A Chat with A.I.

 

                                                        Image: Howcolour on Iconfinder

I recently tested ChatGPT Feb 13 Version for researching topics for this blog site. The first entry was a recent post on the topic of zero: Math Vacation: Zero (jamesmacmath.blogspot.com). The short article written by the bot was the appropriate length and I haven't found errors.

To test the bot's capabilities a little more, I challenged it to write another article on the topic of how Fibonacci found that every other number, starting with 5, in the Fibonacci sequence is the length of the hypotenuse of a right-angled triangle defined by Pythagorean triple (see: Pythagorean Triples, Part II, Part III).

The article produced was interesting; however, I found a few errors. I'm posting the bot's product unedited with errors, which I'll correct in a future post:

A.I. request: Can you write how it is that every other number in the Fibonacci sequence is the length of a Pythagorean triple hypotenuse?

Response: Certainly! The relationship between the Fibonacci sequence and Pythagorean triples is a fascinating one. Let me explain how it works.

First, let's define what we mean by Pythagorean triples. A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the equation a^2 + b^2 = c^2, where c is the length of the hypotenuse of a right triangle and a and b are the lengths of the other two sides.

Now, let's consider the Fibonacci sequence. The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers, starting with 0 and 1. So the sequence goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Now, here's the interesting part: every third number in the Fibonacci sequence is the length of the hypotenuse of a Pythagorean triple! To see why this is the case, let's consider a general Pythagorean triple (a, b, c) where c is odd. We can then write c as the sum of two consecutive integers: c = n + (n+1), where n is a positive integer.

Now, using the Pythagorean theorem, we can express a and b in terms of n as follows:

a = n^2 - (n+1)^2 = -2n-1 b = 2n^2 + 2n

Notice that a and b are both odd integers. We can then express the ratio of a and b as follows:

a/b = -(2n+1)/(2n^2+2n) = -(2n+1)/(2n(n+1))

Now, if we let n be a Fibonacci number, then we have:

n = F_k for some positive integer k.

Therefore, we can express the ratio of a and b in terms of Fibonacci numbers:

a/b = -(2F_k+1)/(2F_k(F_k+1)) = -1/F_{k+1}

This means that the ratio of a and b is a negative reciprocal of a Fibonacci number. And since every third Fibonacci number is even, it follows that every other number in the Fibonacci sequence is the length of a Pythagorean triple hypotenuse.

In summary, every third number in the Fibonacci sequence is the length of a Pythagorean triple hypotenuse because the ratio of the other two sides of the corresponding right triangle can be expressed as a negative reciprocal of a Fibonacci number.