PRIME.CITY

PRIME.TW (a very rare Taiwan domain) redirects to this page.

A website about prime numbers, and possibly about Riemann Hypothesis.

Prime numbers are numbers like 2,3,5,7,11,13,17, 19, 23, 29,..

They have exactly 2 divisors.

Riemann Hypothesis claims that number of primes smaller than x, differs from sum of 1/log n for n smaller than x, not much more than square root of x.

There is a million dollar reward for solving it, but this is actually a trillion dollar problem. If you solve it, don't let your enemies know about your solution, unless it is very unconstructive.

Although Riemann hypothesis is a central problem in mathematics, it may not have any proof or disproof. Probably it is correct, and it may either have no finite proof, or may have one page proof, or may have only ugly proofs. It is something one can believe to be true just for statistical reasons. On the other hand having no zeros not lying on the critical line among the first trillion nontrivial zeros is far from convincing about validity of this conjecture.

Prime number theorem is "equivalent" to absence of zeta zeros on boundaries of the critical strip. RH says all nontrivial zeros are on the middle line, which is much stronger. Just from this one expects that RH must be very difficult. PNT offers an asymptotical average about primes, while RH in addition gives information about deviation from this average. This is similar to expectation and variance in probability theory. Given the average, primes tend to act totally randomly within that scale, up to obvious modular arithmetical relations. Just from this one can make very precise conjectures about distribution of primes. Such as "density" of twin primes, although it is still not known whether there are infinitely many twin primes.

Primality checking requires only a polynomial time algorihm, while factorization into primes is considered a difficult problem algorithmically, and it might be an NP-complete problem (check this). Another million dollar problem asks whether P=NP, meaning, whether problems with nondeterministic polynomial time solutions are solvable in polynomial time. It is guessed that NP is not equal to P, and so NP problems are strictly more difficult than P problems, meaning requires more time to solve. Does RH offer fast factorisation, not sure about this, but maybe your solution will offer a fast factorisation of integers into primes. This is another indication that RH may be too difficult to prove or unprovable.

In short it would be strange if RH was false, but it would also be strange if someone can prove it. But if it is false, of course there is a finite and stupid disproof, but noone may live long enough to do that much computation.

If you know what falsity of RH implies, feel free to let us know.
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