Counterexamples

To show a claim is true you need a proof. To show it is false, one example is enough. That one failing case is a counterexample. Like an honest officer: a hundred good days don't prove anything, but one bribe proves he is not honest.

A famous counterexample. Fermat claimed every number 2^(2ⁿ) + 1 is prime: 3, 5, 17, 257 and 65537 are. Euler showed the next one, 2^(2⁵) + 1 = 4 294 967 297, is not: it equals 641 × 6 700 417.

Same area does not mean congruent. Base 4 and height 3, and base 6 and height 2, both give area 6, but the triangles are different. The converse (congruent ⇒ same area) is true.

No neat formula gives only primes:

• 4n² + 1 fails at n = 4 (65);
• n² + n + 11 fails at n = 10 (121);
• 4ⁿ + 3 fails at n = 4 (259 = 7 × 37).

Many examples are not a proof. 2ⁿ – 1 is prime for n = 2, 3, 5, 7, but n = 11 gives 2047 = 23 × 89. Small cases can fool you.

A statement and its converse can both be false. 'If n is even, n is a multiple of 3' fails at n = 2; 'if n is a multiple of 3, n is even' fails at n = 3.