| Questions | Answers |
|---|---|
| What is the 3x+1 problem? | The 3x+1 problem concerns an iterated function and the question of whether it always reaches 1 when starting from any positive integer. It is also known as the Collatz problem or the hailstone problem. |
| What is the sequence that results from the 3x+1 problem? | Starting from any positive integer, if the integer is even divide it by 2, if odd multiply it by 3 and add 1. This leads to the sequence 3, 10, 5, 16, 4, 2, 1, 4, 2, 1, … which indeed reaches 1. |
| Why is 3×1 impossible? | Read on to find out! |
| Has anyone proven that 3x+1 always reaches 1? | No, this is an unsolved problem in mathematics. |
| What is the significance of the 3x+1 problem? | The 3x+1 problem is a famous example of a conjecture that has not been proven or disproven. It has been studied by many mathematicians and has applications in other areas of mathematics. |
The 3x+1 problem: Why is 3×1 impossible?
The 3x+1 problem, also known as the Collatz problem or the hailstone problem, is a famous unsolved mathematical conjecture. It concerns an iterated function and the question of whether it always reaches 1 when starting from any positive integer. This problem has been studied by mathematicians for decades, but no one has been able to prove that it is true for all positive integers. In this article, we will explore the 3x+1 problem and try to answer the question – why is 3×1 impossible?
The Sequence that Results from the 3x+1 Problem
Starting from any positive integer, if the integer is even divide it by 2, if odd multiply it by 3 and add 1. This leads to the sequence 3, 10, 5, 16, 4, 2, 1, 4, 2, 1, … which indeed reaches 1. This sequence is known as the hailstone sequence or the Collatz sequence.
This sequence is interesting because it raises the question of whether there is some pattern that occurs in all hailstone sequences. Is it true that every positive integer will eventually reach 1 if subjected to the 3x+1 function? We do not know.
Why is 3×1 Impossible?
The question of whether every positive integer reaches 1 in the 3x+1 problem is still an unsolved problem in mathematics. So, we cannot say with certainty whether 3×1 is impossible. However, there are some interesting observations we can make about the 3x+1 function.
For example, if we start with the number 3, we get the sequence 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, … . Notice that we get stuck in a loop of 4, 2, 1. This loop would seem to suggest that the answer to the 3x+1 conjecture is true – every positive integer eventually reaches 1. However, this is not a proof, and there are other examples where we do not fall into a loop.
For example, if we start with the number 5, we get the sequence 5, 16, 8, 4, 2, 1, 4, 2, 1, … . This sequence also seems to suggest that the answer to the 3x+1 conjecture is true. However, there are numbers for which the hailstone sequence is very long and does not seem to converge to 1. For example, if we start with the number 27, we get the sequence:
27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051…
This sequence does not seem to converge to 1, and as of yet, no one has been able to prove that it will. It is possible that there are numbers for which the hailstone sequence is infinitely long and never reaches 1.
Has Anyone Proven that 3x+1 Always Reaches 1?
No, as mentioned earlier, the 3x+1 problem is still an unsolved problem in mathematics. Many mathematicians have worked on this problem and have made some progress, but no one has been able to prove that it is true for all positive integers.
In fact, in 2019, Terence Tao, a prominent mathematician, made a post on his blog about his thoughts on the 3x+1 conjecture. He stated that while he believes the conjecture is true, he does not think there is currently enough evidence to support a proof of this conjecture. This goes to show just how difficult this problem is.
The Significance of the 3x+1 Problem
The 3x+1 problem is a famous example of a conjecture that has not been proven or disproven. It has been studied by many mathematicians and has applications in other areas of mathematics. For example, the study of the 3x+1 problem has led to the development of techniques and ideas that have been useful in other areas of number theory.
Furthermore, the 3x+1 problem has captured the imagination of mathematicians and non-mathematicians alike. It has inspired countless articles, books, and even computer programs that try to explore the properties of the hailstone sequence. The fact that this problem is still unsolved after decades of research also highlights the fact that there are many unsolved problems in mathematics that are waiting to be discovered.
Conclusion
The 3x+1 problem, also known as the Collatz problem or the hailstone problem, is a famous unsolved mathematical conjecture. While we know the sequence that results from this problem, we do not know whether every positive integer reaches 1 when subjected to the 3x+1 function. The 3×1 problem, in particular, remains unsolved. The significance of this problem lies in the fact that it is a famous example of an unsolved conjecture that has captured the imagination of mathematicians and non-mathematicians alike.
References:
- https://en.wikipedia.org/wiki/Collatz_conjecture
- https://mathworld.wolfram.com/CollatzProblem.html
- https://terrytao.wordpress.com/2019/09/18/what-happened-to-the-collatz-conjecture/