This tool helps you calculate the result of Fermat’s Little Theorem for given values of a and p.

## Fermat’s Little Theorem Calculator

## How It Works

Fermat’s Little Theorem states that if *p* is a prime number, then for any integer *a* such that *p* does not divide *a*, the number *a* raised to the power of *p* minus 1 is congruent to 1 modulo *p*. This can be expressed as:

*a*^{p} ≡ *a* (mod *p*).

In other words, the remainder of *a* raised to *p* divided by *p* is the same as the remainder of *a* divided by *p*.

## How to Use the Calculator

- Enter a base integer value in the ‘Enter base (a)’ field.
- Enter a prime modulus in the ‘Enter prime modulus (p)’ field.
- Click the ‘Calculate’ button to see the result. It will compute
*a*^{p}%*p*.

## Limitations

- The modulus
*p*must be a prime number. Otherwise, the calculator will return an error. - JavaScript handles large numbers with floating-point precision, so very large inputs may lead to inaccurate results due to limitations in the language’s number handling.

## Use Cases for This Calculator

### Calculate Modulo Exponentiation

Input a base number, an exponent, and a modulus, to quickly calculate the result of raising the base to the exponent and then taking the modulus. Ideal for fast calculations involving large powers in modular arithmetic.

### Check Prime Number Primality

Enter a prime number candidate and verify its primality by inputting it along with the base and exponent for Fermat’s Little Theorem. Get an instant indication whether the number is likely prime or composite. Precise and reliable for small to moderate-sized numbers.

### Generate Cryptographically Secure Keys

Generate secure keys for cryptographic purposes by inputting random prime numbers and using Fermat’s Little Theorem calculation for fast key generation. Ensure the security of your encryption with ease and efficiency.

### Test for Carmichael Numbers

Effortlessly test if a number is a Carmichael number by utilizing Fermat’s Little Theorem. Input the number and check if it passes the test, providing insights into its primality property. Simplify the detection of these special composite numbers.

### Explore Modular Inverse Possibilities

Discover the modular inverses of numbers by using Fermat’s Little Theorem for efficient computation. Input the number and modulus to find the corresponding inverse. Facilitates quick calculations in number theory and cryptography.

### Study Shortcuts in Modular Arithmetic

Learn and explore shortcuts in modular arithmetic through Fermat’s Little Theorem calculator. Experiment with different numbers and moduli to understand the concept better. Enhance your knowledge with practical calculations.

### Boost Efficiency in Number Theory

Enhance your efficiency in number theory problems by leveraging the power of Fermat’s Little Theorem. Quickly calculate modular exponentiation results without manual calculations. Streamline your work and save time.

### Validate Public Key Cryptography

Validate the strength of public key cryptography systems by performing Fermat’s Little Theorem calculations. Ensure the reliability of your encryption methods with this quick and effective tool. Confirm the security of your communication channels.

### Simplify RSA Encryption Key Generation

Simplify the generation of RSA encryption keys using Fermat’s Little Theorem to compute the totient of a number efficiently. Input the prime factors of the RSA modulus to obtain the totient. Expedite the key generation process for secure communications.

### Unlock Number Theory Concepts

Unlock the mysteries of number theory concepts through practical application with Fermat’s Little Theorem calculator. Experiment with different values to see the theorem in action. Deepen your understanding of modular arithmetic and prime numbers.