On the distribution of arithmetic sequences in the Collatz graph Keenan Monks, Harvard University Ken G. Monks, University of Scranton Ken M. Monks, Colorado State University Maria Monks, UC Berkeley 9 28 14 7 22 11 34 17 52 → → → → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. 9 28 14 7 22 11 34 17 52 → → → → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd 9 28 14 7 22 11 34 17 52 → → → → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Example: (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? 28 14 7 22 11 34 17 52 → → → → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: 9 14 7 22 11 34 17 52 → → → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: 9 28 → 7 22 11 34 17 52 → → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: 9 28 14 → → 22 11 34 17 52 → → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: 9 28 14 7 → → → 11 34 17 52 → → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: 9 28 14 7 22 → → → → 34 17 52 → → → → 26 13 40 20 10 5 16 8 4 2 1 → → → → → → → → → → → 4 2 1 → → ··· (cid:73) Collatz Conjecture: The C-orbit x,C(x),C(C(x)),... of every positive integer x eventually enters the cycle containing 1. (cid:40) x/2 x is even (cid:73) Can also use T(x) = . 3x+1 x is odd 2 The 3x + 1 conjecture (Collatz conjecture) (cid:73) Famous open problem stated in 1929 by Collatz. (cid:40) x/2 x is even (cid:73) Define C : N N by C(x) = . → 3x +1 x is odd (cid:73) What is the long-term behaviour of C as a discrete dynamical system? (cid:73) Example: 9 28 14 7 22 11 → → → → →