Square Pisano, Get Sequence? (Posted on 2007-03-14)

	Submitted by K Sengupta
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At the outset, we observe that S_1 = S_2 = 1^2; S_3 = 2^2; S_4 = 3^2; S_5 = 5^2; S_6 = 8^2, and so on. This leads one to conjecture that S_n = F_n^2, where F_n is the nth term of the Fibonacci Sequence defined by F_1 = F_2 = 1 and: F_(n+1) = F_n + F(n-1) for all n>=2. Suppose this is true for p<= (n+2). We shall prove that this result also holds for p= n+3 Now, S_(n+3) = 2S_(n+2) + 2S_(n+1) - Sn = 2*F_(n+2)^2 + 2*F_(n+1)^2 - F_n^2 = 2*F_(n+2)^2 + 2*F_(n+1)^2 - (F_(n+2)^2 - F_(n+1)^2 = F_(n+1)^2 + F_(n+2)^2 + 2*F_(n+1)*F_(n+2) = (F_(n+1) + F_(n+2))^2 = F_(n+3)^2 Consequently, S_p is a perfect square for all p. ****************************** NOTE : The famous Italian Mathematician, Leonardo Fibonacci who enuciated the Fibonacci sequence was also known as Leonardo Pisano.

	Subject	Author	Date
	Proof	Old Original Oskar!	2007-03-14 14:49:54
	solution	Charlie	2007-03-14 14:15:31
	correction	Dennis	2007-03-14 14:08:19
	some thoughts	Dennis	2007-03-14 13:58:41