In number theory, a Frobenius pseudoprime is a pseudoprime that passes a specific probable prime test described by Jon Grantham in a 1998 preprint and published in 2000.
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It has been studied by other authors for the case of quadratic polynomials.
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Frobenius pseudoprimes w.r.t. quadratic polynomials[edit]

Frobenius pseudoprimes are defined with respect to a fixed monic polynomial. The case of a degree-2 (quadratic) polynomial x2−Px+Q{displaystyle scriptstyle x^{2}-Px+Q}, where D=P2−4Q{displaystyle scriptstyle D=P^{2}-4Q} is not a square, is common and can be expressed in terms of Lucas sequences Un(P,Q){displaystyle U_{n}(P,Q)} and Vn(P,Q){displaystyle V_{n}(P,Q)}, leading to fast implementations for testing pseudoprimality.

A composite number n is a Frobenius (P,Q){displaystyle (P,Q)} pseudoprime if and only if gcd(n,2QD)=1{displaystyle textstyle gcd(n,2QD)=1},

(1)Un−k(P,Q)≡0(modn){displaystyle (1)qquad U_{n-k}(P,Q)equiv 0{pmod {n}}}

and

(2)Vn−k(P,Q)≡{2Q(modn)if k=−12(modn)if k=1,{displaystyle (2)qquad V_{n-k}(P,Q)equiv {begin{cases}2Q{pmod {n}}&{mbox{if }}k=-1\2{pmod {n}}&{mbox{if }}k=1{mbox{,}}end{cases}}}

where k=(Dn){displaystyle scriptstyle k=left({tfrac {D}{n}}right)} is the Jacobi symbol.

Both conditions hold for all primes, hence this constitutes a probable prime test.

Condition (1) is the same condition that defines a Lucas pseudoprime, hence every Frobenius (P,Q){displaystyle (P,Q)} pseudoprime is also a Lucas (P,Q){displaystyle (P,Q)} pseudoprime, but because of the additional condition (2), the converse is not true.

Example[edit]

Frobenius pseudoprimes with respect to the Fibonacci polynomial x2−x−1{displaystyle scriptstyle x^{2}-x-1} are determined in terms of the Fibonacci numbers Fn=Un(1,−1){displaystyle F_{n}=U_{n}(1,-1)} and Lucas numbers Ln=Vn(1,−1){displaystyle L_{n}=V_{n}(1,-1)}. Such Frobenius pseudoprimes form the sequence:

4181, 5777, 6721, 10877, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 90061, 96049, 97921, 100127, 113573, 118441, 146611, 161027, 162133, 163081, 186961, 197209, 219781, 231703, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 330929, 399001, 430127, 433621, 438751, 489601, … (sequence A212424 in the OEIS).

While 323 is the first Lucas pseudoprime with respect to the Fibonacci polynomial x2−x−1{displaystyle scriptstyle x^{2}-x-1}, the first Frobenius pseudoprime with respect to the same polynomial is 4181 (Grantham indicates it is 5777^[2] but multiple authors have noted this is incorrect and is instead the first pseudoprime with (5n)=−1{displaystyle left({tfrac {5}{n}}right)=-1} for this polynomial^[3]).

Another case, Frobenius pseudoprimes with respect to the quadratic polynomial x2−3x−1{displaystyle scriptstyle x^{2}-3x-1} can be determined using the Lucas (3,-1) sequence and are:

119, 649, 1189, 4187, 12871, 14041, 16109, 23479, 24769, 28421, 31631, 34997, 38503, 41441, 48577, 50545, 56279, 58081, 59081, 61447, 75077, 91187, 95761, 96139, 116821, 127937, 146329, 148943, 150281, 157693, 170039, 180517, 188501, 207761, 208349, 244649, 281017, 311579, 316409, 349441, 350173, 363091, 371399, 397927, 423721, 440833, 459191, 473801, 479119, 493697, ….

In this case, the first Frobenius pseudoprime with respect to the quadratic polynomial x2−3x−1{displaystyle scriptstyle x^{2}-3x-1} is 119, which is also the first Lucas pseudoprime with respect to the same polynomial. Besides, (13119)=−1{displaystyle left({tfrac {13}{119}}right)=-1}.

The quadratic polynomial x2−3x−5{displaystyle scriptstyle x^{2}-3x-5}, with (P,Q)=(3,−5){displaystyle scriptstyle (P,Q)=(3,-5)}, has sparse pseudoprimes as compared to many other simple quadratics. Using the same process as above, we get the sequence:

13333, 44801, 486157, 1615681, 3125281, 4219129, 9006401, 12589081, 13404751, 15576571, 16719781, ….

Notice there are only 3 such pseudoprimes below 500000, while there are many Frobenius (1, −1) and (3, −1) pseudoprimes below 500000.

Every entry in this sequence is a Fermat pseudoprime to base 5 as well as a Lucas (3, −5) pseudoprime, but the converse is not true: 642001 is both a psp-5 and a Lucas (3,-5) pseudoprime, but is not a Frobenius (3, −5) pseudoprime. (note that Lucas pseudoprime for a (P, Q) pair need not to be a Fermat pseudoprime for base Q or base −Q, e.g. 14209 is a Lucas (1, −3) pseudoprime, but not a Fermat pseudoprime for base 3.

