Constructing Carmichael numbers through improved subset-product algorithms

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p-1 divides a highly composite \Lambda.Comment: Table 1 fixed; previously the last 30 digits and number of digits were calculated incorrectl

Fibonacci primes, primes of the form 2n−k2^n-k and beyond

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes $u_n$, or else there exists a constant $c_u>0$ (which we can give good approximations to) such that there are $\sim c_u \log N$ primes $u_n$ with $n\leq N$, as $N\to \infty$. We compare our conjecture to the limited amount of data that we can compile.Comment: v2; replace earlier draft inadvertently submitted as v

Representing integers as a sum of three cubes

In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent computation indicates that the method is quite favourable to other methods in terms of time estimates. A hybrid of the method presented here and those in a previous paper is currently underway for unsolved cases.Comment: 4 page

A Probable Prime Test with High Confidence

AbstractMonier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1/4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensures that a compositenwill pass for less than 1/7710 of the polynomialsx2−bx−cwith (b2+4c|n)=−1 and (−c|n)=1. The running time of the test is asymptotically 3 times that of the Strong Probable Prime Test