TriDigital Squares

   Tridigital squares are squares that contain at most 3 unique digits. I came across this interesting problem while looking for sequences in EIS that needed more terms. I saw A053931 which deals with the digits 2,6,9 and decided to see if I can find more terms. It didn't take too long before I found several new terms and extended the sequence to the following: 

   9
   29929
   69696
   929296
   9696996
   996222969
   26629996969
   69926926969
   269996262622969
   9222222699262629962929
   9929662926692269969296
   96296299962926222299669929
   2669292996662992629666262969
   2699929699222292222296696996
   62969966996922666266229669292996
My technique, which turned out to be very effective, involved using quadratic residues and leading significant digits. The next solution is greater than 626969699446323722047902122699929.

For the latest results on various combinations and more information about the problem I recommend the following web-sites:

  • Patrick DeGeest's site
  • Hisanori Mishima's site
    • Joe K. Crump | Number Theory Home