A new kind of Diophantine equations
A specially multiplicative arithmetic function is the Dirichlet convolution of two completely multiplicative arithmetic functions. The aim of this paper is to prove explicitly that two mathematical objects, namely -Fibonacci sequences and specially multiplicative prime-independent arithmetic functions, are equivalent in the sense that each can be reconstructed from the other. Replacing one with another, the exploration space of both mathematical objects expands significantly.
We show that any factorization of any composite Fermat number into two nontrivial factors can be expressed in the form for some odd and , and integer . We prove that the greatest common divisor of and is 1, , and either or , i.e., for an integer . Factorizations of into more than two factors are investigated as well. In particular, we prove that if then and .
It is clear that every rational surgery on a Hopf link in -sphere is a lens space surgery. In this note we give an explicit computation which lens space is a resulting manifold. The main tool we use is the calculus of continued fractions. As a corollary, we recover the (well-known) result on the criterion for when rational surgery on a Hopf link gives the -sphere.