Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence.
Given integers P and Q, where P > 0 and , let Uk(P, Q) and Vk(P, Q) be the corresponding Lucas sequences.
Let n be a positive integer and let be the Jacobi symbol. We define
If this equation does not hold, then n is not prime. If n is composite, then this equation usually does not hold. These are the key facts that make Lucas sequences useful in primality testing.
Some good references are chapter 8 of the book by Bressoud and Wagon (with Mathematica code), pages 142-152 of the book by Crandall and Pomerance, and pages 53–74 of the book by Ribenboim.
Lucas probable primes and pseudoprimes
A Lucas probable prime test is most useful if D is chosen such that the Jacobi symbol is −1 (see pages 1401-1409 of, page 1024 of , or pages 266-269 of ). This is especially important when combining a Lucas test with a strong pseudoprime test, such as the Baillie-PSW primality test. Typically implementations will use a parameter selection method that ensures this condition (e.g. the Selfridge method recommended in and described below).
If then equation (1) becomes
If congruence (2) is false, this constitutes a proof that n is composite.
If congruence (2) is true, then n is a Lucas probable prime. In this case, either n prime or it is a Lucas pseudoprime. If congruence (2) is true, then n is likely to be prime (this justifies the term probable prime), but this does not prove that n is prime. As is the case with any other probabilistic primality test, if we perform additional Lucas tests with different D, P and Q, then unless one of the tests proves that n is composite, we gain more confidence that n is prime.
Examples: If P = 3, Q = −1, and D = 13, the sequence of U's is A006190: U0 = 0, U1 = 3, U2 = 10, U3 = 33, etc.
First, let n = 19. The Jacobi symbol is −1, so δ(n) = 20, U20 = 6616217487 = 19·348221973 and we have
Therefore, 19 is a Lucas probable prime for this (P, Q) pair. In this case 19 is prime, so it is not a Lucas pseudoprime.
For the next example, let n = 119. We have = −1, and we can compute
However, 119 = 7·17 is not prime, so 119 is a Lucas pseudoprime for this (P, Q) pair. In fact, 119 is the smallest pseudoprime for P = 3, Q = −1.
The smallest such pseudoprimes are
- 119, 649, 1189, 1763, 3599, 4187, 5559, 6681, 12095, 12403, 12685, 12871, 14041, 14279, 15051, 16109, 19043, 22847, 23479, 24769, 26795, 28421, 30743, 30889, 31631, 31647, 33919, 34997, 37949, 38503, 39203, 41441, 46079, 48577, 49141, ...
We will see below that, in order to check equation (2) for a given n, we do not need to compute all of the first n + 1 terms in the U sequence.
Let Q = -1, the smallest Lucas pseudoprime to P = 1, 2, 3, ... are
- 323, 35, 119, 9, 9, 143, 25, 33, 9, 15, 123, 35, 9, 9, 15, 129, 51, 9, 33, 15, 21, 9, 9, 49, 15, 39, 9, 35, 49, 15, 9, 9, 33, 51, 15, 9, 35, 85, 39, 9, 9, 21, 25, 51, 9, 143, 33, 119, 9, 9, 51, 33, 95, 9, 15, 301, 25, 9, 9, 15, 49, 155, 9, 399, 15, 33, 9, 9, 49, 15, 119, 9, ...
Strong Lucas pseudoprimes
Now, factor into the form where is odd.
A strong Lucas pseudoprime for a given (P, Q) pair is an odd composite number n with GCD(n, D) = 1, satisfying one of the conditions
for some r < s; see page 1396 of. A strong Lucas pseudoprime is also a Lucas pseudoprime (for the same (P, Q) pair), but the converse is not necessarily true. Therefore, the strong test is a more stringent primality test than equation (1).
We can set Q = -1, then and are P-Fibonacci sequence and P-Lucas sequence, the pseudoprimes can be called strong Lucas pseudoprime in base P, for example, the least strong Lucas pseudoprime with P = 1, 2, 3, ... are 323, 169, 119, ...
for some . An extra strong Lucas pseudoprime is also a strong Lucas pseudoprime for the same pair.
Implementing a Lucas probable prime test
Before embarking on a probable prime test, one usually verifies that n, the number to be tested for primality, is odd, is not a perfect square, and is not divisible by any small prime less than some convenient limit.
In this section, we will assume , so that δ(n) = n + 1.
