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Fibonacci and Square Numbers - The Solution

Author(s): 
Patrick Headley

Leonardo realizes that what he needs is a congruous number that is 5 times a square, because then he can divide through by that square and leave just the 5 [p. 76]. The values ab = 4 and bg = 5 almost work, since 4·5·(4+5)·(5-4) = 180 = 5·62. But 4+5 is not even as is required for a congruous number, the problem being that there are not 5 consecutive odd numbers centered at 36, or 4 centered at 45. Leonardo gets around this by doubling all the values. There are 10 consecutive odd numbers centered at 72, namely those from 63 to 81. There are also 8 consecutive odd numbers centered at 90, those from 83 to 97. Both sequences sum to 720=5·122, and we get

412 - 720 = (1+ 3 + ...  + 81) - (63 + 65 + ...  + 81) = 1 + 3 + ... + 61 = 312,

and

412 + 720 = (1+ 3 + ...  + 81) + (83 + 85 + ...  + 97) = 492.

Finally, Leonardo has his answer. He writes

There is for the first square 6 97/144, with root 2 7/12, which results from dividing 31 by the root of 144, which is 12, and there is for the second, which is the sought square, 11 97/144, with root 3 5/12, which results from dividing 41 by 12, and there is for the last square 16 97/144 with root 4 1/12 [p. 78].

In modern notation, we could write

(41/12)2 - 5 = (31/12)2,

and

(41/12)2 + 5 = (49/12)2.

Patrick Headley, "Fibonacci and Square Numbers - The Solution," Loci (August 2011)

Dummy View - NOT TO BE DELETED