How to Master the World of Conjectures and Counterexamples

A conjecture is a statement you believe true from a pattern but have not proved. A single counterexample disproves it outright, which is why finding one is often faster than attempting a proof.

• They refine and correct conjectures.
• They prevent mathematicians from pursuing false statements.
• They offer clarity on the limitations of a statement’s accuracy.

Worked examples: conjectures and counterexamples

Practice questions on conjectures and counterexamples

1. Conjecture: All positive integers are greater than (0). Is this true or false? If false, provide a counterexample.
2. Conjecture: The square of any integer is positive. Is this true or false? If false, provide a counterexample.
3. Conjecture: All birds can fly. Is this true or false? If false, provide a counterexample.

1. True. By definition, positive integers are greater than (0).
2. False. Counterexample: 0 is an integer, and 0² = 0, which is not positive. The square of any nonzero integer is positive.
3. False. Counterexample: Ostriches, penguins, and kiwis are birds that cannot fly.

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How to Find a Counterexample

A counterexample has to do two things at once. It must fit the “if” part of the statement, and it must break the “then” part. Follow these steps.

1. Split the statement. Find the condition (what is given) and the conclusion (what is claimed).
2. Test the edge cases first. Try 0, 1, negative numbers, fractions, and special shapes such as squares or right angles.
3. Check both parts. Make sure your case meets the condition and fails the conclusion.
4. State it clearly. Write the case and show the work that proves the conclusion fails.

Example: Conjecture: n² + n + 41 is prime for every whole number n. It works for n = 0, 1, 2 and many more values. But at n = 40, you get 1600 + 40 + 41 = 1681 = 41 × 41, which is not prime. Many examples that fit a pattern still do not prove it.

Conjecture and Counterexample Examples in Geometry

Notice that each counterexample really is the kind of shape the statement talks about. A circle is not a counterexample to a claim about quadrilaterals.

Frequently Asked Questions

Can you disprove a conjecture with one counterexample?

Yes. A conjecture that claims something is always true fails as soon as one valid case breaks it. The counterexample must meet the conditions of the statement and make the conclusion false. For example, the conjecture that every multiple of 3 is odd is disproved by 6, which is a multiple of 3 but even. One such case is enough.

Can all conjectures be proven true?

No. Some conjectures turn out to be false once someone finds a counterexample. Others are true and later proved, which turns them into theorems. Some famous ones remain open for decades. The twin prime conjecture, which says there are infinitely many primes that differ by 2, such as 11 and 13, has been checked for huge numbers but has still not been proved.

How do you make a mathematical conjecture?

Look at several examples, find a pattern, and state it as a general rule, often in if-then form. Adding odd numbers gives 3 + 5 = 8, 7 + 9 = 16 and 11 + 1 = 12, so you might conjecture that the sum of two odd numbers is always even. Then test it with more cases and try to prove it: (2a + 1) + (2b + 1) = 2(a + b + 1).

Which conjecture can be disproved by a counterexample?

Any conjecture that is false can be disproved by a counterexample, and those are often statements using words like always, every or never. Take the claim that the product of two numbers is always greater than either number. Choosing 0.5 and 4 gives 0.5 × 4 = 2, which is less than 4, so the conjecture is false. Fractions, zero and negatives are good numbers to test.

Why are conjectures important?

Conjectures give mathematicians a clear target. Stating a pattern as a conjecture invites people to test it, look for counterexamples and search for a proof, and that process often leads to new ideas. Fermat’s Last Theorem was a conjecture for more than 350 years before Andrew Wiles proved it in the 1990s, and the work behind that proof opened whole new areas of mathematics.

Do all conjectures have counterexamples?

No. Only false conjectures have counterexamples. If a conjecture is true, no case can ever break it, so searching for a counterexample will fail and a proof is needed instead. For example, the conjecture that the angles in any triangle add up to 180° has no counterexample in ordinary flat geometry, because it can be proved using parallel lines.