Lucas primality test

In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known.^[1][2] It is the basis of the Pratt certificate that gives a concise verification that n is prime.

Concepts

Let n be a positive integer. If there exists an integer 1 < a < n such that

$a^{n-1}\ \equiv\ 1 \pmod n \,$

and for every prime factor q of n − 1

$a^{({n-1})/q}\ \not\equiv\ 1 \pmod n \,$

then n is prime. If no such number a exists, then n is either 1 or composite.

The reason for the correctness of this claim is as follows: if the first equivalence holds for a, we can deduce that a and n are coprime. If a also survives the second step, then the order of a in the group (Z/nZ)* is equal to n−1, which means that the order of that group is n−1 (because the order of every element of a group divides the order of the group), implying that n is prime. Conversely, if n is prime, then there exists a primitive root modulo n, or generator of the group (Z/nZ)*. Such a generator has order |(Z/nZ)*| = n−1 and both equivalences will hold for any such primitive root.

Note that if there exists an a < n such that the first equivalence fails, a is called a Fermat witness for the compositeness of n.

Example

For example, take n = 71. Then n − 1 = 70 and the prime factors of 70 are 2, 5 and 7. We randomly select an a=17 < n. Now we compute:

$17^{70}\ \equiv\ 1 \pmod {71}.$

For all integers a it is known that

$a^{n - 1}\equiv 1 \pmod{n}\ \text{ if and only if } \text{ ord}(a)|(n-1).$

Therefore, the multiplicative order of 17 (mod 71) is not necessarily 70 because some factor of 70 may also work above. So check 70 divided by its prime factors:

$17^{35}\ \equiv\ 70\ \not\equiv\ 1 \pmod {71}$

$17^{14}\ \equiv\ 25\ \not\equiv\ 1 \pmod {71}$

$17^{10}\ \equiv\ 1\ \equiv\ 1 \pmod {71}.$

Unfortunately, we get that 17¹⁰≡1 (mod 71). So we still don't know if 71 is prime or not.

We try another random a, this time choosing a = 11. Now we compute:

$11^{70}\ \equiv\ 1 \pmod {71}.$

Again, this does not show that the multiplicative order of 11 (mod 71) is 70 because some factor of 70 may also work. So check 70 divided by its prime factors:

$11^{35}\ \equiv\ 70\ \not\equiv\ 1 \pmod {71}$

$11^{14}\ \equiv\ 54\ \not\equiv\ 1 \pmod {71}$

$11^{10}\ \equiv\ 32\ \not\equiv\ 1 \pmod {71}.$

So the multiplicative order of 11 (mod 71) is 70, and thus 71 is prime.

(To carry out these modular exponentiations, one could use a fast exponentiation algorithm like binary or addition-chain exponentiation).

Algorithm

The algorithm can be written in pseudocode as follows:

Input: n > 2, an odd integer to be tested for primality; k, a parameter that determines the accuracy of the test 
Output: prime if n is prime, otherwise composite or possibly composite;
determine the prime factors of n−1.
LOOP1: repeat k times:
   pick a randomly in the range [2, n − 1]
      if $a^{n-1} \not\equiv 1 \pmod n$ then
          return composite
      else # $\color{Gray}{a^{n-1} \equiv 1 \pmod n}$
         LOOP2: for all prime factors q of n−1:
            if $a^\frac{n-1}q \not\equiv 1 \pmod n$ then
               if we checked this equality for all prime factors of n−1 then
                  return prime
               else
                  continue LOOP2
            else # $\color{Gray}{a^\frac{n-1}q \equiv 1 \pmod n}$
               continue LOOP1
return possibly composite.

See also

• Édouard Lucas
• Fermat's little theorem

Notes

1. Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: a Computational Perspective (2nd edition). Springer. p. 173. ISBN 0-387-25282-7.
2. Křížek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics 9. Canadian Mathematical Society/Springer. p. 41. ISBN 0-387-95332-9.