Bounded Apollonian circle packings (ACP’s) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In [S1], Sarnak proves that there are inf...

Let p be any prime and a be a positive integer.

The security of public key encryption such as RSA scheme relied on the integer factoring problem. The security of RSA algorithm is based on positive integer N, because each transmitting node generates pair of keys such as public and private. Encryption and decryption of any message depends on N. Where, N is the product of two prime numbers and pair of key generation is dependent on these prime ...

Let p be any prime and a be a positive integer.

In this article we classify actions of the groups Z pk (p is a prime integer) on compact oriented surfaces.

Primality test, prime In this paper we prove the following simplification of Miller's primality criterion [2]. Theorem 1. Assume that for every integer d that is 1 mod 4 and either prime or the product of two primes, the L-function I= & (k/d) l kmS satisfies the generalized Riemann hypothesis, where {kid) denotes the Jacobi symbol, defined below. Let n be an odd integer, n > I , and write n-I-=...