Why 89 is the real deal? The 11th Fibonacci number.

I have decided to devote some time to Mathematical reading, the kind of play and explorations which mathematicians do, and as a result, end up making some beautiful discoveries.

I started out by the reading and playing with Fibonacci series, and very soon in my pursuit, I read and discovered something beautiful I couldn’t resist sharing.

I won’t go into too much details about the Fibonacci series (which is amazing by the way from a historical, mathematical beauty and applications point of view) for now, and share crisply something that just blew my mind. I know this article isn’t doing justice to the series in any way, but the that’s not the point of this article.

Fibonacci series is the series with the 1st and 2nd term as 1, and the all the further terms obtained by adding the previous 2 terms. So, the series turns out be :

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ……

As per the series, the 11th Fibonacci number is 89. Following are some of the observations and discoveries made by mathematicians, students and which make this occurrence really amazing.

1. 11 and 89 are both prime numbers. (Ok….Not that great)
2. 8/9 = 0.89… (Somewhat interesting)
3. If we see 11 as (8+3), 89 is the (8+3)rd Fibonacci number, and it is also (8*3)rd prime number. (Tell me something which is more beautiful)
4. 8+9 = 17, which is the sum of all 4 prime numbers smaller than 11. Interestingly enough , 8*9 = 72, which is the sum of 4 prime numbers greater than 11 (13,17,19,23) . (This seems to be getting somewhere)
5. 8*9 + (8 + 9) = 89

These are nice. But what really blew my mind is the following observation.

Squaring the digits of a number and adding them

Start with 89. Square the digits and add them. This gives us a new number. Let’s keep doing that and see what happens.

If we keep doing this, what we get is the following :

89 –> 145 -> 42 –> 20 –> 4 –> 16 –> 37 –> 58 –> 89

Wow. We come back to 89.

What if we start at a different number, and repeat the process. Would we arrive at the initial number always. Lets check that out with say…43.

43 –> 25 –> 29 –> 85 –> 89

And it’s 89 again.

What’s really cool is that, you could start with any number, with any number of digits, and you would always end at 89 or 1. Isn’t this just amazing? Try that out for a few numbers, and you could share your sequence in the comments.

Now, a really curious mind would never be satisfied here. The real question now would be…….Why does this happen? Could you dare to explore that?

It may be hard to define mathematical beauty, but that is true of beauty of any kind.
—
G.H Hardy

PS: 89 is the 11th Fibonacci number, and there are 1189 gospels in the Bible. Seems like a part of the God’s plan.

PSS:I have skipped some of the more technical discoveries, for the sake of simplicity, so this is not an exhaustive list is any way. I might share them in a separate post.