An infinite family of pairs of imaginary quadratic fields with both class numbers divisible by five

We construct a new infinite family of pairs of imaginary quadratic fields with both class numbers divisible by five. Let n be a positive integer that satisfy n≡±3(mod500) and n≢0(mod3). We prove that 5 divides the class numbers of both Q(2−Fn) and Q(5(2−Fn)), where Fn is the nth Fibonacci number.