Unsolved Millennium Problems

The Kissing Number Problem
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In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
When a bunch of spheres are packed in some region, each sphere has a Kissing Number, which is the number of other spheres it’s touching; if you’re touching 6 neighboring spheres, then your kissing number is 6. Nothing tricky.

(check the images on the comment to be clear.)
✅In one dimension, the kissing number is 2
✅In two dimensions, the kissing number is 6
✅In three dimensions, the kissing number is 12
✅In four dimensions, it was known for some time that the answer was either 24 or 25
✔️The kissing number in n dimensions is unknown for n > 4, except for n = 8 (where the kissing number is 240), and n = 24 (where it is 196,560).
There are several hurdles to a full solution, including computational limitations. So expect incremental progress on this problem for years to come.

Collatz conjecture
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This is the simple math problem we still can’t solve!!
Just pick a number (any number). If it's even, divide it by 2. If it's odd, multiply it by 3 and add 1. Take that new number and repeat the process, again and again. If you keep this up, you’ll eventually get stuck in a loop. At least, that’s what we think will happen.

Take 10 for example: 10 is even, so we cut it in half to get 5. Since 5 is odd, we triple it and add 1. Now we have 16, which is even, so we halve it to get 8, then halve that to get 4, then halve it again to get 2, and once more to get 1. Since 1 is odd, we triple it and add 1. Now we’re back at 4, and we know where this goes: 4 goes to 2 which goes to 1 which goes to 4, and so on. We’re stuck in a loop.

Or try 11: It’s odd, so we triple it and add 1. Now we have 34, which is even, so we halve it to get 17, triple that and add 1 to get 52, halve that to get 26 and again to get 13, triple that and add 1 to get 40, halve that to get 20, then 10, then 5, triple that and add 1 to get 16, and halve that to get 8, then 4, 2 and 1. And we’re stuck in the loop again. it is a mathematical black hole.

As simple as it sounds, it actually works. But the problem is that even though mathematicians have shown this is the case with millions of numbers, they haven't found any numbers out there that won't stick to the rules.

"It's possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1," explains Thompson. "But no one has ever been able to prove that for certain."

Riemann Hypothesis
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Riemann Hypothesis is one of unsolved millennium problem. if you can solve this problem, you will get $1 Million. Good Luck!!

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + . . .

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

ζ(s) = 0

lie on a certain vertical straight line.

This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

References:https://www.claymath.org/sites/default/files/official_problem_description.pdf
https://www.claymath.org/sites/default/files/sarnak_rh_0.pdf