Question 1165285: Please Help me to solve this
Evaluate: Let Fz be the zth Fibonacci number where in z=x^2+y^2 such that x and y are coprime integers between 2 and 6, inclusive.
Thank you in advance
Answer by solver91311(24713)   (Show Source):
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Coprime integers are sets of integers such that 1 is the only positive integer that divides all elements of the set. In this instance we are looking for 2-element subsets of the set {2, 3, 4, 5, 6} that are coprime.
Those two element sets are {2, 3}, {2, 5}, {3, 4}, {3, 5}, and {5, 6}
Now we want to find which of the above pairs of integers where the square of one added to the square of the other produces a Fibonacci number.
In review, the first twelve Fibonacci numbers are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
The sum of the squares of the elements of the two-element sets identified above:
4 + 9 = 13, 4 + 25 = 29, 9 + 16 = 25, 9 + 25 = 34, 25 + 36 = 61
Of these, only 13 and 34 are numbers in the Fibonacci sequence.
So the pairs you seek are {2, 3} and {3, 5}, hence x, y, z are either 2, 3, 13 or 3, 5, 34
John
My calculator said it, I believe it, that settles it
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