Counterexamples

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A counterexample is an example that refutes a universal (“for all”) statement.

For example, consider the following statement:

An even, prime-indexed triangular number plus 1 equals a prime number, while an odd, prime-indexed triangular minus 2 also equals a prime.

The statement is false and the smallest counterexample is the 31st triangular number, 496. We verify that 496 + 1 = 497 = 7 × 71. See A097785 for more counterexamples to this particular statement.

Searching for counterexamples

Beal's Conjecture

See Beal's Conjecture: a search for counterexamples.

See also

• Disproofs