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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 $n\in \mathbb{N},$ n$ does there exist a positive integer $N$, dependent on $n$ so such that $T^N(n)=1$?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.
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The Collatz $3n+1$ 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 $n\in \mathbb{N},$ does there exist a $N,$ N$, dependent on $n,$ n$ so that for every $T^N(n)=1?$T^N(n)=1$?

The origin of this precise question seems to be obscure, although Lothar Collatz Conjectured made similar conjectures during the question 1930s1. For example, Bryan Thwaites claims to have been the first to make this conjecture in 1937 19522, and it has been open since. Paul Erdos believing this does not seem to be a tough problem offered \$500 for ithave been decisively refuted.

Edit: Fixed some dumb errors (The 1937 dates in the answer. FirstWikipedia 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 not Paul Halmos offered the 50 dollars (Canadian?) from H.S.M. Coxeter1.

  1. Lagarias, The $500 as per wikipedia 3x+1$ problem and several other informal referencesits generalizations Am. SecondMath. Monthly 92 (1985) 3-23.
  2. Bryan Thwaites, there was an elementary logic error in the statementTwo conjectures or how to win £1100. Math. Gazette 80 (1996) 35-36.
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The Collatz Conjecture has some money assigned to it.

Define $T(n) = \begin{cases} n/2 & n=0 \mod 2 \ 3n+1 & n=1\mod 2 \end{cases}$ n/2$ when $n$ is even and $3n+1$ when $n$ is odd.

For $n\in \mathbb{N},$ does there exist a $N,$ dependent on $n,$ so that for every $T^N(n)=1?$

Lothar Collatz Conjectured the question in 1937 and it has been open since. Paul Erdos believing this to be a tough problem offered \$500 for it.

Edit: Fixed some dumb errors in the answer. First, Paul Erdos and not Paul Halmos offered the $500 as per wikipedia and several other informal references. Second, there was an elementary logic error in the statement.
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