b,c is defined as. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α b,c .—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Normal Number.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Stoneham number. The reviewed identity is: In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Stoneham number Domain-specific

Parents (1) — more general patterns this builds on

Stoneham number is a kind of Normal Number Domain-specific

Stoneham numbers are the constructed real numbers whose entire mathematical significance is that they are provably normal to their base.

Hierarchy path (1) — routes to 1 parentless root

Stoneham number → Normal Number → Convergence

Neighborhood in Abstraction Space¶

Stoneham number sits in a sparse region of the domain-specific corpus (78^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Algebraic Structures & Order Relations (18 abstractions)

Nearest neighbors

Integral part — 0.84
Binade — 0.83
Coprime integers — 0.83
Absolute value — 0.83
Terminal singularity — 0.82

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Classification. The parent omits the specialist differentia. Tell: Can the case establish In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G?
Dudeney number. A base-dependent natural number that is a perfect cube whose digit sum equals its cube root. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Normal Number. A real number whose base-b expansion gives every finite length-k digit block its uniform limiting frequency b^-k, for every k, with the base kept explicit. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Woodall number. A natural number of the form n times two to the n minus one. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Stoneham number remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside number theory lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Stoneham_number (revision 1321323656).
Preserved source candidate: https://mathworld.wolfram.com/StonehamNumber.html
Preserved source candidate: https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704
Preserved source candidate: http://www.emis.de/journals/EM/expmath/volumes/11/11.4/pp527_546.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.