All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Subtract 1, get a square (Posted on 2009-06-24)
By subtracting 1 from the positive base N integer having the form XYXYXYZY, we get a perfect square. It is known that each of X, Y and Z represents a different base N digit from 0 to N-1, and X is nonzero.

What are the integer value(s) of N, with 3 ≤ N ≤ 16 for which this is possible?

 No Solution Yet Submitted by K Sengupta No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 extended computer list | Comment 3 of 4 |
` `
`                  Decimal n    x  y  z  square       square root 3    1  2  0  4096          64`
` 5    3  2  0  276676        526`
` 6    4  2  0  1247689       1117`
` 7    5  2  0  4443664       2108`
` 8    6  2  0  13315201      3649`
` 9    7  2  0  34975396      5914`
`10    4  5  6  45454564      674210    8  2  0  82828201      9101`
`11    9  2  0  180418624     13432`
`12   10  2  0  366837409     19153`
`13   11  2  0  704053156     26534`
`14   12  2  0  1286585161    35869`
`15   13  2  0  2253970576    47476`
`16   14  2  0  3806519809    61697`
`17   15  2  0  6224894404    78898`
`18   16  2  0  9894081961    99469`
`19   17  2  0  15332382976   123824`
`20   18  2  0  23226064801   152401`
`21   19  2  0  34470378244   185662`
`22   20  2  0  50217672649   224093`
`23   21  2  0  71933385616   26820423   21 19 20  74454762496   272864`
`24   22  2  0  101460723841  318529`
`25   23  2  0  141094891876  375626`
` `
`DEFDBL A-ZCLSFOR n = 2 TO 25  psn(0) = 1  FOR i = 1 TO 7    psn(i) = n * psn(i - 1)  NEXT  FOR x = 1 TO n - 1   xpart = x * (psn(7) + psn(5) + psn(3))  FOR y = 0 TO n - 1   ypart = y * (psn(6) + psn(4) + psn(2) + 1)  IF y <> x THEN  FOR z = 0 TO n - 1  IF z <> y AND z <> x THEN    zpart = z * n    xyxyxyzy = xpart + ypart + zpart    sr = INT(SQR(xyxyxyzy - 1) + .5)    IF sr * sr + 1 = xyxyxyzy THEN     IF n <> nPrev THEN PRINT     PRINT USING "##   ## ## ##"; n; x; y; z;     PRINT , xyxyxyzy - 1, SQR(xyxyxyzy - 1)     nPrev = n    END IF  END IF  NEXT  END IF  NEXT  NEXTNEXT n`

 Posted by Charlie on 2009-06-24 17:03:37

 Search: Search body:
Forums (0)