1, 4, 9, 7, 7, 9, 4, 1, 9, …
This is really a story about pattern recognition, and discovery. The best part about discovery is that for you it’s as though you were the first to see it!
Back in high school history class, my mind wandering, I took out a piece of paper and wrote down the squares of the positive integers and started summing up the digits for each square.
1², 2², 3², … which is 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …
became
1, 4, 9, 7, 7, 9, 13, 10, 9, 1, 4, 9, …
If a result contained more than one digit, I added the digits in the result and repeated this process until there was one digit left in each result. So I ended up with:
x²: 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, …
Wow! A pattern?
Yes, it keeps repeating, forever. I called the single digit result SOD, for “Sum Of Digits”. It turns out that the results are almost identical to performing modular arithmetic (see the 2nd Pause! below).
The pattern contains a palindrome too. It has 1 4 9 7 7 9 4 1 repeated.
Pause! All numbers in this article are positive integers. Both the base and the exponent. In x², x is the base and 2 is the exponent. Unpause!
Then I took cubes and did the same thing.
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, …
got me
x³: 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9 …
Another pattern!
And fourth, fifth, sixth and seventh powers got me
x⁴: 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, …
x⁵: 1, 5, 9, 7, 2, 9, 4, 8, 9, 1, 5, 9, …
x⁶: 1, 1, 9, 1, 1, 9, 1, 1, 9, 1, 1, 9, …
x⁷: 1, 2, 9, 4, 5, 9, 7, 8, 9, 1, 2, 9, …
There’s all sorts of patterns in and among each sequence. I’ll talk about them in a potential 1, 4, 7, 9, 9, 7, 4, 1, 9, … Part II article.
Then I got to eighth powers:
x⁸: 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, …
And so the sequence groups themselves repeat! x² and x⁸ have the same sequence. So do x³ and x⁹. And so on.
Pause! The results of performing SOD arithmetic and performing modular arithmetic, in this case “mod 9", are identical except when a SOD result is 9, the “mod 9” result is 0, and vice versa. Modular arithmetic is simply obtaining the remainder as a whole number after division.
After we complete pattern generation, the SOD table is given by:
Mathematically, I discovered the following equivalence relationship:
Among the amazing stuff you can say after looking at the table is this: Take any positive integer. Find its SOD. If it’s 3 or 6, then the number is not the power of any number! 123,456,789,012 is not the square, cube, fourth power, etc. of any number. Two out of every nine numbers are like this. And we know exactly which ones. The sequence:
3, 6, 12, 15, 21, 24, 30, 33, …
contains all such numbers. None are squares, cubes, etc.
3 + 9n and 6 + 9n are the numbers in the sequence.
What about powers of 3? Surely there must be a number that violates this rule. Take a look at the table. Right off the bat, all powers of 3 (third column in the table) have a SOD of 9. They miss 3 and 6 altogether.
You can make this grid into another grid with a different base. Right now it is base 10. If you start going down that road, then you get deeper into number theory with primitive roots and all sorts of pure mathematics stuff. Check it out:
I attempted to solve Fermat’s Last Theorem (before a proof was found) by working within this system.
I didn’t find a solution, but independently re-discovered a number of techniques in mathematics, including Fermat’s method of infinite decent.
It was a wild ride back then with new discoveries around each corner, and it all started from writing numbers on a piece of paper and thinking “what if”.
Maybe I should continue looking for another proof of Fermat’s Last Theorem. Somewhere in Table 1 lies…something! Maybe one of you can find it? If you find something interesting, get in touch with me; I’d like to hear from you. I have all my notes from that time period and if I dig them out I’ll return and write about some of the things I found out in a Part II.
Until then, happy exploring!
