∞ Open Problems

Famous unsolved problems

These are well-known problems with no agreed solution. Anyone can post an attempt. Explain what you tried, any partial results, and where you got stuck.

The badges describe how much background you need to understand the question. All of these problems are still unsolved.

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Number Theory

Needs more backgroundMillennium Prize: $1,000,000since 1859

Riemann Hypothesis

The Riemann Hypothesis says that every non-trivial zero of the Riemann zeta function has real part 1/2. These zeros are tied to how the prime numbers are spread out.

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Easy to understandsince 1742

Goldbach's Conjecture

Every even integer greater than 2 can be expressed as the sum of two primes. Stated in 1742 and verified computationally for all even numbers up to 4 × 10¹⁸, yet no general proof exists.

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Easy to understand

Twin Prime Conjecture

There are infinitely many pairs of primes differing by 2, such as (11, 13) or (17, 19). In 2013 Zhang proved there are infinitely many pairs of primes less than 70 million apart. That gap has since been brought down to 246, but a gap of 2 is still open.

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Easy to understandsince 1937

Collatz Conjecture

Take any positive integer. If even, divide by 2. If odd, multiply by 3 and add 1. Repeat. The conjecture: you always eventually reach 1. Verified for all starting values up to 2⁶⁸, yet no proof is known. Erdős said of it: "Mathematics is not yet ready for such problems."

submission

Needs more backgroundsince 1985

ABC Conjecture

For coprime positive integers a, b, c with a + b = c, the radical of abc (the product of its distinct prime factors) is rarely much smaller than c. A claimed proof by Mochizuki (2012) remains contested by the mathematical community.

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Complexity Theory

Needs more backgroundMillennium Prize: $1,000,000since 1971

P versus NP

If a solution to a problem can be checked quickly, can a solution also be found quickly? Here "quickly" means the time needed grows at most like a polynomial in the size of the input.

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Analysis / PDEs

Needs more backgroundMillennium Prize: $1,000,000since 1845

Navier-Stokes Existence and Smoothness

Do smooth, globally defined solutions to the Navier-Stokes equations always exist for all time, or can they develop singularities? These equations describe how fluids move, and they are used across physics and engineering. Whether their solutions always stay smooth is not known.

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Mathematical Physics

Needs more backgroundMillennium Prize: $1,000,000since 1954

Yang-Mills Existence and Mass Gap

Prove that quantum Yang-Mills theory exists as a rigorous mathematical theory and has a mass gap: a positive lower bound on the energy of anything above the vacuum. The theory is part of the Standard Model of particle physics.

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Algebraic Geometry

Needs more backgroundMillennium Prize: $1,000,000since 1965

Birch and Swinnerton-Dyer Conjecture

The rank of an elliptic curve over the rationals (the number of independent rational points of infinite order) equals the order of vanishing of its L-function at s = 1. It links the curve's rational points to an analytic function.

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Needs more backgroundMillennium Prize: $1,000,000since 1950

Hodge Conjecture

For a smooth complex projective algebraic variety, every rational Hodge class is a rational linear combination of the cohomology classes of algebraic cycles. It asks whether certain classes that describe a space's topology can be represented by algebraic cycles.

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Algebra

Needs more backgroundsince 1939

Jacobian Conjecture

If a polynomial map from ℂⁿ to ℂⁿ has a constant nonzero Jacobian determinant everywhere, must it have a polynomial inverse? Stated in 1939, it has been "proved" and retracted multiple times, and remains open for n ≥ 2.

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Functional Analysis

Needs more background

Invariant Subspace Problem

Does every bounded linear operator on a separable infinite-dimensional Hilbert space over ℂ have a non-trivial closed invariant subspace? It has been answered for many special kinds of operators, but the general case is still open.

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