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Monday Mathematics: Power Play
Sep. 24th, 2012 08:09 pmI give up; I'm not going to be able to say anything more about my polygons research without going into a horrendous amount of detail. Sorry about that, the one or two of you who were waiting for more....
Instead of that, I'm going to talk about a nice little tidbit that I learned from Dr. Chung-wu Ho, who was then the Chair of the department. (He retired and moved west years ago.) Here it is. Let A and B be two positive whole numbers, and assume that A is not a power of 10. Then there is some power of A whose decimal representation begins with the digits of B. For example, suppose A is 5 and B is 7. The successive powers of 5 are 5, 25, 125, 625, 3125, 15625, 78125, and there we are: 57 begins with 7. But the same thing would work with A=13 and B=215, or any other pair, so long as A is not a power of 10. Unfortunately, the proof doesn't give any information about what power will do the trick; it may be very very large. The proof - which really isn't that hard, if you've had enough math to understand the concept of greatest lower bound - is under the cut.
( Mantissa Hunt )
Instead of that, I'm going to talk about a nice little tidbit that I learned from Dr. Chung-wu Ho, who was then the Chair of the department. (He retired and moved west years ago.) Here it is. Let A and B be two positive whole numbers, and assume that A is not a power of 10. Then there is some power of A whose decimal representation begins with the digits of B. For example, suppose A is 5 and B is 7. The successive powers of 5 are 5, 25, 125, 625, 3125, 15625, 78125, and there we are: 57 begins with 7. But the same thing would work with A=13 and B=215, or any other pair, so long as A is not a power of 10. Unfortunately, the proof doesn't give any information about what power will do the trick; it may be very very large. The proof - which really isn't that hard, if you've had enough math to understand the concept of greatest lower bound - is under the cut.
( Mantissa Hunt )
