>>13630570You basically found an idempotent in the ring of 10-adic numbers, meaning a number which when you square, you get itself back.
There is another one that starts with 5 (from the right), which you can find here:
https://www.physicsforums.com/threads/interesting-property-of-idempotent-10-adic-number.877983/Note how when you add the 6-digit number 109376 to the 6-digit number 890625 in the link, you get 1000001. Not a coincidence ;)
High-level explanation:
The ring of 10-adics is a product of the 2-adics and the 5-adics. Each of these has two idempotents, which are 0 and 1 (as with every ring).
But then the product ring will have four idempotents: 0 = (0,0), 1 = (1,1), ...109376 = (0,1), and ...890625 = (1,0). I put the ellipses on the left because in the p-adics, the digis go infinitely to the LEFT.
You can do operations on them component-wise, and so ...109376 + ...890625 = (0,1) + (1,0) = (1,1) = 1, hence why we got that 1000001 earlier (the more digits you have, the closer it will be to 1 in the p-adic 10-adic topology, so 100000000001 is even closer to 1 than 1000001, because higher powers of 10 are small). In other words, if we call the number you found x, then the number in the link is just 1-x.
And if you multiply, you get ...109376 * ...890625 = (0,1) * (1,0) = (0,0) = 0. Indeed, when you multiply the two 6-digit numbers, you get 97413000000, which is approximately zero in the 10-adic topology (because of all the zeros on the right).
Keep playing around with math :)
