Is there any integer multiple of N=2^2004 that includes no zeroes in its decimal representation?

2^2004 is a 604-digit number: 183700911243880723877...194957618384470016.

The rightmost digit of its multiples goes through a cycle of 5: 6, 2, 8, 4, 0. Four fifths of these are not zero.  Its multiples' rightmost two digits go through a cycle of 25: 16  32  48  64  80  96  12  28  44  60  76  92  08  24  40  56  72  88  04  20  36  52  68  84  00.  Eighteen twenty-fifths of these do not contain a zero.

Similarly, considering the last 3 digits, 81/125 of the last-3-digit sets lack any zeros.  364/625 of the last 4 digits lack any zeros, as do 1638/3125 of the last 5 digits and 7371/15,625 of the last 6 digits.  It appears to be approaching what would be expected of a random distribution of digits.  By the time we get to 604 digits, there is a cycle of 5^604 values out of the 10^604 possible combinations of digits.  The probability that none of 604 random digits is zero is (9/10)^604 or about 1/10^27.64. But 5^604 is about 10^422.18, indicating there are about 10^394.5 of the numbers in the cycle that lack zeros.  Some digits spill over into the next decimal positions to the left, but those are controllable by adding more times 2^2004.

So, probabilistically speaking, its likely that such a multiple exists, barring some quirk about the number that prevents such.

 

Posted by Charlie on 2004-08-27 15:13:48