Teaser: What The Remainder Sequence Holds In Store
Friday August 11, 2006 by Noam Samuel
I swear that I was going to post a full article, but my family is going on a camping trip in a very short span of time. Thus, since I do not want to leave the Underground again without an article for a week, I’ve decided to post a little about an article soon to come.
You see, I was thinking about prime numbers for quite a while, and eventually I came to the conclusion that if we were to take the remainder left from divinding a number by each prime that comes before it, the number would be prime only if none of those were 0. While this might seem obvious and, quite frankly, useless, it can be quite interesting.
My next idea was to take any number n and construct a sequence of numbers representing the remainders from the division of n by each prime that is smaller than it, starting from 2 (which is, well, the smallest prime). This sequence was eventually called the remainder sequence of a number.
So, for example, the remainder sequence of 5 would be “1,2”. The remainder sequence of 15 would be “1,0,0,1,4,1”. The remainder sequence of 23 would be “1,2,3,2,1,10,6,4”.
At this point, most people wouldn’t get excited. This is, after all, just another representation of numbers. Nothing new here. But before you dismiss it as an idea deemed to the trashcan of mathematics, think of the following two question:
First of all, is it unique? We have no indication that only one number can hold each remainder sequence, but this is not to say that it isn’t true.
And, just as important, what would the implications be if it were unique?
I’ll explore both of those questions in my next article.