Sunday, 7 June 2009
New Insights via Old Problems
While sending me some old documents recently in my quest to track down the first use of !n for the subfactorial symbol, Dave Renfro sent me an electronic copy of an old "Mathematical Questions from the Educational Times", 1880. With time on my hands I was looking through them when I came upon the one above. Not a remarkable problem, but it did remind me of a recent problem I came across that said something like "Prove that 2009 can be written as the difference of squares of two integers."
A no brainer, every odd number is the difference of two squares since (n+1)^2 -n^2 = 2n+1. But after reading the old problem, I began to wonder if 2009 could not be done as the difference of two squares in more than one way. As it turned out, it could... in fact, it could be expressed in that form in three ways... 10052 - 10042, and 1472-1402, and again as 452-42. Perhaps it would not have registered on me, but I noticed that the differences between the squares, 1, 7, and 41 were all factors of 2009.
So I tried 2007 = 32(223) and sure enough it could be written as 10042-10032, or as 3362-3332, and also as 1162-1072; differences of 1, 3, and 9. The remaining factors, 223, or 669 (and of course 2007) were too big to be differences (later I realized they were sums, duh).
SO why??.. a moment's thought made me blush at my own ignorance... of course, algebra one... if a2-b2 = n then factoring the difference of the squares gave us (a-b)(a+b)= n so a-b must be a factor of n... and suddenly I knew that 1999and 2011 could only be expressed as a difference of squares in one way; they are both prime. [Followup question for students, prove that neither of 1999 nor 2011 can be expressed as the sum of two squares... also show that 2009 CAN be expressed as the sum of squares, but in how many ways?]
Its a simple idea, so why did I have to struggle with a very old, very much harder problem to have it fall apart for me??? Now I'm going to go think about the even numbered years..... hmmm... but ONLY the leap years.