Random Cyclic Triangle-Free Graphs of Prime Order

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

Factorization of Odd Integers with a Divisor of the Form $2^au \pm 1$

The paper proves that an odd composite integer $N$ can be factorized in at most $O( 0.125u(log_2N)^2)$ searching steps if $N$ has a divisor of the form $2^a{u} +1$ or $2^a{u}-1$ with $a > 1$ being a positive integer and $u > 1$ being an odd integer. Theorems and corollaries are proved with detail mathematical reasoning. Algorithms to factorize the kind of odd composite integers are designed and tested by factoring certain Fermat numbers. The results in the paper are helpful to factorize the related kind of odd integers as well as some big Fermat numbers

On strong pseudoprimes in arithmetic progressions

A composite integer N is said to be a strong pseudoprime for the base C if with N – 1 = 2sd, (2, d) = 1 either Cd = 1, or C2r ≡ 1 (mod N) some r, 0 ≤ r < s. It is shown that every arithmetic progression ax+b (x = 0,1, …) where a, b are relatively prime integers contains an infinite number of odd strong pseudoprimes for each base C ≤ 2.1980 Mathematics subject classification (Amer. Math. Soc.): 10 A 15.

On Numbers Analogous to the Carmichael Numbers

A base a pseudoprime is an integer n such that1A Carmichael number is a composite integer n such that (1) is true for all a such that (a, n ) = l. It was shown by Carmichael [1] that, if n is a Carmichael number, then n is the product of k(>2) distinct primes P1,P2,P3, … Pk, and Pi-1|n-1(i=1, 2, 3, …, k).