Monday, October 12, 2015

The Thrill of Victory...

I've been looking forward to doing Problem #9 for a couple days.  It's the last problem in the problem set and I finally got to it today, having completed all the others.  It seemed simple enough:  show that the reciprocals of all (positive) divisors of a perfect number sum to 2.  For example, the divisors of the perfect number 28 are 1, 2, 4, 7, 14, and 28; 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 =2.

Seems pretty easy, when you know that a perfect number is one in which all the proper divisors (all the divisors except the number itself) add up to the number itself (1+2+4+7+14=28), so the sum of all the divisors is twice the number.

I had an intuitive sense on how to do this, but my first attempt didn't lead anywhere.  I erased it all and tried a second time.  I just didn't seem to make any progress.

Third time's a charm, right?  I had been moving in the right direction all along, I just need a small insight in order to solve the problem.  A couple of steps later, accompanied by a couple of explanatory steps, the problem was complete.

Once it was done it looked like the easiest proof in the world, but lacking the "insight" step it was next to impossible.

I felt victorious!

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