Is there any integer multiple of N=2^2004 that includes no zeroes in its decimal representation?
2^2004 is a 604digit 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 twentyfifths of these do not contain a zero.
Similarly, considering the last 3 digits, 81/125 of the last3digit 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 20040827 15:13:48 