Sums of Fourth Powers of Digits

I have played around with the sums of digits of numbers, in particular I have written about the patterns that emerge with the sums of the squares, and the sums of the cubes of the digits of a number.

So recently Derek Orr started playing around with the fourth powers. Rather than try to keep up with him, I just asked him to guest post his work in progress. Derek found, as I had in a brief survey, that for most numbers, the sum of the fourth powers fall into an orbit cycling through 13,139.  Here is what he had so far:

Digits to the Fourth Power
By Derek J. Orr, University of Pittsburgh

I was perusing through Pat’s blog here and I noticed two pages that involved summing the squares of digits and summing the cubes of digits. I, then, proposed taking the fourth power of these digits. When I did so, I found some pretty interesting results. Unfortunately, it is time consuming to do each of these numbers manually but I plan to get up to 1,000 and maybe even 10,000 (if motivation and free time permit).

Some numbers that I found went to 1 and are called “happy numbers”. Unfortunately, the only ones I’ve found were 1, 10 and 100 (I went up to 225 so far). However, I am assuming the next smallest happy number is 1,000 just because it’s hard to find one. I wanted to find one and, since I couldn’t, I forced myself to.

I know that a happy number is a number that will eventually reduce to the number 1 and repeat forever. So, the guaranteed happy numbers are 1, 10, 100, 1000, 10000, etc. Thus, I figured out what numbers can get me to these guaranteed happy numbers and, though they aren’t small, I found a few. First, I’ll write out the fourth powers of the single digit numbers:

Using these values, I found different combinations that could get me to the guaranteed happy numbers. Also, since we have 1 as a possibility, it’s impossible to skip any number I choose. The table below lists a few combinations that I found.



The numbers inside the table represent how many of the digits we need. So, for the first line of the table, the number 1,111,111,111 is a happy number because added together 10 times gets us 10. From this table, the smallest happy number that is not a multiple of 10 (and that is not 1) is 11,123. However, in just two steps, this number can reach 1. What if there are longer iterations that still get us to 1? What if there is a number that can get us 11,123? So, I experimented:

(the other equation I found has 18 digits, so I won’t write it out)

So, we see that there is a new happy number, 22,233,489. Once again, what numbers get to this number? I could do this all day but I won’t, mainly because 22,233,489 can be divided by over 3,388 times. So, this means that the next number has more than 3,388 digits, which is far too many. I’ve assumed that up to now, the numbers will only get bigger. However, what if we tried a different number instead of 11,123? Since the digits are added, we can always change the order. So, 12113, 13112, 21113, 31112, 12131, 13121, 12311, 13211, etc. are also happy numbers. Again, I could experiment on these but I won’t just to save time. I believe it is safe to say that the smallest happy number without a zero (and that is not 1) is 11,123. Since the happy numbers are so hard to find, I looked at different numbers.

One loop that I found was with the number 2,178. I saw it for the first time when I tried the number 127. Here is the iteration for 127.

127 -- 2418 -- 4369 -- 8194 --10914 -- 6819 -- 11954 -- 7444 -- 3169 -- 7939 -- 15604 -- 2178 -- 6514 -- 2178 -- 6514 --…

So, we can see it goes through this 2,178----6,514 loop. Now, I’ve only seen this work for 127, 172, and 217 (so far) but the four- and five-digit numbers above will also bring about this loop.

Also, I found that there seems to be a fixed point where some numbers end up at, similar to what Pat found when cubing the digits. This number was 8,208, and satisfies the condition . What numbers gave me 8,208? I computed a bit more than the first 200 digits (225 to be exact) and found the numbers 12, 17, 21, 46, 64, 71, 102, 107, 120, 137, 145, 154, 170, 173, 201, 210, and 224 (and of course 288) will get to 8,208 and stay there. Now, past these, it’s obvious that 317, 415, 514, 710, 701, 713, 422, etc. also work. When you cubed the digits, Pat found that you reach the number 153 and it repeats forever. However, Pat found that if you have any multiple of three, you will reach 153. There is a pattern there. I am sadly not able to find any pattern with these; they seem to be random, like the happy numbers when you square the digits (1, 7, 10, 13, 19, 23, 28…etc.).

Another loop number these could go to is 13,139. With 13,139, there is a loop involved (13139 -- 6725 -- 4338 -- 4514 -- 1138 -- 4179 -- 9219 -- 13139…). This has happened with every number I haven’t mentioned (around 90% of the numbers I’ve tried).

Going back to 8208, I keep wondering if there is another fixed point. When cubing the digits, there are five numbers that equal the sum of their digits cubed: 1, 153, 370, 371, and 407. When squaring the digits, there is only one number that equals the sum of its digits squared: 1. But, when taking the fourth power, I have only found two numbers that work: 1 and 8,208. I do wonder if there are more or not; perhaps a good problem for a computer programmer because doing these manually, though possible, takes up a lot of time.

I would like to thank Pat for posting this