Number theory · Maths EE idea · Solid
Perfect numbers and Mersenne primes
A research question to start from
Why must every even perfect number have the form 2^(p−1)(2^p − 1) with 2^p − 1 prime, and what can be proved about odd perfect numbers?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A complete classification with a proof you can own, plus an open problem where you can prove partial results.
Mathematics you would need
- The sum-of-divisors function and multiplicativity
- Mersenne numbers
- Proof by contradiction
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Prove the Euclid direction and then the Euler direction of the classification.
- Prove that 2^n − 1 prime forces n prime.
- Prove simple necessary conditions for an odd perfect number (for example about its form modulo small numbers).
Scope and difficulty
Solid. Solid; keep the odd-perfect section to results you can prove yourself.
Pitfalls
- A history essay with little mathematics.
- Overclaiming about open problems.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Euclid–Euler theorem; sigma function multiplicative; odd perfect numbers. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).