Examples of Collatz Conjecture

We can take any starting number and put the value in the condition until we get 1 at the end with end loop of 4-2-1.

Here we will see two more examples one with n = 5 and other is n = 7. Let’s discuss about these.

Starting with n = 5

If n is odd, we apply the function f(n)=3n+1. So, f(5) = 3(5)+1 = 16.

If n is even, we apply the function f(n) = n/2 . So, f(16) = 16/2 = 8

f(8) = 8/2 = 4

f(4) = 4/2 = 2

f(2) = 2/2 = 1

5 → 16 → 8 → 4 → 2 → 1

The sequence reaches 1 after 5 steps, confirming the Collatz conjecture for n = 5.

Starting with n = 7

If n is odd, we apply the function f(n)=3n+1. So, f(7) = 3(7)+1 = 22.

If n is even, we apply the function f(n)= n/2 So, f(22) = 22/2 = 11.

f(11) = 3(11) + 1 = 34

f(34) = 34 /2 = 17

f(17) = 3(17) + 1 = 52

f(52) = 52/2 = 26

f(26) = 26/2 = 13

f(13) = 3(13) + 1 = 40

f(40) = 40/2 = 20

f(20) = 20/2 = 10

f(10) = 10/2 = 5

f(5) = 3(5) + 1 = 16

f(16) = 16/2 = 8

f(8) = 8/2 = 4

f(4) = 4/2 = 2

f(2) = 2/2 = 1

7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

The sequence reaches 1 after 16 steps, confirming the Collatz conjecture for n = 7.

Collatz Conjecture or 3n + 1 Conjecture or Ulam Conjecture, is the problem in mathematics for almost a decade. It is proposed in 1937 by Lothar Collatz. Although extensively tested and always found true, this conjecture remains unproven, making it a persistent and enticing mystery in the world of mathematics.

Famous mathematicians Paul Erdős said about the Collatz Conjecture, “Mathematics may not be ready for such problems,” highlighting its deceptive simplicity and deep complexity. In this article, we will discuss this conjecture which seems true but still not proven by scholars.