Mersenne Primes
Primes of the form 2ᵖ−1.
Key idea
Primes of the form 2ᵖ−1.
In depth
A Mersenne number is 2^p − 1; primes among them are Mersenne primes, requiring p to be prime. They link to even perfect numbers and give the largest known primes, found via the Lucas–Lehmer test.
Worked example
2³ − 1 = 7 is prime; 2¹¹ − 1 = 2047 = 23 × 89 is not.
Formula
Real-world uses
- The largest known primes, found with distributed computing projects and used in testing hardware
Common mistake
Concluding that every prime p gives a prime 2^p − 1.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Mersenne Primes?
Primes of the form 2ᵖ−1. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Mersenne Primes?
The key formula is 2ᵖ − 1.
Where is Mersenne Primes used in real life?
Real-world examples: The largest known primes, found with distributed computing projects and used in testing hardware.
What should I learn before Mersenne Primes?
Make sure you are comfortable with Perfect and Amicable Numbers, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Mersenne Primes?
Concluding that every prime p gives a prime 2^p − 1.
How can I practise Mersenne Primes?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Mersenne Primes?
Move on to Pythagorean Triples Parametrisation next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.