Finding -strong pseudoprimes

Let be odd primes and . Put

Then we call the kernel, the triple the signature, and the height of , respectively. We call a -number if it is a Carmichael number with each prime factor . If is a -number and a strong pseudoprime to the bases for , we call a -spsp . Since -numbers have probability of error (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of (the smallest strong pseudoprime to all the first prime bases). If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement.

In this paper, we first describe an algorithm for finding -spsp(2)'s, to a given limit, with heights bounded. There are in total -spsp's with heights . We then give an overview of the 21978 - spsp(2)'s and tabulate of them, which are -spsp's to the first prime bases up to ; three numbers are spsp's to the first 11 prime bases up to 31. No -spsp's to the first prime bases with heights were found. We conjecture that there exist no -spsp's to the first prime bases with heights and so that

which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those -spsp's is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates of -spsp's and their prime factors to Miller's tests, and obtain the desired numbers. At last we speed our algorithm for finding larger -spsp's, say up to , with a given signature to more prime bases. Comparisons of effectiveness with Arnault's and our previous methods for finding -strong pseudoprimes to the first several prime bases are given.

     

Finding strong pseudoprimes to several bases. II

Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Upper bounds for were first given by Jaeschke, and those for were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).

In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) to the first five or six prime bases, which have the form with odd primes and ; then we tabulate all Carmichael numbers , to the first six prime bases up to 13, which have the form with each prime factor . There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for and are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

We conjecture that

and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor and propose necessary conditions on to be a strong pseudoprime to the first prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.

     

Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.     

用Dickson多项式构造差集

最近,Dillon和Dobbertin证明了在有限域Fq(q=2m)的乘法群中,多项式(x+1)d+xd+1(其中d=22k-2k+1)的像集是一个新的具有Singer参数的循环差集.利用有限域上的Fourier分析,本文证明了在有限域Fq(q=2m)的乘法群中,一些用Dickson多项式构造的集合是具有Singer参数的循环差集.     

Embeddings of

We show that there is only one embedding of in at the prime , up to self-maps of . We also describe the effect of the group of self-equivalences of at the prime on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of whose homotopy fixed point set coincide with

     

The Rabin-Monier theorem for Lucas pseudoprimes

We give bounds on the number of pairs with such that a composite number is a strong Lucas pseudoprime with respect to the parameters .

     

Reducibility of translates of Dickson polynomials

Let be a field and . The Dickson polynomial is characterized by the equation . We prove that is reducible if and only if there is a prime such that for some , or and for some . This result generalizes the well-known reducibility criterion for binomials; and it provides a reducibility criterion for where denotes the Chebyshev polynomial of degree .