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MathidiaMathematics Academy
Mathematical reasoning and modelling
Lesson 72 of 75
Easy practice

Conjectures and counterexamples

Objective. Test statements and disprove false claims efficiently.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A mathematical conjecture is a statement believed to be true but not yet proven, and it can be disproven by finding just one counterexample where the statement fails. To test a conjecture efficiently, try small or unusual numbers first, such as zero, negatives or fractions, since these often reveal exceptions quickly. A single valid counterexample is enough to disprove a general claim, even if the statement holds for many other cases. Example: the conjecture 'squaring a number always makes it bigger' fails for 0.5, since 0.5² = 0.25, which is smaller than 0.5.

Worked example

Test the conjecture 'the sum of two prime numbers is always even' and give a counterexample if the conjecture is false.

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  1. 1Try two odd prime numbers first, such as 3 and 5, giving 3 + 5 = 8, which is even.
  2. 2Now try the prime number 2, which is even, combined with another prime such as 3.
  3. 3Calculate 2 + 3 = 5, which is odd, not even.
  4. 4This counterexample, 2 + 3 = 5, disproves the conjecture.

Answer: 2 + 3 = 5 is a counterexample

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Practice

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Common mistake. Pupils sometimes try to prove a conjecture true by testing several examples that happen to work, without searching for exceptions; always try unusual cases like negatives or fractions before accepting a conjecture as true.

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