Test an arbitrary supported integer and report its integer square-root floor.
Number Theory
Enter integer data
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1EnterProvide the known values
2CalculateResults update automatically
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Try an example
Enter base-10 integers only. General integer fields accept up to 100 digits; factorization-based tools explicitly limit magnitudes to 10¹², bounded searches to 100,000 or 1,000,000 as labelled, and primality tests to unsigned 64-bit integers. Inputs stay on this device.
Result
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Number theory studies whole-number structure: factors, divisibility, primes, remainders, and exact integer relationships.
Use it to inspect exact whole-number patterns without replacing a proof or a stated theorem condition.
The relationship
Write the rule before calculating
n is a perfect square iff floor(√n)²=n.
See the structure
What the calculation is doing
602 × 302 × 3 × 1060 ≡ 0 (mod 3)n is a perfect square iff floor(√n)²=n.
Worked interpretation
Read the result in context
144=12²; negative integers are not squares.
Interpret with care
Important boundary
A numerical check can support a pattern but does not replace the hypotheses or proof of a number-theory theorem.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator
Choose the required modular or symbol mode when one is available.
Enter the labelled base-10 integers or comma-separated integer lists.
Read the exact classification, residue, factorization, solution family, or bounded sequence.
Calculation method
Apply the exact integer algorithm
n is a perfect square iff floor(√n)²=n.
The engine uses exact BigInt arithmetic and explicit work limits. Undefined inverses, incompatible congruences, invalid prime assumptions, and unsupported exhaustive searches return an error or a mathematically distinct no-solution result.
Worked example
Worked example
144=12²; negative integers are not squares.
n is a perfect square iff floor(√n)²=n.
Supported inputs
Precision and limits
Exact arithmetic
Integer arithmetic and displayed residues are exact. No floating-point conversion is used in the calculation engine.
Work limits
Factorization and divisor tools support positive inputs through 10¹². Enumeration tools label limits of 100,000 or 1,000,000; deterministic primality testing supports n<2⁶⁴. General non-enumerated integer fields accept up to 100 digits.
Conventions
Modulo results use the least non-negative residue. Divisors are positive unless a theorem explicitly uses signed or directed integers. Primality begins at 2.
Scope
Exploratory calculators compute only the entered bounded case. A Goldbach pair, Carmichael classification, or other result does not claim to settle a general conjecture beyond that input.
