Fermat's Near Miss Finder

Overview

This project implements a Java program to search for near misses to Fermat's Last Theorem. The program takes two inputs from the user:

1. n (the exponent, where 2 < n < 12)
2. k (the upper limit for x and y values, where k > 10)

The program iterates over values of x and y (from 10 to k), calculating x^n + y^n and comparing the result to z^n or (z+1)^n. It then identifies the smallest "relative miss" where the sum x^n + y^n closely matches an integer power.

Purpose

This project demonstrates numerical techniques to explore potential near-miss solutions to Fermat's Last Theorem for specific cases. While Fermat's Last Theorem famously states that there are no integer solutions for x^n + y^n = z^n for n > 2, this program finds cases where the sum is close to an integer power.

Project Structure

• Main file: FermatNearMiss.java
• Input: Two user-provided inputs: n (exponent) and k (limit for x, y values).
• Output: The program outputs the smallest "relative miss" and the corresponding values of x, y, and z.

Features

• Supports any integer exponent n where 2 < n < 12.
• Supports large ranges of x and y (up to a value of k as determined by the system's ability to handle large integers).
• Displays detailed output of misses and relative errors for each combination of x, y, and z.
• Prevents overflow by limiting k based on n and the system's maximum long value.

How to Run the Program

1. Prerequisites:

   • Ensure you have Java JDK installed.
   • Basic knowledge of command-line/terminal commands.

2. Clone the Repository:

   git clone https://github.com/DeviSreeDornala/Assignment-1
   cd Fermat-Near-Miss-Finder

3. Compile the Program:

   javac FermatNearMiss.java

4. Run the Program:

   java FermatNearMiss

   You will be prompted to enter two values:

   • n (Exponent, where 2 < n < 12)
   • k (Upper limit for x and y, where k > 10)

5. Example Output:

   Enter the value of n (2 < n < 12): 3
   Enter the value of k (k > 10): 20
   x: 10, y: 10, z: 14, Miss: 4, Relative Miss: 0.0001830
   x: 10, y: 11, z: 14, Miss: 31, Relative Miss: 0.0012379
   ...
   Smallest Relative Miss: x: 10, y: 10, z: 14, Miss: 4, Relative Miss: 0.0001830
   

Limitations

• The program is computationally intensive for large values of k.
• Java's double precision may introduce rounding errors for very large calculations.
• The current implementation caps k to avoid overflow based on the system's Long.MAX_VALUE.

Contact

Programmer 1: Aasritha Thota Programmer 2: Devi Sree Dornala