Schnirelmann Basis Calculator
Exactfiniteh-fold sum set of actual basis with repetition and prefix coverage, no infiniteSchnirelmannadditivebasisdensity proof. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
A finite h-fold sumset records sums of exactly h elements, allowing repetition from a supplied basis.
A finite h-fold sumset records sums of exactly h elements, allowing repetition from a supplied basis.1
This tool reports representable and missing integers in a bounded prefix without proving an infinite additive-basis property.
When to use Schnirelmann Basis Calculator
This tool reports representable and missing integers in a bounded prefix without proving an infinite additive-basis property.
How to use Schnirelmann Basis Calculator and how it works
- Prepare finite basis elements, sumset term count, observed prefix cutoff, using the input definitions and matching units below.
- Enter the required fields and run Schnirelmann Basis Calculator.
- Read result.observedRepresentablePrefixIntegers, result.observedMissingPrefixIntegers using the units, signs and ordering explained below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualFiniteBasisElements | Actual Finite Basis Elements. Use the units and matching context stated in the calculation below. | array of number | Required; other constraints below |
| actualSumsetTermCount | Actual Sumset Term Count. Use the units and matching context stated in the calculation below. | number | Required; integer 1–16; finite; other constraints below |
| actualObservedPrefixCutoff | Actual Observed Prefix Cutoff. Use the units and matching context stated in the calculation below. | number | Required; integer 0–10000; finite; other constraints below |
Require a nonempty list of at most 128 distinct integer basis elements 0–10000, term count h integer 1–16 and prefix cutoff integer 0–10000. Start with sumset {0}; perform h rounds adding every basis element with repetition, keeping only sums at most cutoff. Return representable and missing prefix integers in ascending order. Exactly h terms are used, so shorter sums require zero-padding available in the basis; no infinite basis theorem follows.
S0={0}; S_j={s+a<=cutoff:s in S_(j-1),a in basis}; output=prefix intersect S_h Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.observedRepresentablePrefixIntegers | Ascending prefix integers represented by exactly the supplied number of basis terms. | array of number |
| result.observedMissingPrefixIntegers | Ascending prefix integers absent from that exact-term-count sumset. | array of number |
{
"input": {
"actualFiniteBasisElements": [
0,
1,
3
],
"actualSumsetTermCount": 2,
"actualObservedPrefixCutoff": 6
},
"output": {
"result": {
"observedRepresentablePrefixIntegers": [
0,
1,
2,
3,
4,
6
],
"observedMissingPrefixIntegers": [
5
]
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
Alternative or Complementary methods
Use Syracuse Function Calculator for a complementary operation: The accelerated odd Syracuse step removes every factor of two from 3n+1 after an odd input.
Use Infinite Product Cosine Approximation for a complementary operation: A repeated half-angle cosine product has a sinc limit, but a finite depth remains a finite product.
References
-
Schnirelmann Basis Calculator — tool.io
Similar or alternative tools
- Syracuse Function Calculator
The accelerated odd Syracuse step removes every factor of two from 3n+1 after an odd input.
- Infinite Product Cosine Approximation
A repeated half-angle cosine product has a sinc limit, but a finite depth remains a finite product.