The Syracuse problem, also known as the Collatz conjecture or the 3n+1 conjecture or Ulam conjecture, is a very simple problem of arithmetics that is still unsolved today. It can be stated as follows.

**Syracuse problem** : being an integer, repeat the following operations

- If the number is even then divide it by two
- If the number is odd then multiply it by and add

**Conjecture** : This process always reaches the number

Example : starting with the sequence is

If we continue the process after then it indefinitely repeats a cycle

The following graph represents the sequence for until it reaches .

n = 11

The following graph represents the sequence for until it reaches .

n = 27

The following graph represents the number of steps to reach for between and .

Number of steps until 1 is reached

The algorithm to compute the sequence was implemented in SciLab as follows :

result = [];
while n>1
result = [result ; n];
if modulo(n,2)==0
n = n/2;
else
n = 3*n+1;
end
end
result = [result ; 1];

This program halts when is reached. If the conjecture is true then it must always stops. At this time, the conjecture has been checked for all initial values up to as explained on this web site.

Surprisingly, the story has inspired a novel (in French)

La Conjecture de Syracuse, Antoine Billot, Editions Gallimard

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The novel is surprising. Not sure I understood the link between the conjecture and the drama. Someone could explain?