  In 2016, in the work related to Galois representations, Greenberg conjectured
the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$
for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria
provided by him to check $p$-rationality for abelian number fields, certain
infinite families of quadratic, biquadratic and triquadratic $p$-rational
fields have been shown to exist in recent years. In this article, for any
integer $k \geq 1$, we build upon the existing work and prove the existence of
infinitely many prime numbers $p$ for which the imaginary quadratic fields
$\mathbb{Q}(\sqrt{-(p - 1)}),\ldots,\mathbb{Q}(\sqrt{-(p - k)})$ and
$\mathbb{Q}(\sqrt{-p(p - 1)}),\ldots, \mathbb{Q}(\sqrt{-p(p - k)})$ are all
$p$-rational. This can be construed as analogous results in the spirit of
Iizuka's conjecture on the divisibility of class numbers of consecutive
quadratic fields. We also address a similar question of $p$-rationality for two
consecutive real quadratic fields by proving the existence of infinitely many
$p$-rational fields of the form $\mathbb{Q}(\sqrt{p^{2} + 1})$ and
$\mathbb{Q}(\sqrt{p^{2} + 2})$. The result for imaginary quadratic fields is
accomplished by producing infinitely many primes for which the corresponding
consecutive discriminants have large square divisors and the same for real
quadratic fields is proven using a result of Heath-Brown on the density of
square-free values of polynomials at prime arguments.
