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 3digit nor 6digit 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 AZ
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 20110311 13:42:19 