We study the solutions of the equation φ(Cm)/φ(Cn) = r, where r is a fixed rational number, Ck is the kth Catalan number and φ is the Euler function. We note that the number r = 4 is special for this problem and for it we construct solutions (m,n) to the above equation which are related to primes p such that 2p− 1 or 4p− 3 is also prime. 1. An observation concerning φ(Cn+1)/φ(Cn) For a positive...

This paper examines winding number of a closed planar curve through various aspects. It is related to range on an arc in the complex plane and point not range. Functions such natures are considered be continuous real-valued functions. Conclusion was drawn by naming areas applications.

Let cn = 1 n + 1 ( 2n n ) be the nth Catalan number. In this paper, we look at some of the arithmetic properties of cn. For example, we show that w2(cn) ! log log n for all n ! 3 provided that cn is odd, where w2(m) is the Hamming weight (or the binary sum of digits) of the positive integer m. We also determine all instances in which cn is a base 2 palindrome, and prove that the first k odd Cat...

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm×Cn with n even is strongly hamiltonianlaceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn−2 unless both faulty elements are contained in the sa...