Mathematics Atlas

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Open Questions

Genuinely unresolved questions in mathematics, phrased as the doubt itself rather than as a formal proposed claim. Belongs here: a question the mathematical community has not settled, framed in plain language, such as whether the Collatz process always reaches one or whether infinitely many twin prime pairs exist. Does not belong here: the formal conjecture that proposes a specific answer to such a question, such as the Collatz Conjecture or the Twin Prime Conjecture, filed as its own conjecture entity and linked to the open question it addresses rather than merged with it.

Open Questions

Can every even integer greater than two really always be written as the sum of two primes?

Open Questions
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Do all nontrivial zeros of the Riemann zeta function really lie on the critical line?

Open Questions
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Do infinitely many twin prime pairs really exist, or does the last one eventually appear?

Open Questions
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Do smooth, globally defined solutions to the three dimensional Navier-Stokes equations always exist, or can a solution break down in finite time?

Open Questions
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Does a nontrivial quantum Yang-Mills theory really exist for every compact simple gauge group in four dimensions, with a strictly positive mass gap?

Open Questions
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Does every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?

Open Questions
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Does every runner on a circular track, moving at its own constant and distinct speed, really become lonely at some moment, for every number of runners n?

Open Questions
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Does the Collatz process really always reach one, from every possible starting number?

Open Questions
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Does the Erdos-Straus equation 4/n = 1/x + 1/y + 1/z really have a positive-integer solution for every integer n of 2 or more?

Open Questions
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Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?

Open Questions
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Is every efficiently checkable problem also efficiently solvable, or is checking genuinely easier than solving?

Open Questions
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Is every Hodge class on a smooth complex projective algebraic variety really a rational combination of the cohomology classes of actual algebraic subvarieties?

Open Questions
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Is the abc conjecture actually true, and does Shinichi Mochizuki's inter-universal Teichmuller theory really prove it?

Open Questions
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Is there really a size of infinity strictly between the integers and the real numbers, and if not, does that fact hold absolutely or only relative to which further axioms mathematicians choose to accept?

Open Questions
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Is there really always at least one prime number between n squared and (n plus one) squared, for every positive integer n?

Open Questions
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Whenever A to the x plus B to the y equals C to the z, with x, y and z all greater than two, do A, B and C really always share a common prime factor?

Open Questions
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