A counterexample is a single case that satisfies the conditions of a statement but makes its conclusion false. One valid counterexample disproves a conjecture completely, whereas no number of agreeing cases can prove it.
This lesson follows forming a conjecture and leads to explaining a rule with algebra. Together they sit inside International Mathematics investigations.
Why is one counterexample stronger than many examples?
A general statement such as “for all positive whole numbers n, this is true” makes a promise about infinitely many cases. One failing case breaks that promise permanently.
Agreeing cases only fail to break it yet. So when you test a conjecture, your aim should be to try to break it, not to confirm it.
How to test a conjecture with a counterexample
- Write the conjecture exactly, including the values it claims to cover.
- List natural test values: small numbers first, then special ones such as 0, 1, even and odd numbers, negatives and fractions if allowed.
- Substitute and work each one out in full.
- Stop at the first failure and double-check the working.
- State the counterexample clearly: the value, the calculation and why the conclusion is false.
- Say what is now known: the conjecture is false as written, and you may suggest a corrected version.
Worked example
Conjecture: for every positive whole number n, the number n² + n + 1 is prime.
Step 1, test small values:
| n | n² + n + 1 | Prime? |
|---|---|---|
| 1 | 3 | yes |
| 2 | 7 | yes |
| 3 | 13 | yes |
| 4 | 21 | no |
Step 2, check the failure: for n = 4, 4² + 4 + 1 = 16 + 4 + 1 = 21. Since 21 = 3 × 7, it is not prime.
Step 3, state the counterexample: n = 4 is a positive whole number, yet n² + n + 1 = 21, which is composite. So the conjecture is false.
Continuing shows the pattern is misleading for a while: n = 5 gives 31 and n = 6 gives 43, both prime, but n = 7 gives 49 + 7 + 1 = 57 = 3 × 19. Five agreeing cases out of six would still not have made the statement true.
The mistake to watch for
A common slip is to keep testing values that agree, and then declare the conjecture proved.
Mistaken working: n = 5 gives 31, n = 6 gives 43, both prime, so the conjecture is true.
This is agreement, not proof. It also skipped n = 4, where the statement fails.
The correction is to choose test values on purpose, in order, and to accept a single failure as decisive. A second slip is to offer a value the statement does not cover, such as n = 0.5 for a claim about whole numbers. A counterexample must meet the conditions of the statement.
Check yourself
1. Conjecture: 2ⁿ − 1 is prime for every whole number n ≥ 2. Find a counterexample.
Show answer
n = 2 gives 3, n = 3 gives 7, both prime. n = 4 gives 2⁴ − 1 = 16 − 1 = 15 = 3 × 5.
n = 4 is a counterexample, so the conjecture is false.
2. Conjecture: for every positive number x, x² is greater than x. Find a counterexample.
Show answer
Try x = 1: 1² = 1, which is equal to x, not greater. Also try x = 0.5: 0.5² = 0.25, which is less than 0.5.
x = 1 is a counterexample (and so is x = 0.5).
3. Conjecture: for every odd whole number n, n² + 2 is prime. Find a counterexample.
Show answer
n = 1 gives 3 and n = 3 gives 11, both prime. n = 5 gives 25 + 2 = 27 = 3 × 9.
n = 5 is a counterexample.
Where this leads next
A conjecture that survives honest testing still needs a reason, so move on to explaining a general rule with algebra. Then use the investigations practice set. The non-calculator working trainer is useful for checking whether a number such as 57 or 1681 is prime.
If you find it hard to decide which test values to try, a teacher can model that search with you in online one-to-one Mathematics tuition.