These three numbers are the only known solutions y^q of the Nagell-Ljunggren equation (b^m-1)/(b-1) = y^q with y > 1, q > 1, b > 1, m > 2. Yann Bugeaud and Maurice Mignotte propose two alternative conjectures:
A) The Nagell-Ljunggren 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 Nagell-Ljunggren equation has only a finite number of solutions.
This last conjecture is true if the abc conjecture is true (see article Bugeaud-Mignotte 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
From David A. Corneth, Apr 29 2019: (Start)
Intersection of A001597 and A053696.
a(4) > 10^25 if it exists using constraints above.
In the Nagell-Ljunggren 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)