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Frontiers

Nobody has solved these. Each page states the problem precisely, translates it into plain language, lists the exact prerequisites from the stages above, breaks it into the parts mathematicians actually work on, and says what is known. The point is not to solve them today; it is to be able to read them.

Opetuksia

Advanced The Riemann Hypothesis Every non-trivial zero of the zeta function has real part ½ — and with it, the primes obey their expected rhythm. Advanced Navier–Stokes existence and smoothness Do the equations of fluid flow always have smooth solutions, or can a fluid blow up in finite time? Advanced P versus NP If a solution can be checked quickly, can it always be found quickly? Advanced Goldbach's conjecture Every even number greater than 2 is the sum of two primes. Core The Collatz conjecture Halve it if even, triple-and-add-one if odd: does every start reach 1? Advanced The twin prime conjecture Are there infinitely many primes two apart — 3 and 5, 11 and 13, 101 and 103?

Symbols used here

\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\zeta(s)
Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
\frac{\partial u}{\partial t},\ \nabla^2 u
partial derivative in time, Laplacian
Rate of change in time; sum of second partials (the diffusion operator).

Questions people ask

Can I actually work on these?

You can understand them, which is the honest first step and what these pages are for. Working on them means the full path through the stages above and then the research literature — but every person who has made progress started by reading the statement.

Why are they unsolved if so many people have tried?

Usually because the existing tools provably cannot work (the barriers in P vs NP), or because the problem mixes two structures mathematics handles separately (additive and multiplicative in Goldbach and Collatz). A solution needs a genuinely new idea.

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