maths – that endless game

…or may be not?

the fact that in the previous game we got 4 different graphs is of course a consequence of using a decimal positional system to write our numbers. what if we were using an hexadecimal system? or if we decided to work in base 13 instead? how many distinct graphs would there be for each base? how long would the transient states be before the graphs start to repeat, if they were actually guaranteed to repeat?

damn, see?, there’s always new questions to answer. that’s the (beautiful) problem with playing around with maths – once you start, you can’t stop, cause there is always a new game to be played as soon as you finish with the current one.

of course i couldn’t resist answering these questions. so, here we go, i computed all the graphs for all bases from base 2 (binary) to base 31. and indeed, for a given base, the sequence of graphs seems to get periodic. these are the results:

 base: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 … # graphs: 1 2 3 4 2 6 4 7 4 10 3 12 6 4 7 16 7 18 5 6 10 22 4 21 12 20 7 28 4 30 … period: 1 2 2 4 2 6 2 6 4 10 2 12 6 4 4 16 6 18 4 6 10 22 2 20 12 18 6 28 4 30 …

observation: for bases with simple factorization, the number of graphs and period seem to equal the biggest factor minus one. it also seems that those bases don’t have transient phase. for the bases with multiplicity in their factors (bases 4, 8, 9, 12, 16, 18, 20, 24, 25, 27 and 28), i cannot explain the numbers yet.

seems like i have not been the first human being in this territory, the sequence is apparently known as the Carmichael function, and is referenced to as A002322 in OEIS. still, it’s amazing how far you can go in the depths of maths in a matter of minutes by making a couple of simple questions to youself!