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March 18th, 2017, 08:47 AM   #1
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On the Collatz Conjecture

If you don't know what the Collatz Conjecture is, Wikipedia has a great description.

I've noticed two things about numbers one less than a power of 2[1,3,7,15,31,...]

The first is that (apart from 127)their height is equal to or less than the height of all previous numbers. I've checked up to (2^16)-1, so it still may be incorrect.

The second is that the (2n-1)'th term of a number (2^n)-1 will be 2f^n(2n-1) where f(n)=(3n+1)/2. This term will be divisible by 4 as well, but not by 8. For this, I've checked up to (2^22)-1 since after that Google Spreadsheets breaks.

If anybody can prove these, it would be much appreciated. Thank you.
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