Apologies if I came off as dismissive here - was not my intent at all - there’s obviously tons of fascinating little patterns to dig into here.
Like, factorizing numbers of the form 111…111 is a whole thing…
If there’s a composite number of digits in the number, then you can always factor it into a smaller 11…11 number, and a number consisting of 1s separated by repeating 0s - a 100…00100…001 type number.
And you can do the same to that number if it has a composite number of 1s in it (turns out the 1111 case is just a ‘number with a bunch of ones separated by sequences of no zeroes’).
And it’s not a coincidence that the numbers of ones in those factors match the prime factors of 12.
So numbers in this form tend to have factors like 11, 101, and 111… and of course that means they tend to have 37 as a factor as a result. Like, in the above example, it means 100010001 must have 37 as a factor too.
Lot of weird patterns to be found in here, for sure.
Like, factorizing numbers of the form 111…111 is a whole thing…
If there’s a composite number of digits in the number, then you can always factor it into a smaller 11…11 number, and a number consisting of 1s separated by repeating 0s - a 100…00100…001 type number.
And you can do the same to that number if it has a composite number of 1s in it (turns out the 1111 case is just a ‘number with a bunch of ones separated by sequences of no zeroes’).
To be concrete:
And it’s not a coincidence that the numbers of ones in those factors match the prime factors of 12.So numbers in this form tend to have factors like 11, 101, and 111… and of course that means they tend to have 37 as a factor as a result. Like, in the above example, it means 100010001 must have 37 as a factor too.
Lot of weird patterns to be found in here, for sure.