Squares with the same digits

A study of square numbers up to 10^5 has revealed that …
 
 
… there is only one set of six squares that have the same digits:

          103^2 = 10609     130^2 = 16900     140^2 = 19600
          247^2 = 61009     301^2 = 90601     310^2 = 96100
 
 
 
… there is one set of five squares that have the same digits:

          128^2 = 16384     178^2 = 31684     191^2 = 36481
                   196^2 = 38416     209^2 = 43681
 
 
… there are two sets of four squares that have the same digits:

              102^2 = 10404     120^2 = 14400
              201^2 = 40401     210^2 = 44100

              133^2 = 17689     137^2 = 18769
              281^2 = 78961     286^2 = 81796
 
 
… there are nine sets of three squares that have the same digits:

          13^2 = 169     14^2 = 196     31^2 = 961

          36^2 = 1296     54^2 = 2916     96^2 = 9216

          108^2 = 11664     129^2 = 16641     204^2 = 41616

          112^2 = 12544     211^2 = 44521     229^2 = 52441

          135^2 = 18225     159^2 = 25281     285^2 = 81225

          148^2 = 21904     203^2 = 41209     302^2 = 91204

          154^2 = 23716     181^2 = 32761     269^2 = 72361

          160^2 = 25600     245^2 = 60025     250^2 = 62500

          207^2 = 42849     222^2 = 49284     288^2 = 82944
 
 

… there are 30 pairs of squares that have the same digits

we exclude trivial solutions such as   (100, 001),   (400,004),   etc.

         12^2 = 144         21^2 = 441

         16^2 = 256         25^2 = 625

         32^2 = 1024         49^2 = 2401

         33^2 = 1089         99^2 = 9801

         37^2 = 1369         44^2 = 1936

         42^2 = 1764         69^2 = 4761

         64^2 = 4096         98^2 = 9604

         101^2 = 10201         110^2 = 12100

         106^2 = 11236         146^2 = 21316

         113^2 = 12769         311^2 = 96721

         122^2 = 14884         221^2 = 48841

         144^2 = 20736         174^2 = 30276

         152^2 = 23104         179^2 = 32041

         153^2 = 23409         198^2 = 39204

         155^2 = 24025         205^2 = 42025

         157^2 = 24649         158^2 = 24964

         166^2 = 27556         275^2 = 75625

         172^2 = 29584         233^2 = 54289

         186^2 = 34596         213^2 = 45369

         190^2 = 36100         251^2 = 63001

         202^2 = 40804         220^2 = 48400

         192^2 = 36864         294^2 = 86436

         206^2 = 42436         208^2 = 43264

         216^2 = 46656         258^2 = 66564

         224^2 = 50176         226^2 = 51076

         225^2 = 50625         255^2 = 65025

         228^2 = 51984         309^2 = 95481

         266^2 = 70756         274^2 = 75076

         273^2 = 74529         282^2 = 79524

         314^2 = 98596         316^2 = 99856

 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Math Beauty, Number Puzzles and tagged , . Bookmark the permalink.

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