Unsolved equations

What it asks: For an elliptic curve E over the rational numbers, with rank r, the associated L-function L(E, s) vanishes to order r at s = 1. In plain terms: a simple way to tell, from the curve's L-function, whether its rational solutions are finite or infinite in number, and how the rank shows up in that function.

\[ \operatorname{ord}_{s=1} L(E,s) = r \]

Official statement by: Andrew Wiles.

Gatekeeper / prize holder: Clay Mathematics Institute. Prize $1,000,000.

Steps to collect: Prove the Millennium statement exactly as posed; publish in a refereed publication of worldwide repute; wait at least two years; gain general acceptance; CMI Scientific Advisory Board recommendation and directors' decision. No direct submission to CMI.

Verification links: Clay Millennium problems · Clay prize rules

What it asks: Let X be a non-singular complex projective variety. Then every Hodge class on X is a linear combination, with rational coefficients, of the cohomology classes of complex subvarieties of X. In short: for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

\[ \operatorname{Hdg}^k(X) = H^{2k}(X,\mathbb{Q}) \cap H^{k,k}(X) \]

Official statement by: Pierre Deligne.

Gatekeeper / prize holder: Clay Mathematics Institute. Prize $1,000,000.

Steps to collect: Same Clay steps as above: exact statement, refereed publication, two-year wait, general acceptance, board recommendation, directors' decision.

Verification links: Clay Millennium problems · Clay prize rules

What it asks: For the three-dimensional, incompressible Navier–Stokes equations, prove either that smooth, globally defined solutions always exist under the stated conditions, or that they do not always exist and the equations break down. The official statement, by Charles Fefferman, asks for one of four statements (A), (B), (C), (D): (A) and (B) are existence of smooth global solutions without an external force, (C) and (D) are construction of a solution with a singularity in finite time with a smooth force. (A) and (C) are on three-dimensional Euclidean space R³; (B) and (D) are on the three-dimensional torus T³.

\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{g} \]

Official statement by: Charles Fefferman.

Gatekeeper / prize holder: Clay Mathematics Institute. Prize $1,000,000.

Current claim, status: On September 8, 2026, OpenAI posted a claimed solution satisfying statements (C) and (D), with a formalized proof. As of October 2026 it awaits independent scientific validation and is subject to a priority dispute. OpenAI said it does not intend to claim the Millennium Prize for this result. Because no qualifying refereed publication has been established, the Clay two-year clock has not started on this claim.

Steps to collect: Same Clay steps as above, starting with refereed publication — a preprint or formal proof file alone does not start the clock.

Verification links: Clay Millennium problems · Clay prize rules · Millennium overview, including the Navier–Stokes claim section

What it asks: For all problems for which an algorithm can verify a given solution quickly (in polynomial time), can an algorithm also find that solution quickly? That is, is P = NP, or is P ≠ NP? Most mathematicians and computer scientists expect P ≠ NP, but it remains unproven. A common example of an NP problem not known to be in P is Boolean satisfiability.

\[ P = NP \quad \text{or} \quad P \ne NP \; ? \]

Official statement by: Stephen Cook.

Gatekeeper / prize holder: Clay Mathematics Institute. Prize $1,000,000.

Steps to collect: Same Clay steps as above.

Verification links: Clay Millennium problems · Clay prize rules

What it asks: The real part of every non-trivial zero of the Riemann zeta function ζ(s) is 1/2. Proposed by Bernhard Riemann in 1859, it concerns the distribution of prime numbers. The zeta function is defined for Re(s) > 1 by ζ(s) = sum_n=1^∞ 1/n^s, analytically continued elsewhere; its trivial zeros are at the negative even integers, and the hypothesis is about all the other, non-trivial zeros.

\[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \qquad \text{non-trivial zeros: } \Re(s) = \frac{1}{2} \]

Official statement by: Enrico Bombieri.

Gatekeeper / prize holder: Clay Mathematics Institute. Prize $1,000,000.

Steps to collect: Same Clay steps as above.

Verification links: Clay Millennium problems · Clay prize rules

What it asks: Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R⁴ and has a mass gap Δ > 0. The mass gap is the difference in energy between the vacuum (energy zero by definition) and the next lowest energy state — the mass of the lightest particle. Existence includes establishing axiomatic properties at least as strong as those in Streater & Wightman (1964) and Osterwalder & Schrader (1973, 1975).

\[ \Delta > 0, \qquad \langle \phi(0,t)\phi(0,0) \rangle \sim \sum_n A_n e^{-\Delta_n t} \]

Official statement by: Arthur Jaffe and Edward Witten.

Gatekeeper / prize holder: Clay Mathematics Institute. Prize $1,000,000 — for existence and mass gap together.

Current claim, status: In 2026, OpenAI claimed a rigorous existence proof for certain interacting quantum field theories in four dimensions, including quantum chromodynamics, with pure Yang–Mills existence as a subproblem. It did not claim the mass gap, and the physics material has not been publicly released; a group led by David Gross is writing the work up first. Existence alone, even if verified, does not trigger the Clay prize.

