0, 25, 2025, 13225…are squares that remain squares if every digit in the number defining them is augmented by 1.
Let's call them squarish numbers.

a. List two more samples of squarish numbers.
b. Prove that all such numbers are evenly divisible by 25.
c. Why are there neither 3-digit nor 6-digit squarish numbers?
d. Prove that between 10^k and 10^(k+1) there is at most one squarish number.

Here are three more added to the list:

squarish     sqrt          augmented      sqrt
                             square
25           5              36            6
2025         45             3136          56
13225        115            24336         156
4862025      2205           5973136       2444
60415182025  245795         71526293136   267444
207612366025 455645         318723477136  564556

It seems lengths 8, 9 and 10 are missing also.

DEFDBL A-Z

repu = 1: rep9 = 9 * repu
FOR n = 1 TO 9999999
  sq = n * n
  WHILE sq > rep9
    repu = repu * 10 + 1
    rep9 = 9 * repu
  WEND
  s$ = LTRIM$(STR$(sq))
  IF INSTR(s$, "9") = 0 THEN
    tst = sq + repu
    sr = INT(SQR(tst) + .5)
    IF sr * sr = tst THEN
      PRINT sq; n, tst; sr
    END IF
  END IF
NEXT n

 

Posted by Charlie on 2011-03-11 13:42:19