Some J verbs for handling repunits

is_integral =: = <.

repunit =: 3 : 0 "0
  10 repunit y
:
  NB. Gives the y-th repunit in the x base, with the result expressed
  NB. in base 10. E.g., 3 repunit 4 gives 40, because 40 in base 3 is 1111.
  ((x ^ y) - 1) % x - 1
)

which_repunit =: 3 : 0 "0
  10 which_repunit y
:
  NB. An integer solution to this equation means it's a repunit,
  NB. and returns which one it is for that base. E.g., which_repunit 111
  NB. gives the integer 3, so 111 is the third repunit in base 10.
  NB. By contrast, 112 gives 3.00389, which means it is not a repunit.
  NB.
  NB. repunit which_repunit y will always give back y, regardless of whether
  NB. y is a repunit in the given base.
  x ^. (y * (x - 1)) + 1
)

is_repunit =: 3 : 0 "0
  10 is_repunit y
:
  is_integral x which_repunit y
)