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The Six-Digit Repunit

Issue 26 Core Published 12 Oct 2026 By The Chalkboard Desk

Tier Core — The Six-Digit Repunit

The problem

Factor the number completely into primes, and explain how the factorisation follows from the identity rather than from trial division.
  • A complete factorisation lists only primes.
  • The argument should use the algebraic structure of the number, not a search.

Take a minute

Reveal hint 1
factors as a difference of cubes *and* as a difference of squares — exploit both.
Reveal hint 2
with and splits the number immediately.

Solution

Answer.

Derivation

  1. Rewrite as a quotient.

  2. Factor each cube-sum/difference.

  3. Combine and cancel the .

  4. Verify each factor is prime.

  5. Check the product.

  6. Note the general pattern.