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Lucas primality test

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The Lucas primality test invented in 1891 by Édouard Lucas, determines whether a number N is prime or not, using the complete factorization of N-1.

If, for some integer b, the quantity bN-1 is congruent to 1 modulo N, and if b(N-1)/q is not congruent to 1 modulo N for any prime divisor q of N-1, then N is a prime.


Prove that N = 811 is prime knowing that N-1 = 2 × 34 × 5

Let's start with b = 3.

[math]3^{810/2}\,= \,3^{405}\,\equiv \, 810\,\pmod{811}[/math]
[math]3^{810/3}\,= \,3^{270}\,\equiv \, 680\,\pmod{811}[/math]
[math]3^{810/5}\,= \,3^{162}\,\equiv \, 212\,\pmod{811}[/math]
[math]3^{810}\,\equiv \, 1\,\pmod{811}[/math]

All conditions of the test hold so 811 is prime.

The fourth computation is not needed: compute b(N-1)/2 as done in the example, if it is congruent to 1, the value b must be changed, if it is congruent to N-1, the first condition of the test holds, otherwise N is composite.

Notice the b = 7 is a bad choice because:

[math]7^{810/2}\,= \,7^{405}\,\equiv \, 1\,\pmod{811}[/math]

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