Characterizing Pseudoprimes for Third-Order Linear Recurrences
نویسندگان
چکیده
This paper continues the work begun by D. Shanks and myself in [1] where certain cubic recurrences were used to give a very strong primality test. A complete characterization of the pseudoprimes for this test is given in terms of the periods of the corresponding sequences. Then these results are used to produce various types of pseudoprimes. A discussion of open problems is included.
منابع مشابه
Pseudoprimes for Higher-order Linear Recurrence Sequences
With the advent of high-speed computing, there is a rekindled interest in the problem of determining when a given whole number N > 1 is prime or composite. While complex algorithms have been developed to settle this for 200-digit numbers in a matter of minutes with a supercomputer, there is a need for simpler, more practical algorithms for dealing with numbers of a more modest size. Such practi...
متن کاملOn the Generalized Fibonacci Pseudoprimes
In this paper the results established by the first two authors in [3], [4], and [5] are extended and generalized. After defining (in this section) classes of generalized Lucas numbers, {Vn(m)}9 governed by the positive integral parameter/??, the Fibonacci pseudoprimes of the m kind (/77-F.Psps.) are characterized in Section 2. A method for constructing them is discussed in Section 3, while some...
متن کاملNonexistence of Even Fibonacci Pseudoprimes of The
With regard to this problem, Di Porto and Filipponi, in [4], conjectured that there are no even-Fibonacci pseudoprimes of the 1 kind, providing some constraints are placed on their existence, and Somer, in [12], extends these constraints by stating some very interesting theorems. Moreover, in [1], a solution has been found for a similar problem, that is, for the sequence {Vn(2,1)}, defined by F...
متن کاملOn the Number of Elliptic Pseudoprimes
For an elliptic curve E with complex multiplication by an order in K = Q('A-Ai), a point P of infinite order on E, and any prime p with (-d \ p) = —1, we have that (p + 1) • P = O(modp), where O is the point at infinity and calculations are done using the addition law for E. Any composite number which satisfies these conditions is called an elliptic pseudoprime. In this paper it is shown that, ...
متن کاملSOME BOUNDARY VALUE PROBLEMS FOR A NON-LINEAR THIRD ORDER O.D.E.
Existence of periodic solutions for non-linear third order autonomous differential equation (O.D.E.) has not been investigated to as large an extent as non-linear second order. The popular Poincare-Bendixon theorem applicable to second order equation is not valid for third order equation (see [3]). This conclusion opens a way for further investigation.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010