theorem on Collatz sequences starting with Mersenne numbers
Theorem. Given a Mersenne number (with a nonnegative integer), the Collatz sequence starting with reaches in precisely steps. Also, the parity of such a sequence consistenly alternates parity until is reached. For example, given gives the Collatz sequence 3, 10, 5, 16, 8, 4, 2, 1, in which is reached at the fourth step. Also, the least significant bits of this particular sequence are 1, 0, 1, 0, 0, 0, 0, 1.
As you might already know, a Collatz sequence results from the repeated application of the Collatz function for odd and for even . If I may, I’d like to introduce the iterated Collatz function notation as a recurrence relation thus: and for all . In our example, , , , etc. (We could choose to have instead with only slight changes to the theorem and its proof).
Of course the generalized formulas do not work when , nor does any of this give any insight into when a Collatz sequence starting with a Mersenne number reaches a power of 2. Likewise, the pattern of consistently alternating parity usually breaks down on or right after the th step.
|Title||theorem on Collatz sequences starting with Mersenne numbers|
|Date of creation||2013-03-22 17:34:32|
|Last modified on||2013-03-22 17:34:32|
|Last modified by||PrimeFan (13766)|