Largest Mersenne Prime Number Calculator
Enter an exponent p, calculate the size of 2ᵖ − 1, test the exponent, run Lucas–Lehmer when practical, and compare the candidate with the largest known Mersenne prime.
Largest Mersenne Prime Number Calculator inputs
The controls and limits below belong only to this prime calculation.
Largest Mersenne Prime Number Calculator results
How the largest mersenne prime number tool works
This page targets one distinct largest-prime search intent. It separates finite answers, known records, probable-prime status and formal proof instead of merging them into a generic prime checker.
Current largest known Mersenne prime
The current record is M136279841 = 2^136,279,841 − 1. It contains 41,024,320 digits and is the largest known prime, not the final prime.
Prime exponent requirement
If p is composite, 2ᵖ − 1 is composite. Therefore, testing the exponent is an efficient first filter. The converse is false: many prime exponents produce composite Mersenne numbers.
Lucas–Lehmer test
For prime p greater than 2, the Lucas–Lehmer sequence starts at 4 and repeatedly applies s ← s² − 2 modulo Mₚ. After p − 2 iterations, Mₚ is prime exactly when the residue is zero. The browser limit is visible because the test becomes expensive as p grows.
Digit calculation without expanding the number
The digit formula uses p log₁₀2. This is exact for the digit count when evaluated with sufficient floating precision and lets the calculator measure a record-size Mersenne expression without storing its full decimal expansion.
Record comparison
The visual compares the entered exponent and digit estimate with M136279841. A candidate larger than the record is not automatically prime; it is merely larger in form and requires a valid primality proof.
Why record primes often use this form
Specialized algorithms and efficient modular arithmetic make Mersenne candidates attractive for distributed searches. Arbitrary integers of the same digit length are much harder to certify.
Largest Mersenne Prime Number Calculator questions
What is the largest known Mersenne prime?
M136279841, equal to 2^136,279,841 − 1.
Must p be prime?
Yes, for Mₚ to be prime; however a prime p does not guarantee a prime Mₚ.
Can the calculator run Lucas–Lehmer on the record exponent?
No. That computation is far beyond a normal browser session. The page provides the verified record and limits live tests to manageable exponents.
Is 2³¹ − 1 prime?
Yes. It equals 2,147,483,647.