Prime Number Calculator

Number Theory & Prime Factorization Theorems

A prime number is an integer greater than 1 whose only positive divisors are 1 and itself:

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• Deterministic 6k ± 1 Primality Test**:

Every prime greater than 3 can be expressed in the form 6k ±

• Testing trial division up to √(N) across 6k ± 1 candidates guarantees deterministic primality in O(√(N)) steps.

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• Canonical Prime Factorization**:

By the Fundamental Theorem of Arithmetic:

N = p_1^α_1 · p_2^α_2 · · p_k^α_k

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• Number of Divisors τ(N)**:

τ(N) =

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• Euler's Totient Function φ(N)**:

Counts the count of positive integers up to N that are coprime to N:

φ(N) = N Π_p N (1 - (1) / (p))

How to Use This Calculator

Enter your Input Integer to Test (N: 1 to 10,000,000) (e.g. 97) and Number Theory Benchmark — choose from Custom Integer (N), 97 (Prime / 25th Prime Number), 561 (Composite: 3 × 11 × 17 / Carmichael Number). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Prime Number Calculator

Input variables:

inputIntegerN — Input Integer to Test (N: 1 to 10,000,000)

Computed outputs:

primalityVerdictStatus — Primality Status

canonicalPrimeFactorization — Prime Factorization

totalDivisorsCountTau — Total Divisors Count (τ(N))

eulersTotientFunctionPhi — Euler's Totient Function (φ(N))

nearestPrimeNumbers — Nearest Previous & Next Primes

Mathematical relationships extracted from calculation logic:

n = Math.floor(n / d)

d = d === 2 ? 3 : d + 2

phi = Math.floor(phi * (1.0 - (1.0 / p)))

nextP = n + 1

Reference constants:

i = 5

Standard: NIST / ISO mathematical definitions

Worked Example

Testing 561 reveals it is composite=8 divisors and φ(561)=320), identifying the famous smallest Carmichael pseudoprime.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.