In 2009, Luca and Nicolae [[Formula: see text], Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another number are [Formula: text]. 2015, Faye [Pell a Pell number, Publ. Inst. Math. 101(115) (2017) 231–245] text] Here, we add to these two results prove for any fixed natural if define sequence as all then solution Diophantine equation

It is well-known that the Euler product for the Riemann zeta function ζ(s) is still valid for !(s) = 1 and s "= 1. In this paper, we extend this result to zeta functions of number fields. In particular, the Dedekind zeta function ζk(s) for any algebraic number field k and the Hecke zeta function ζ(s,χ) for the rational number field are shown to have the Euler product on the line !(s) = 1 except...

Recently (see [1]) I has introduced an interesting the Euler-Barnes multiple zeta function. In this paper we construct the q-analogue of Euler-Barnes multiple zeta function which interpolates the q-analogue of Frobenius-Euler numbers of higher order at negative integers. §