These numbers have been on my mind for 45 years, since I was a senior in High School. I can’t just let them go, they will probably be the last thing on my mind as a draw my final breath -
I first saw these numbers in the book "Fun with Mathematics" by Jerome S. Meyer, published in 1961. I have had them taped to my computer terminal at work for years.The first two numbers are my favorites, I have been waiting for inspiration or insight, I want these numbers to talk to me - but nothing. They both have four sets of two numbers that should be telling me something 58, 13/31, 27/72, 64, some simple idea of shape, form or pattern. Mathematics, after all, has been described as the study of all possible patterns, therefor I should be able to ask and logically expect an answer to why these patterns occur and what they signify. This isn’t just my flight-of-fancy, W. W. Sawyer wrote in "Prelude to Mathematics," (1982) "Where there is pattern there is significance. If in mathematical work of any kind we find a certain striking pattern recurs, it is always suggested that we should investigate why it occurs." The second two numbers just hate me and I don’t expect anything out of their apparent chaos.
Now for some mathematical mysticism. You may have noticed that each of the four numbers has eight digits, 1 through 8 that do not repeat. If you multiply any of these numbers by 9 your answer will be nine digits, 1 through 9 that do not repeat. Take it up a step, if you multiply any of the original four numbers by 18 you will get a ten digit answer with numbers 1 through 0 that do not repeat. Now press your luck, try to go three for three - a pattern is emerging, will it continue? Unfortunately no, no other multiples of nine appear to give any results other than chaos, but I didn’t try everything.
Back to the original problem, the four sets of numbers that appear in the first two numbers. My gut feeling is, this is more than random chance, but I do not have a clue as to why. If you have any ideas – reply.
p.s. note, this is a "Recreational Number Theory" post, emphasis on "recreational," not a hard core mathematics post.