Pure maths · Accessible · Dissertation

Proofs that the primes never run out: a maths EPQ idea

Why it works as an EPQ

Several short, genuinely different proofs exist, so you can compare them against criteria you choose (length, what else each proves, how far each generalises). That comparison is the judgement an EPQ needs.

Scope and difficulty

Accessible. Very manageable. Keep to three or four proofs and one extension, such as primes of the form 4k + 3.

The maths

Builds on these A Level topics: Proof · Sequences and series.

You would learn:

• Modular arithmetic
• Pairwise coprime sequences (Fermat numbers)
• Why the harmonic series diverges

Learning maths beyond the course is expected in a maths EPQ, but you must understand and explain everything you use.

One possible plan

1. Explain Euclid's argument carefully, including the common misreading of it.
2. Work through a second proof using Fermat numbers, and a third using the divergence of a series.
3. Set your criteria for 'powerful' and judge each proof against them.
4. Use one proof to show there are infinitely many primes of the form 4k + 3.

Pitfalls

• Claiming that the product of the first n primes plus 1 is always prime: 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509.
• Listing proofs without ever comparing them.

Where to start reading

• Proofs from THE BOOK (Aigner and Ziegler), the chapter on primes
• Search for: Fermat numbers coprime proof infinitude of primes

Prefer books, lecture notes, journal articles and official data to a single website, and record every source as you read (how to reference).

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Your EPQ must be your own work. These pages coach: ideas, structure, checklists and planning. Submitting text, proofs, code or analysis written by someone else or by an AI tool as your own is malpractice. How to use help and AI honestly.