THE 10 HARDEST MATH PROBLEMS THAT REMAIN UNSOLVED

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The Riemann Hypothesis is indeed a fascinating and profound conjecture in mathematics with deep implications for number theory, particularly concerning the distribution of prime numbers. Let's break down some of the key elements in more detail.

### The Riemann Hypothesis

The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function, $\zeta(s)$, lie on the critical line in the complex plane, $\Re(s) = \frac{1}{2}$. These zeros are of the form $s = \frac{1}{2} + it$, where $t$ is a real number.

### The Riemann Zeta Function

The zeta function $\zeta(s)$ is initially defined for complex numbers $s = \sigma + it$ with $\Re(s) > 1$ as:

\[
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
\]

This series converges absolutely for $\Re(s) > 1$. However, the zeta function can be extended to other values of $s$ (except $s = 1$) through a process called analytic continuation. The function has trivial zeros at the negative even integers ($-2, -4, -6, \ldots$).

### Importance of the Riemann Hypothesis

The hypothesis has significant implications for understanding the distribution of prime numbers. It suggests that primes are distributed as regularly as possible, given their known asymptotic density described by the Prime Number Theorem.

### Implications

If the Riemann Hypothesis is true, it would lead to a more precise understanding of the error term in the Prime Number Theorem. This would refine our knowledge about the density and gaps between successive prime numbers.

### History and Status

- **Proposed by Bernhard Riemann**: Riemann introduced this hypothesis in his 1859 paper, "On the Number of Primes Less Than a Given Magnitude."
- **Unproven**: Despite extensive efforts by many mathematicians, no one has yet proven or disproven the hypothesis.
- **Millennium Prize Problem**: It is one of the seven Millennium Prize Problems, with a reward of $1 million for a correct proof.

### Non-trivial Zeros

Non-trivial zeros are the complex zeros of the zeta function that lie in the critical strip, $0 < \Re(s) < 1$. The hypothesis claims that these zeros all have their real part equal to $\frac{1}{2}$.

### Example Problems and Exercises

1. **Convergence of the Zeta Function's Series for $\Re(s) > 1$**:
- The series $\sum_{n=1}^{\infty} \frac{1}{n^s}$ converges absolutely for $\Re(s) > 1$ because each term $\frac{1}{n^s}$ diminishes rapidly as $n$ increases. This can be shown by comparison to the integral test or by noting that the series resembles a p-series $\sum \frac{1}{n^p}$ with $p > 1$.

2. **Evaluating $\zeta(s)$ for Simple Cases**:
- For $s = 2$, we have the well-known result:
\[
\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
\]
- For $s = 1$, the series diverges, leading to a pole at $s = 1$.

3. **Understanding the Critical Strip**:
- The critical strip is defined by the region $0 < \Re(s) < 1$ in the complex plane. The Riemann Hypothesis asserts that all non-trivial zeros of $\zeta(s)$ within this strip lie on the line $\Re(s) = \frac{1}{2}$.

These example problems help illustrate some foundational aspects of the Riemann zeta function and why the Riemann Hypothesis is such a central topic in number theory. If you have any more specific questions or need further clarification on any aspect, feel free to ask!

Ruleew

I solved the first question in 5 min and I have 240 IQ btw

tommygg

This video should be called "The 10 hardest math problems that RIEMANNS unsolved" XD

ptaszor

Ai has I think a prediction to 1 I copied it so not my work

Unfortunately, the Collatz conjecture is an unsolved problem in mathematics, which means that we currently do not know whether the conjecture is true or false. The conjecture states that if we start with any positive integer "n" and repeatedly apply the function "f(n)" defined as "n/2" if "n" is even and "3n + 1" if "n" is odd, we will eventually reach the value "1" regardless of the starting value of "n". Despite much effort, no one has been able to prove the conjecture, nor has anyone found a counterexample.

bassilgeorge

Obviously number 1 is Transjecture, it's both even and odd, gotta read between the lines!

HansensUniverseT-A

Conject if f and n is 2(even)
Then
4=1/7
7*4=28
1/28answer

PowerYT

1 is the answe to the collate conjuncture

Uthman-yn

Yes, you're correct. This problem falls into the field of "recreational maths" which is maths that is seen as entertaining and engaging rather than purely for solving real world problems. As a result, some very complicated theories have been developed to try and explain why the Collatz Conjecture works in the way it does. However, no one has managed to come up with a general proof for why the Collatz Conjecture is always true. In fact, the Collatz Conjecture is still a problem that is considered to be unsolved by mathematicians today.
It may be that the Collatz Conjecture is simply too difficult to solve using the tools and techniques that we currently have available. However, it is also possible that the Conjecture is true for all numbers, but is so difficult to prove that we may never be able to find a general proof. Either way, the Collatz Conjecture remains an intriguing and fascinating problem that has captivated mathematicians for over 60 years now.

corderofam

Hello po Sir.. gumagawa din ako ng mga simpleng math content.. nice meeting you po..

KuyaRenren

The collatz conjecture seems to be more of a law really, that says that all numbers are rooted in one.

andrewkosinski

9.
Pi+e
(Pi+e)^-infinity
(1/pi +1/e)^positive infinity
(1/infinity)+(1/infinity)
0+0
0 answer
When they both are raised towards the negative infinity power, it leads toward a very small number so zero

nonoja

i figured out the twin prime. first u use the rules of exponents to simplify it so u get pi^2 x^1. to divide powers of the same base, you have to subtract the denominators exponent from the numerator exponent. pi^2-2 x^1-1 is what you will have to write. you subtract 2 and 2 so you now have pi^0 x^1-1. for any number a except for 0, a^0=1 so you get x^1-1. subtract 1-1 and that gets you x^0 or 1

pikachu