smooth number
Let an agent define smooth numbers precisely, test smoothness for a given bound, explain their distribution and their role in algorithms, and relate regular numbers to music, computing and history.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define The definition.
Definition
Definition.
Definition
Definition.
- What does B-smooth mean, and what are regular numbers? definition
- Is the question about smooth, rough or squarefree numbers? boundary
Test
Testing.
Test
Test.
- How is smoothness tested for a given number and bound? action
- What is the result for the number in question? measurement
Distribution How common they are.
Density
Density.
Density
Density.
- How many smooth numbers are there up to x, and what does the Dickman function describe? provenance
- Which references are standard? provenance
Results
Known results.
Results
Results.
- What is proven versus heuristic about smooth number distribution? provenance
- What is proven versus conjectured? boundary
Algorithms Uses in computation.
Factoring
Factoring and logarithms.
Factoring
Factoring.
- How do the quadratic sieve, number field sieve and index calculus use smooth numbers? provenance
- Which entry fits integer factorization? action
Other
Other uses.
Other
Other.
- How are regular numbers used in FFT sizes, music theory and Babylonian tables? provenance
- Which entry fits the fast Fourier transform? action
Context History and teaching.
History
History.
History
History.
- How did regular numbers appear in Babylonian mathematics and in later theory? provenance
- Which entry fits Babylonian mathematics? action
Teach
Teaching.
Teach
Teaching.
- How are smooth numbers taught in number theory and cryptography courses? action
- Which misconceptions arise? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
- Other names and narrower kinds
- regular number
What it is Filled
A positive integer whose prime factors are all at most some bound B, called B-smooth; for example 5-smooth or regular numbers have only the factors 2, 3 and 5; smooth numbers are central to factoring and discrete logarithm algorithms, to fast Fourier transform sizes and to ancient Babylonian arithmetic.
Why it exists Filled
Let an agent define smooth numbers precisely, test smoothness for a given bound, explain their distribution and their role in algorithms, and relate regular numbers to music, computing and history.
Distinguishing features Filled
- Bounded largest prime factor
- Density described by the Dickman function
- Key to sieve-based algorithms
- Regular numbers as the classic case
What robots and AI may and may not do Filled
Must not
- Call a number smooth without stating the bound.
- Confuse smooth numbers with rough or squarefree numbers.
- Present heuristic estimates of how many smooth numbers exist as proven results.
- Overstate what smooth-number methods imply for the security of deployed cryptography.
Only with a human decision
- Choosing cryptographic key sizes on the basis of smoothness estimates.
May
- Test whether a number is B-smooth and state the bound used.
- Explain the role of smooth numbers in factoring, discrete logarithms and transform sizes.
Moral aspects Filled
- Results about smooth numbers bear on how secure some cryptographic systems are.
Who is affected
- Learners of number theory
- People relying on cryptography
Owners Filled
Steward
Nobody owns the concept; it belongs to the shared body of mathematics.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - a property of integers
- integer factorization - sieve algorithms
- rough number - the opposite property
- On-Line Encyclopedia of Integer Sequences - sequence A051037 for regular numbers
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.smooth-number
- Wikidata: Q1529876 (https://www.wikidata.org/wiki/Q1529876)
- OEIS: A051037 regular numbers
Direct properties not applicable Not applicable
- regular numbers below 60: 1 to 60 with factors 2, 3, 5 only integers - used in Babylonian reciprocal tables
Plane XCT: no invented physical properties.
Recognition optional Filled
- All prime factors at most B
- Regular numbers are 5-smooth: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12
- Rough numbers have only large prime factors, the opposite property
- Not visible; a property of integers.
Capabilities and actions required Filled
- test whether a number is B-smooth
- explain the definition and examples
- explain the role in factoring algorithms
- relate regular numbers to their uses
Hazards and failure modes required Filled
- Omitting the bound B when calling a number smooth
- Confusing smooth with rough or with squarefree
- Presenting heuristic distribution results as exact
Standards and interfaces required Filled
- No regulation; definitions follow the literature
Context of use required Filled
- Not for anything in itself; a property used in algorithms and computation.
- B-smooth numbers for a bound B
- regular or 5-smooth numbers
- 7-smooth humble numbers
- powersmooth numbers
- smooth numbers in algorithm analysis
Sources Filled
- Wikidata item Q1529876: smooth number - identity and sense of the item
- Wikipedia: Smooth number - general description of the item
Open questions
- Should regular numbers be a separate entry?
- How should OEIS entries be linked?
- How should the Dickman function be represented?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1529876/spec.json