Conjectures and counterexamples

After looking at a few prime factorisations, Anshu makes a claim: 'The larger a number is, the longer its prime factorisation will be.'

It sounds plausible. Bigger numbers do often break into more pieces. What do you think of the claim?

It is not true, and one pair of numbers settles it. Look at 96 and 121. 96 = 2 × 2 × 2 × 2 × 2 × 3 — six prime factors. 121 = 11 × 11 — two prime factors.

121 is the LARGER number, and its factorisation is the SHORTER one. So Anshu's claim fails.

There are names for what just happened, and they are worth having. In mathematics, a statement or claim made WITHOUT proof or verification is called a CONJECTURE. Anshu's claim is a conjecture.

We disproved it by finding a COUNTEREXAMPLE — an example where the conjecture is false. And notice how little work it took: one pair of numbers was enough. That is the asymmetry at the heart of general claims. To PROVE one you must cover every case; to DISPROVE one you need only a single case where it fails.

This is not a warning against making conjectures. While exploring or solving problems you will form your own, and you should — try to reason about them, verify them, and share them with the class. Just remember that a conjecture is a question, not yet an answer.