These three numbers are the only known solutions y^q of the NagellLjunggren equation (b^m1)/(b1) = y^q with y > 1, q > 1, b > 1, m > 2. Yann Bugeaud and Maurice Mignotte propose two alternative conjectures:
A) The NagellLjunggren equation has only these three solutions.
Considering the current state of our knowledge, this conjecture seems too ambitious, while the next one seems more reasonable.
B) The NagellLjunggren equation has only a finite number of solutions.
This last conjecture is true if the abc conjecture is true (see article BugeaudMignotte in link (p. 148).
Consequence: 121 is the only known square of prime which is Brazilian.
There are no other solutions for some base b < 10000.
Some theorems and results about this equation:
With the exception of the 3 known solutions,
1) for q = 2, there are no other solutions than 11^2 and 20^2,
2) there is no other solution if 3 divides m than 7^3,
3) there is no other solution if 4 divides m than 20^2.  Bernard Schott, Apr 29 2019
a(4) > 10^25 if it exists using constraints above.
In the NagellLjunggren equation, we need b > 2. If b = 2, we get y^q = 2^m  1 which by Catalan's conjecture has no solutions (see A001597). (End)
