Riding the subway home on the last leg of our return from the Staten
Island Zoo yesterday, my seven year old son Dan asked me to do the
following (Dan doesn't know much terminology, so I'm including it
parenthetically):
1. Pick a Fibonacci number (Fn)
2. Square it
3. Square the Fibonacci "two before the one you picked" (F(n-2))
4. Subtract it from the first square
5. You get the Fibonacci when you add the other two (F(2n-2)).
Dan giggled triumphantly when my choice of 13 (F7) led to an answer
of 55 (F12). After trying a few more numbers, I couldn't generate a
counter-example*
Generally, when Dan sounds precocious, and I ask him if he'd read
what he said in a book, his answer is in the affirmative. This time,
however, Dan said he "made it up in his mind" (although he did credit a
book he had which included a list of many Fibonacci numbers as helping
him think it up).
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* playing a bit at home, I realized that Dan's (Fn)^2 - (F(n-2))^2 was
always equal to F(2n-2), and that (Fn)^2 - (F(n-2))^2 could be
re-expressed as F(n-1) times [Fn+F(n-2)]. So, in effect, Dan's
conjecture is that, for all n greater than 2, F(n-1) times [Fn+F(n-2)]
is always F(2n-2). I'm sure that's well-know to people who aren't me,
and that even I could probably prove why it is true. But I was still
pretty amazed that he came up with it on his own.
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