Given n, one technique for choosing D is to use trial and error to find the first D in the sequence 5, −7, 9, −11, ... such that the Jacobi symbol is −1. (If D and n have a factor in common, then ). Then set P = 1 and . Once we have P and Q, it is a good idea to check that n has no factors in common with P or Q.
Given D, P, and Q, there are recurrence relations that enable us to quickly compute and without having to compute all the intermediate terms; see Lucas sequence-Other relations. First, we can double the subscript from to in one step using the recurrence relations
Next, we can increase the subscript by 1 using the recurrences
(If either of these numerators is odd, we can make it be even by increasing it by n, because all of these calculations are carried out modulo n.) Observe that, for each term that we compute in the U sequence, we compute the corresponding term in the V sequence. As we proceed, we also compute powers of Q.
We use the bits of the binary expansion of n + 1, starting at the leftmost bit, to determine which terms in the U sequence need to be computed. For example, if n + 1 = 44 (= 101100 in binary), we compute U1, U2, U4, U5, U10, U11, U22, and U44. We also compute the same-numbered terms in the V sequence and those powers of Q.
By the end of the calculation, we will have computed Un+1, Vn+1, and Qn+1. We then check equation (2) using our known value of Un+1.
The strong versions of the Lucas test can be implemented in a similar way.
To calculate a list of extra strong Lucas pseudoprimes, set Q = 1. Then try P = 3, 4, 5, 6, ..., until a value of is found so that the Jacobi symbol . With this method for selecting D, P, and Q, the first 10 extra strong Lucas pseudoprimes are 989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, and 72389 (sequence A217719 in the OEIS)
Checking additional congruence conditions
If we have checked that equation (2) is true, there are additional congruence conditions we can check that have almost no additional computational cost. If n happens not to be prime, these additional conditions may help discover that fact.
Although this congruence condition is not, by definition, part of the Lucas probable prime test, it is almost free to check this condition because, as noted above, the value of Vn+1 was computed in the process of computing Un+1.
If either equation (2) or (3) is false, this constitutes a proof that n is not prime. If both of these conditions are true, then it is even more likely that n is prime than if we had checked only equation (2).
If the above method for choosing D happened to set Q = −1, then we can adjust P and Q so that D and remain unchanged and P = Q = 5 (see Lucas sequence-Algebraic relations). If we make this adjustment, there is only one composite n < 108 for which equation (3) is true (see page 1409 and Table 6 of; this n is 913 = 11·83).
Here is yet another congruence condition that is true for primes and that is trivial to check.
First, recall that is computed during the calculation of . It would be easy to save the previously-computed power of , namely, .
Next, if n is prime, then, by Euler's criterion,
(Here, is the Legendre symbol; if n is prime, this is the same as the Jacobi symbol). Therefore, if n is prime, we must have
The Jacobi symbol on the right side is easy to compute, so this congruence is easy to check. If this congruence does not hold, then n cannot be prime.
Comparison with the Miller–Rabin primality test
k applications of the Miller–Rabin primality test declare a composite n to be probably prime with a probability at most (1/4)k.
Aside from two trivial exceptions (see below), the fraction of (P,Q) pairs (modulo n) that declare a composite n to be probably prime is at most (4/15).
Therefore, k applications of the strong Lucas test would declare a composite n to be probably prime with a probability at most (4/15)k.
There are two trivial exceptions. One is n = 9. The other is when n = p(p+2) is the product of two twin primes. Such an n is easy to factor, because in this case, n+1 = (p+1)2 is a perfect square. One can quickly detect perfect squares using Newton's method for square roots.
As noted above, when P = 1 and Q = −1, the numbers in the U sequence are the Fibonacci numbers.
A Fibonacci pseudoprime is often (page 264 of, page 142 of, or page 127 of ) defined as a composite number n for which equation (1) above is true with P = 1 and Q = −1 (but n is not divisible by 5). By this definition, the first ten Fibonacci pseudoprimes are 323, 377, 1891, 3827, 4181, 5777, 6601, 6721, 8149, and 10877 (sequence A081264 in the OEIS). The references of Anderson and Jacobsen below use this definition.
If n is congruent to 2 or 3 (mod 5), then Bressoud (, pages 272-273) and Crandall and Pomerance (, page 143 and exercise 3.41 on page 168) point out that it is rare for a Fibonacci pseudoprime to also be a Fermat pseudoprime base 2. However when n is congruent to 1 or 4 (mod 5), the opposite is true, with over 12% of Fibonacci pseudoprimes under 1011 also being base-2 Fermat pseudoprimes.
