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The $3n+1$ Conjecture has some money assigned to it.

Define $T(n) = n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

For any positive integer $n$ does there exist a positive integer $N$, such that $T^N(n) = 1$?

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s¹. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1952², and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Rewards offered to date include 1000 UK pounds from Bryan Thwaites, 500 US dollars from Paul Erdos, and 50 dollars (Canadian?) from H.S.M. Coxeter¹.

Update: In July 2021, a Japanese firm offered 120 million yen for a proof.

1. Lagarias, The $3x+1$ problem and its generalizations Am. Math. Monthly 92 (1985) 3-23.
2. Bryan Thwaites, Two conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.

The $3n+1$ Conjecture has some money assigned to it.

Define $T(n) = n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

For any positive integer $n$ does there exist a positive integer $N$, such that $T^N(n) = 1$?

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s¹. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1952², and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Rewards offered to date include 1000 UK pounds from Bryan Thwaites, 500 US dollars from Paul Erdos, and 50 dollars (Canadian?) from H.S.M. Coxeter¹. Tim Roberts offered 1000 dollars and a bottle of champagne.

1. Lagarias, The $3x+1$ problem and its generalizations Am. Math. Monthly 92 (1985) 3-23.
2. Bryan Thwaites, Two conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.

The $3n+1$ Conjecture has some money assigned to it.

Define $T(n) = n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

For any positive integer $n$ does there exist a positive integer $N$, such that $T^N(n) = 1$?

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s¹. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1952², and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Rewards offered to date include 1000 UK pounds from Bryan Thwaites, 500 US dollars from Paul Erdos, and 50 dollars (Canadian?) from H.S.M. Coxeter¹.

1. Lagarias, The $3x+1$ problem and its generalizations Am. Math. Monthly 92 (1985) 3-23.
2. Bryan Thwaites, Two conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.

The $3n+1$ Conjecture has some money assigned to it.

Define $T(n) = n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

For any positive integer $n$ does there exist a positive integer $N$, such that $T^N(n) = 1$?

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s¹. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1952², and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Rewards offered to date include 1000 UK pounds from Bryan Thwaites, 500 US dollars from Paul Erdos, and 50 dollars (Canadian?) from H.S.M. Coxeter¹.

Update: In July 2021, a Japanese firm offered 120 million yen for a proof.

1. Lagarias, The $3x+1$ problem and its generalizations Am. Math. Monthly 92 (1985) 3-23.
2. Bryan Thwaites, Two conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.

The $3n+1$ Conjecture has some money assigned to it.

Define $T(n) = n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

 

For any positive integer $n$ does there exist a positive integer $N$, such that $T^N(n) = 1$?

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s¹. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1952², and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Rewards offered to date include 1000 UK pounds from Bryan Thwaites, 500 US dollars from Paul Erdos, and 50 dollars (Canadian?) from H.S.M. Coxeter¹.

1. Lagarias, The $3x+1$ problem and its generalizations Am. Math. Monthly 92 (1985) 3-23.
2. Bryan Thwaites, Two conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.

The $3n+1$ Conjecture has some money assigned to it.

Define $T(n) = n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

For any positive integer $n$ does there exist a positive integer $N$, such that $T^N(n) = 1$?

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s¹. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1952², and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Rewards offered to date include 1000 UK pounds from Bryan Thwaites, 500 US dollars from Paul Erdos, and 50 dollars (Canadian?) from H.S.M. Coxeter¹.

1. Lagarias, The $3x+1$ problem and its generalizations Am. Math. Monthly 92 (1985) 3-23.
2. Bryan Thwaites, Two conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.