Talk:Frobenius pseudoprime

Merger proposal
This article and strong Frobenius pseudoprime are closely related and the latter is very thin. I propose to merge them. Richard Pinch (talk) 20:22, 20 August 2008 (UTC)
 * merged phoebe / (talk to me) 01:09, 2 November 2008 (UTC)

explains nothing
This article does not define what a Frobenius pseudoprime is at all. For example, reading this article does not help in any way to check the claimed result that 4181 is an example.--Hagman (talk) 15:10, 16 October 2013 (UTC)


 * A definition is given on page 9 in the first reference at http://mathworld.wolfram.com/FrobeniusPseudoprime.html. It reads:


 * A Frobenius pseudoprime with respect to a monic polynomial f(x) is a composite which is a Frobenius probable prime with respect to f(x).


 * Perhaps that definition should be added to the article. It should be noted that the exact definition depends on the parameters chosen for the quadratic Frobenius test. --  Toshio   Yamaguchi  15:52, 16 October 2013 (UTC)

The first Frobenius pseudoprime to x^2-3x-1
649, 4187, 12871, 14041, 23479, 24769, 28421, 34997, 38503, 41441, 48577, 50545, 58081, 59081, 61447, 95761, 96139, 116821, 127937, 146329, 150281, 157693, 170039, 180517, 188501, 281017, 321441, 349441, 363091, 397927, 423721, 440833, 473801, 478401, 479119, 493697, 507529, 545273, 550551, 558145, 561601, 587861, 597871, 625049, 665473, 711361, 712481, 749057, 841753, 842401, 860161, 888445, 930151, 979473, 1019041, 1034881, 1115687, 1152271, 1153741, 1184401, 1241633, 1252033, 1270801, 1351633, 1361837, 1373633, 1374649, 1477909, 1493857, 1531531, 1548481, 1553473, 1578977, 1596403, 1599329, 1699201, 1703677, 1755001, 1758121, 1822285, 1854841, 1879809, 1920985, 1987021, 2030341, 2132737, 2250767, 2260021, 2288209, 2290289, 2320501, 2480689, 2508013, 2525563, 2538251, 2590981, 2639329, 2908181, 2931673, 2984983, 3024505, 3057601, 3141841, 3362083, 3383353, 3384001, 3385041, 3575121, 3581761, 3629857, 3717419, 4082653, 4137251, 4224533, 4231681, 4335241, 4411837, 4555651, 4682833, 4835377, 4972033, 5000449, 5002013, 5160013, 5252101, 5263553, 5419751, 5430643, 5434153, 5604161, ... — Preceding unsigned comment added by 101.14.112.72 (talk) 14:38, 14 March 2015 (UTC)

Is 58081 a Frobenius pseudoprime to x^2-3x-1 ?
According to the Quadratic Frobenius test article, if sqrt(n) is an integer, then it return composite immediately, and sqrt(58081) is an integer, so 58081 should not be a Frobenius pseudoprime. — Preceding unsigned comment added by 101.8.115.219 (talk) 16:19, 13 August 2015 (UTC)
 * The QFT is a different test, and should not be confused with the Frobenius test with respect to a quadratic polynomial. It adds numerous extra conditions on top of the Frobenius criteria.  The naming is unfortunate.  I don't see anything about a perfect square test of n in either this article's description or in his paper.  It's an interesting point though, as these tests aren't uncommon.  DAJ NT (talk) 19:57, 18 August 2015 (UTC)

Probable prime test

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

This statement raises some stochastics 101 red flags. My understanding is that a probabilistic test must be parameterized (check) but also have the property that no composite number can pass it for all possible choices of parameters (which I think holds for Frobenius but is not stated). The important part is that the probability in probabilistic is defined as the relative frequency of parameters for which a given composite number $$n$$ is detected as such, over all possible parameter choices, and it must be greater than zero for all $$n$$ in order to have a truly probabilistic test. E.g. if you take the (original?) definition $$\forall a: a^n \equiv a \pmod n$$ of Fermat's little theorem for prime numbers $$n$$, then I would not consider this a probabilistic test because the probability of detecting a Carmichael number as composite for random choice of parameter $$a$$ is zero. If the test is rephrased as $$\forall a: a \not \equiv 0 \pmod n \implies a^{n-1} \equiv 1 \pmod n$$, however, then it is a probabilistic test because any $$a$$ that is not coprime to $$n$$ will be a witness against it, and any Carmichael number then has such witnesses. (For a probabilistic test to be strong, there must also be a lower probability bound that holds for all $$n$$, e.g. Solvay-Strassen or Miller-Rabin.) Aragorn2 (talk) 11:05, 28 April 2019 (UTC)