Posted: Apr 09, 2010 7:28 pm
Ok. I see there has been some continuity with previous conversation (or something). As I said, I am presenting details promised a week ago in a series of posts. This post continues where the last left off, with a palindrome of marked coincidence. This time--the first thing--the discovery is only a partial re-discovery. After that, I transition to the subject of things known prior to my involvement.

You can quickly find the details of the one that I partially re-discovered by going to the website I linked to in one of the overnight posts. The number is 98689, and my partial re-discovery was that this is the first prime which remains prime upon placing 1, 2, 3, 4, or 5 zeros between each pair of consecutive digits.

I have definitely found that others before me have certainly discovered some outlandish mathematical coincidences in base ten previously. Here is the biggest one that perhaps says something about how our digits are drawn: The largest power which uses no digit twice is the square of 99066 ( 9814072356), and the number of squares (not counting 1) of that sort is 609. The base ten also has the nice feature on this subject that 229--a much higher power than for any other small base--is of this type (though this seems barely worth mention).

Any mathematically informed reader will know the first ten digits of the base of the natural logarithms. In the context of all of the coincidences, the question can almost be raised as to whether logarithms are a 'natural' thing for mathematicians to use. In any case, I speculate that e=2.718281828... (It does not continue repeating) is the first real notable base-ten coincidence that was ever found, though defining that notability would be an outrageous task. On a related note, mathematicians will generally know the fact that log102=0.30102999... (and log105=0.69897000..., et cetera). These facts are enough or almost enough in themselves for a little bit of speculation on the true origin of our having gravitated in large measure toward base-ten usage.

At this point, I'm going to break and organize the next post. I recommend a look at the rather poor article (currently) http://en.wikipedia.org/wiki/mathematical_coincidences. You will find an additional nice one involving the reciprocal of 17 and perhaps a few others on base ten that I wrongly think of as insignificant. The article is mostly about other things which the reader may find interesting too.

Edit: Oh, yeah! While you're at the article, read the article linked to by the Noam Elkies I have earlier mentioned. It won't be easy-going for a non-mathematician, but it's well worthwhile. It's about 'why' the square root of 10 and pi are reasonably close to each other.