Lucas pseudoprime

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.

Basic properties

Given integers P and Q, where P > 0 and $D=P^2-4Q$, let U_k(P, Q) and V_k(P, Q) be the corresponding Lucas sequences.

Let n be a positive integer and let $\left(\tfrac{D}{n}\right)$ be the Jacobi symbol. We define

$\delta(n)=n-\left(\tfrac{D}{n}\right).$

If n is a prime such that the greatest common divisor of n and Q (that is, GCD(n, Q)) is 1, then the following congruence condition holds (see page 1391 of ):

$\text{ } (1) \text{ } U_{\delta(n)} \equiv 0 \pmod {n}.$

If this equation does not hold, then n is not prime. If n is composite, then this equation usually does not hold (see, page 1392). 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 for a given (P, Q) pair is any positive integer n for which equation (1) above is true (see, page 1398).

A Lucas pseudoprime for a given (P, Q) pair is a positive composite integer n for which equation (1) is true (see, page 1391).

A Lucas probable prime test is most useful if D is chosen such that the Jacobi symbol $\left(\tfrac{D}{n}\right)$ 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 $\left(\tfrac{D}{n}\right)=-1,$ then equation (1) becomes

$\text{ } (2) \text{ } U_{n+1} \equiv 0 \pmod {n}.$

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 $\left(\tfrac{13}{19}\right)$ is −1, so δ(n) = 20, U₂₀ = 6616217487 = 19·348221973 and we have

$U_{20} = 6616217487 \equiv 0 \pmod {19} .$

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 $\left(\tfrac{13}{119}\right)$ = −1, and we can compute

$U_{120} \equiv 0 \pmod {119}.$

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 $\delta(n)$ into the form $d\cdot2^s$ where $d$ 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

$U_d \equiv 0 \pmod {n}$

or

$V_{d \cdot 2^r} \equiv 0 \pmod {n}$

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 $U_n$ and $V_n$ 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, ...

An extra strong Lucas pseudoprime is a strong Lucas pseudoprime for a set of parameters (P, Q) where Q = 1, satisfying one of the conditions

$U_d \equiv 0 \pmod {n} \text{ and } V_d \equiv \pm 2 \pmod {n}$

or

$V_{d \cdot 2^r} \equiv 0 \pmod {n}$

for some $r<s-1$. An extra strong Lucas pseudoprime is also a strong Lucas pseudoprime for the same $(P,Q)$ 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 $\left(\tfrac{D}{n}\right)=-1$, 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 $\left(\tfrac{D}{n}\right)$ is −1. (If D and n have a factor in common, then $\left(\tfrac{D}{n}\right)=0$). Then set P = 1 and $Q=(1-D)/4$. 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 $U_{n+1}$ and $V_{n+1}$ without having to compute all the intermediate terms; see Lucas sequence-Other relations. First, we can double the subscript from $k$ to $2k$ in one step using the recurrence relations

$U_{2k}=U_k\cdot V_k$

$V_{2k}=V_k^2-2Q^k$.

Next, we can increase the subscript by 1 using the recurrences

$U_{2k+1}=(P\cdot U_{2k}+V_{2k})/2$

$V_{2k+1}=(D\cdot U_{2k}+P\cdot V_{2k})/2$.

(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 U_n+1, V_n+1, and Qⁿ⁺¹. We then check equation (2) using our known value of U_n+1.

When D, P, and Q are chosen as described above, the first 10 Lucas pseudoprimes are (see page 1401 of ): 323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, and 10877 (sequence A217120 in OEIS)

The strong versions of the Lucas test can be implemented in a similar way.

When D, P, and Q are chosen as described above, the first 10 strong Lucas pseudoprimes are: 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519 (sequence A217255 in OEIS)

To calculate a list of extra strong Lucas pseudoprimes, set Q = 1. Then try P = 3, 4, 5, 6, ..., until a value of $D=P^2-4Q$ is found so that the Jacobi symbol $\left(\tfrac{D}{n}\right)=-1$. 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 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.

If n is an odd prime and $\left(\tfrac{D}{n}\right)=-1$, then we have the following (see equation 2 on page 1392 of ):

$\text{ } (3) \text{ } V_{n+1} \equiv 2 Q \pmod {n} .$

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 V_n+1 was computed in the process of computing U_n+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 $\left(\tfrac{D}{n}\right)$ 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 $Q^{n+1}$ is computed during the calculation of $U_{n+1}$. It would be easy to save the previously-computed power of $Q$, namely, $Q^{(n+1)/2}$.

Next, if n is prime, then, by Euler's criterion,

$Q^{(n-1)/2} \equiv \left(\tfrac{Q}{n}\right) \pmod {n}$ .

(Here, $\left(\tfrac{Q}{n}\right)$ is the Legendre symbol; if n is prime, this is the same as the Jacobi symbol). Therefore, if n is prime, we must have

$\text{ } (4) \text{ } Q^{(n+1)/2} \equiv Q \cdot Q^{(n-1)/2} \equiv Q \cdot \left(\tfrac{Q}{n}\right) \pmod {n}$ .

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.

Additional congruence conditions that must be satisfied if n is prime are described in Section 6 of. If any of these conditions fails to hold, then we have proved that n is not 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.

There is a similar probability estimate for the strong Lucas probable prime test.

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.

By combining a Lucas pseudoprime test with a Fermat primality test, say, to base 2, one can obtain very powerful probabilistic tests for primality, such as the Baillie-PSW primality test.

Fibonacci pseudoprimes

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 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.

If n is prime and GCD(n, Q) = 1, then (see equation 4 on page 1392 of ) we also have

$\text{ } (5) \text{ } V_n \equiv P \pmod {n} .$

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).

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 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:

1. n is also a Carmichael number
2. 2(p_i + 1) | (n − 1) or 2(p_i + 1) | (n − p_i) for every prime p_i dividing n.

The smallest example of a strong Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331.

Pell pseudoprimes

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 U_n 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:

$\text{ } U_n \equiv \left(\tfrac{2}{n}\right) \pmod {n}$

with (P, Q) = (2, -1) again defining U_n 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, ...

External links

• 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.
• Weisstein, Eric W., "Lucas Pseudoprime", MathWorld.
• Weisstein, Eric W., "Strong Lucas Pseudoprime", MathWorld.
• Weisstein, Eric W., "Extra Strong Lucas Pseudoprime", MathWorld.