Lehmann test is a primality test – it determines whether the given integer is composite or a prime.
Little Fermat's theorem states that for every prime it holds that
Hence it also holds
If we use the formula to expand the expression, we get
From divisibility of numbers, we know
So, if the equation holds, than one of the following conditions must also hold
Finally, provided that
than may be a prime. In any other case is a composite number, because it contradicts the Little Fermat's theorem. It can be shown that every iteration of Lehmann test eliminates at least fifty percent of composite numbers.
Probability that the number is a prime after iterations of Lehmann test can be expressed as