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We argue about the (non) existence of {\it superregular} scalar clouds (i.e.,bound states of a massive and complex-valued scalar field $\Psi$) around exact{\it extremal} ($a = M$) Kerr black holes (BH's) possessing {\it bounded radialderivatives at the horizon} (in Boyer-Lindquist coordinates) as opposed tosimilar cloud solutions that exist but with unbounded derivatives in the samecoordinate system. The latter solutions have been reported recently bothanalytically and numerically. The superregular clouds cannot be obtained fromthe regular clouds around subextremal Kerr BH's ($|a|< M$) in the limit ofextremality $(a\rightarrow M)$ as in this limit the radial derivatives of$\Psi$ at the horizon $r_H$ diverge when $r_H\rightarrow r_H^{\rm ext}:=M=a$,thus, such superregular clouds must be analyzed separately. We conclude thatthe superregular clouds, which are found in the {\it exact} extremal scenario($a = M$), are not continuously connected with the regular ones in the limit ofextremality $(a\rightarrow M)$. Remarkably, the spectrum leading to theexistence of the radial part of the full solution of these superregular clouds(which obeys a Teukolsky equation) is given by the exact formula$M=a=\frac{1}{2\mu}\sqrt{m^2 + \left[-\kappa+\sqrt{\kappa^2+m^2}\,\right]^2}$,which depends on three (positive) integers: the principal number $n$, the {\itmagnetic number} $m$, and an integer $j$, related with the {\it type} ofregularity at the horizon. Here $\kappa= j +n$, and $\mu$ is the massassociated with $\Psi$. This spectrum depends implicitly on the {\it orbital}number $l$, an integer number that determines the existence of well behavedspheroidal harmonics which are associated with the angular part of the cloudsolution. Since the separation constants that are obtained from thesuperregularity conditions in the radial part of the solution do {\it not}coincide in general with the standard separation constants required for thespheroidal harmonics to be well behaved on the axis of symmetry, we concludethat non-trivial boson clouds having such superregularity conditions cannotexist in the background of an exact extremal Kerr BH. The only exception tothis conclusion is in the limit $n\rightarrow \infty$ and $m\ll n$. In such alarge $n$ limit consistency in the separation constants leads to a quadraticDiophantine equation of Pell's type for the integer numbers $(l,m)$. SuchPell's equation can be readily solved using standard techniques. In thatinstance well behaved spheroidal harmonics are obtained, and thus, well behavednon-trivial superregular clouds can be computed. Of course, this situation,does not preclude the existence of other kind of smooth cloud solutions for anyother $n$, not necessarily large (e.g. clouds with a non-integer $\kappa$) whenusing a better behaved coordinate system at the horizon (e.g. Wheeler'stortoise coordinate or proper radial distance).

On the (non) existence of superregular boson clouds around extremal Kerr black holes and its connection with number theory