Perfect numbers and Mersenne primes: proving Euclid's rule

AA HL Ambitious Also in: Pure maths

Research question

Why does 2ᵖ⁻¹(2ᵖ − 1) give a perfect number whenever 2ᵖ − 1 is prime, why must p itself be prime, and how rare are such numbers among those you can check?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

An ancient result with a short, satisfying proof that uses geometric series and divisors. The second question (why p must be prime) is a neat proof by contradiction.

The mathematics you'll need

Sum of divisors and geometric series
Proof of Euclid's rule
Factorising 2ᵃᵇ − 1 to show p must be prime (proof by contradiction)
Counting how many cases you can verify
Logarithmic growth of the known examples

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; check lists of perfect numbers and Mersenne exponents against OEIS.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Find the first perfect numbers by hand.
Prove Euclid's rule with a geometric series.
Prove the exponent must be prime.
Test which prime exponents fail.
Reflect on what is known and unknown (odd perfect numbers).

Pitfalls that cost marks

Claiming the converse (Euler's result) without proof or acknowledgement.
History instead of mathematics.
Not checking examples.

Showing personal engagement

Find the perfect numbers yourself before reading about them.
Investigate abundant and deficient numbers.
Make and test your own conjecture.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Sum of divisors and geometric series; Proof of Euclid's rule
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Build and break your own small cipher or code, invent examples to test each result, and record the conjectures you made and the ones that turned out to be false.

Reflection (D)

Reflect on what each result guarantees and what it doesn't: which errors a check digit misses, which attacks a cipher survives, and how the answer depends on the size of the numbers. For this idea, start with: claiming the converse (euler's result) without proof or acknowledgement — say how it affects your answer.

Use of mathematics (E)

SL: Counting principles, probability or frequency statistics used correctly; any number theory (modular arithmetic, primes) introduced with your own small worked examples and explained, not quoted.

HL: Rigorous proofs (by contradiction or induction) of the number-theory facts you rely on, counting arguments made general, or a statistical attack tested formally.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Prove that every even perfect number has Euclid's form (a harder, known result), or investigate amicable pairs.

Extending it for HL

Prove the key result in general (why the check digit catches every single-digit error, why the decryption undoes the encryption) rather than checking examples.

See a complete IA, marked

Our annotated exemplar How far can a stack of books lean over the edge of a table? (AA HL) asks a different question, but shows how a complete cryptography exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).