Using parameter selection ideas first laid out in Baillie and Wagstaff (1980)^[5]
as part of the Baillie-PSW primality test and used by Grantham in his quadratic Frobenius test,^[6]
one can create even better quadratic tests. For instance, for the parameters (P,2), where P is the first odd integer that satisfies (Dn)=−1{displaystyle scriptstyle left({tfrac {D}{n}}right)=-1}, there are no pseudoprimes below 264{displaystyle scriptstyle 2^{64}}.

Relations to other pseudoprimes[edit]

For quadratic polynomials x2−Px+Q{displaystyle scriptstyle x^{2}-Px+Q}, every Frobenius (P,Q) pseudoprime is also a Lucas (P,Q) pseudoprime.
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This immediately follows from condition (1) which defined a Lucas (P,Q) pseudoprime.
The converse is not true, making the Frobenius pseudoprimes a subset of the Lucas pseudoprimes for a given (P,Q). The condition on Vk{displaystyle scriptstyle V_{k}} indicate it must also be a Dickson pseudoprime of the second kind.^[7]

Every Frobenius pseudoprime to x3−rx2+sx−1{displaystyle x^{3}-rx^{2}+sx-1} is also a Perrin pseudoprime.^[2]

Alternate formulations[edit]

An alternate formulation is given by Khashin.^[8] Given a number n, not a perfect square, where c is the smallest odd prime with Jacobi symbol (cn)=−1{displaystyle left({tfrac {c}{n}}right)=-1}, n is either a prime or Frobenius pseudoprime if:

(1+c)n≡(1−c)(modn){displaystyle (1+{sqrt {c}})^{n}equiv (1-{sqrt {c}}){pmod {n}}}.

Note the additional condition of choosing a parameter based on the Jacobi symbol finding a quadratic non-residue. This is similar to the method from Baillie and Wagstaff shown in the examples section.^[5] This makes far stronger tests, and is one reason for the success of the Baillie-PSW primality test. Similar to the example, Khashin notes that no pseudoprime has been found for his test. He further shows that any that exist under 2⁶⁰ must have a factor less than 19 or have c > 128.

Strong Frobenius pseudoprimes[edit]

Strong Frobenius pseudoprimes are also defined.^[2] Details on implementation for quadratic polynomials can be found in Crandall and Pomerance.^[3]

Properties[edit]

The computational cost of the Frobenius pseudoprimality test with respect to quadratic polynomials is roughly three times the cost of a strong pseudoprimality test (e.g. a single round of the Miller-Rabin primality test), 1.5 times that of a Lucas pseudoprimality test, and slightly more than a Baillie-PSW primality test.

Note that the quadratic Frobenius test is stronger than the Lucas test. For example, 1763 is a Lucas pseudoprime to (P, Q) = (3, -1) since U₁₇₆₄(3,-1) ≡ 0 (mod 1763) (U(3,-1) is given in OEIS: A006190), and it also passes the Jacobi step since (131763)=−1{displaystyle left({tfrac {13}{1763}}right)=-1}, but it fails the Frobenius test to x² - 3x - 1. This property can be clearly seen when the algorithm is formulated as shown in Crandall and Pomerance Algorithm 3.6.9^[3] or as shown by Loebenberger,^[4] as the algorithm does a Lucas test followed by an additional check for the Frobenius condition.

While the quadratic Frobenius test does not have formal error bounds beyond that of the Lucas test, it can be used as the basis for methods with much smaller error bounds. Note that these have more steps, additional requirements, and non-negligible additional computation beyond what is described on this page. It is important to note that the error bounds for these methods do not apply to the standard or strong Frobenius tests with fixed values of (P,Q) described on this page.

Based on this idea of pseudoprimes, algorithms with strong worst-case error bounds can be built. The quadratic Frobenius test,^[6] using a quadratic Frobenius test plus other conditions, has a bound of 17710{displaystyle {tfrac {1}{7710}}}. Müller in 2001 proposed the MQFT test with bounds of essentially 1131040t{displaystyle {tfrac {1}{131040^{t}}}}.
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Damgård and Frandsen in 2003 proposed the EQFT with a bound of essentially 256331776t{displaystyle {tfrac {256}{{331776}^{t}}}}.
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Seysen in 2005 proposed the SQFT test with a bound of 14096t{displaystyle {tfrac {1}{{4096}^{t}}}} and a SQFT3 test with a bound of 16336442t{displaystyle {tfrac {16}{336442^{t}}}}.
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Given the same computational effort, these offer better worst-case bounds than the commonly used Miller-Rabin primality test.

See also[edit]

• Pseudoprime


• Lucas pseudoprime


• Ferdinand Georg Frobenius


• Quadratic Frobenius test

References[edit]

External links[edit]

• Weisstein, Eric W. "Frobenius Pseudoprime". MathWorld.


• Weisstein, Eric W. "Strong Frobenius Pseudoprime". MathWorld.


• Jacobsen, Dana Pseudoprime Statistics, Tables, and Data (data for Frobenius (1,-1) and (3,-5) pseudoprimes)