This leads to an alternate definition of Fibonacci pseudoprime. By this definition, a Fibonacci pseudoprime is a composite number n for which equation (5) is true with P = 1 and Q = −1. Using this definition, the first ten Fibonacci pseudoprimes are 705, 2465, 2737, 3745, 4181, 5777, 6721, 10877, 13201, and 15251 ( page 129) (sequence A005845 in the OEIS).
It has been shown that there are no even Fibonacci pseudoprimes with the second definition using equation (5). Using the more common first definition with equation (1), they do exist (sequence A141137 in the OEIS).
A strong Fibonacci pseudoprime may be defined as a composite number for which equation (5) holds for all P. It follows ( page 460) that an odd composite integer is a strong Fibonacci pseudoprime if and only if:
- n is also a Carmichael number
- 2(pi + 1) | (n − 1) or 2(pi + 1) | (n − pi) for every prime pi dividing n.
The smallest example of a strong Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331.
A Pell pseudoprime may be defined as a composite number n for which equation (1) above is true with P = 2 and Q = −1; the sequence Un then being the Pell sequence. The first pseudoprimes are then 35, 169, 385, 779, 899, 961, 1121, 1189, 2419, ...
This differs from the definition in A099011 which may be written as:
with (P, Q) = (2, -1) again defining Un as the Pell sequence. The first pseudoprimes are then 169, 385, 741, 961, 1121, 2001, 3827, 4879, 5719, 6215 ...
A third definition uses equation (5) with (P, Q) = (2, -1), leading to the pseudoprimes 169, 385, 961, 1105, 1121, 3827, 4901, 6265, 6441, 6601, 7107, 7801, 8119, ...
- Robert Baillie; Samuel S. Wagstaff, Jr. (October 1980). "Lucas Pseudoprimes" (PDF). Mathematics of Computation. 35 (152): 1391–1417. doi:10.1090/S0025-5718-1980-0583518-6. MR 583518.
- David Bressoud; Stan Wagon (2000). A Course in Computational Number Theory. New York: Key College Publishing in cooperation with Springer. ISBN 978-1-930190-10-8.
- Richard E. Crandall; Carl Pomerance (2005). Prime numbers: A computational perspective (2nd ed.). Springer-Verlag. ISBN 0-387-25282-7.
- Paulo Ribenboim (1996). The New Book of Prime Number Records. Springer-Verlag. ISBN 0-387-94457-5.
- Carl Pomerance; John L. Selfridge; Samuel S. Wagstaff, Jr. (July 1980). "The pseudoprimes to 25·109" (PDF). Mathematics of Computation. 35 (151): 1003–1026. doi:10.1090/S0025-5718-1980-0572872-7.
- Jon Grantham (March 2000). "Frobenius Pseudoprimes". Mathematics of Computation. 70 (234): 873–891. doi:10.1090/S0025-5718-00-01197-2. MR 1680879.
- F. Arnault (April 1997). "The Rabin-Monier Theorem for Lucas Pseudoprimes". Mathematics of Computation. 66 (218): 869–881. doi:10.1090/s0025-5718-97-00836-3.
- Di Porto, Adina; Filipponi, Piero; Montolivo, Emilio (1990). "On the generalized Fibonacci pseudoprimes". Fibonacci Quarterly. 28: 347–354. CiteSeerX 10.1.1.388.4993.
- Di Porto, Adina (1993). "Nonexistence of Even Fibonacci Pseudoprimes of the First Kind". Fibonacci Quarterly. 31: 173–177. CiteSeerX 10.1.1.376.2601.
- Müller, Winfried B.; Oswald,Alan (1993). "Generalized Fibonacci Pseudoprimes and Probable Primes". In G.E. Bergum; et al. Applications of Fibonacci Numbers. 5. Kluwer. pp. 459–464. doi:10.1007/978-94-011-2058-6_45.
- Anderson, Peter G. Fibonacci Pseudoprimes, their factors, and their entry points.
- Anderson, Peter G. Fibonacci Pseudoprimes under 2,217,967,487 and their factors.
- Jacobsen, Dana Pseudoprime Statistics, Tables, and Data (data for Lucas, Strong Lucas, AES Lucas, ES Lucas pseudoprimes below 1014; Fibonacci and Pell pseudoprimes below 1012)
- Weisstein, Eric W. "Fibonacci Pseudoprime". MathWorld.