Description
For nonnegative decimal-integer value use exact binary search to find lowerRoot=floor(sqrt(value)).
Exact integer arithmetic and finite numeric arithmetic have different range limits. A safe integer number is bounded by +/-9007199254740991, while supported decimal-integer text can represent larger values. Division and remainder also need a signed convention: truncating quotients and Euclidean nonnegative remainders can give different results.
For nonnegative decimal-integer value use exact binary search to find lowerRoot=floor(sqrt(value)).
When to use this tool
Use Integer Square Neighbor Gap Pair when you need this exact stated integer sign, bound, division, comparison, or factor component.
How to use the tool and how it works
- Enter value. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read lowerRoot, lowerGap, upperGap. Use the output convention below.
For nonnegative decimal-integer value use exact binary search to find lowerRoot=floor(sqrt(value)). Return lowerRoot, lowerGap=value-root^2, and upperGap=(root+1)^2-value as decimal strings.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| value | Supplied scalar or returned value under the stated operation. | Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| lowerRoot | Lower Root | Required. Text; maximum UTF-16 length 8192. |
| lowerGap | Lower Gap | Required. Text; maximum UTF-16 length 8192. |
| upperGap | Gap to the next square (lowerRoot+1)^2, including for a perfect-square input. | Required. Text; maximum UTF-16 length 8192. |
{
"input": {
"value": "10"
},
"output": {
"lowerRoot": "3",
"lowerGap": "1",
"upperGap": "6"
}
}Limitations and assumptions
- The upper neighbor is always the next integer square, so a perfect square gives lowerGap zero but a positive upperGap. No floating-point root or rounded decimal approximation is returned.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Alternative or Complementary methods
Use exact decimal-integer tools when the range exceeds safe numeric integers. State the division, remainder, midpoint, and zero convention before combining results from different operations.
References
Similar or alternative tools
- Divisor Pair Square Duplication Check
Return partner=integer/divisor and squarePair=(partner===divisor).
- Integer Square Bracket Containment
Test integer membership in a supplied numeric interval with independently open or closed endpoints.