Steps to collect: Prove both existence and mass gap as posed; refereed publication; two-year wait; general acceptance; board recommendation; directors' decision.

Verification links: Clay Millennium problems · Clay prize rules

What it asked: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. The two-dimensional version — a closed, simply connected surface is a sphere — was the model.

\[ \text{Every simply connected, closed 3-manifold is homeomorphic to } S^3 \]

Official statement by: John Milnor.

Outcome: Proved by Grigori Perelman in 2002–2003, completing Richard Hamilton's Ricci-flow program. Perelman was awarded the Millennium Prize on March 18, 2010, and declined the award and the $1,000,000, stating Hamilton's contribution was no less than his own. Included here so readers do not mistake the remaining six for seven.

What it asks: Proposed by Christopher Sogge in 1991, in harmonic analysis. Roughly: the most extreme focusing of waves must be short-lived — at a typical random time, the amount of focusing is much smaller than at the worst-case time. Averaged over time, wave energy stays far more contained than its worst instant suggests.

Prior progress: Proved in two spatial dimensions by Guth, Wang, and Zhang in 2019.

Current claim, status: OpenAI claims a proof in every dimension, obtained October 18, 2026 and formalized in Lean October 19, 2026, using a novel auxiliary partial differential equation. Not peer reviewed as of October 2026. No designated verification group was announced for the mathematics result. A Lean formalization checks the formal statement step by step; experts still have to confirm the formal statement faithfully captures Sogge's conjecture, with no quiet simplifications.

Steps to count as solved: Independent peer review and general acceptance by the harmonic-analysis community, including confirmation that the formal proof matches the conjecture as posed. There is no prize to collect and no prize clock.

Verification links: Reporting on the October 2026 claims, including local smoothing · Reporting on the Kakeya / ETH / Yang–Mills claims

What it asks: A set in Euclidean space containing a unit line segment in every direction must possess full Hausdorff dimension, even if its Lebesgue measure is zero. Picture a shape that can hold a one-unit needle pointing in every direction: the conjecture says such a shape is never "thinner" than the space it sits in, in the Hausdorff-dimension sense.

Prior progress: Proved in three dimensions by Hong Wang and Joshua Zahl in 2025.

Current claim, status: Claimed in all dimensions as a consequence of the claimed local-smoothing proof above. Same verification status: not peer reviewed as of October 2026, formal proof still to be checked against the conjecture as posed.

Steps to count as solved: Independent peer review and general acceptance, as above. No prize, no prize holder.

What it asks: How functions behave when their frequency information is restricted to curved surfaces — in particular, restriction estimates for the sphere, paraboloid, and cone. It sits at the heart of modern Fourier analysis.

Current claim, status: Claimed as a consequence of the claimed local-smoothing proof, with reporting noting the claimed estimates hold throughout the interior of the conjectured range. Not peer reviewed as of October 2026.

Steps to count as solved: Independent peer review and general acceptance, as above. No prize, no prize holder.

What it asks: Proposed by Russell Impagliazzo and Ramamohan Paturi in 1999, formalized with Francis Zane in 2001. ETH says 3-SAT cannot be solved in subexponential time, 2^o(n), where n is the number of Boolean variables. It is a stronger claim than P ≠ NP: it says SAT stays truly exponential, not merely non-polynomial. Hundreds of papers use ETH as a foundation for conditional hardness results.

\[ \text{ETH: } 3\text{-SAT} \text{ cannot be solved in } 2^{o(n)} \text{ time} \]

Current claim, status: OpenAI and Anthropic each claimed, on October 26, 2026, a deterministic subexponential algorithm counting satisfying assignments, with Lean formalizations. Neither result has been peer reviewed as of October 2026. A verification group led by Impagliazzo and Paturi, with Ryan Williams of MIT, is organizing an open, independent examination, and has stated that its involvement should not be taken as certification of the claim.

Steps to count as solved (in this case, refuted): Independent peer review and general acceptance, including checking that the formal proof matches the claimed algorithm and that the algorithm genuinely beats the ETH bound. Disproving ETH does not settle P vs NP, and does not by itself give practical faster SAT solvers. No prize, no prize holder.

Verification links: Reporting on both October 26 ETH claims and the verification group

\[ \text{Every even } n > 2 \text{ is } n = p + q \text{ for primes } p, q \]

What it asks: Every even integer greater than 2 is the sum of two primes, for example 28 = 11 + 17.

Status: Unsolved. Checked computationally to very large bounds, but no proof covering every even integer.

Do not confuse with: the weak / ternary Goldbach conjecture (every odd number greater than 5 is the sum of three primes), proved by Harald Helfgott in 2013. That one is not unsolved.

Gatekeeper / prize holder: Research community peer review. No prize.

Steps to count as solved: A proof covering every even integer greater than 2, published and accepted by the community.

Verification link: Goldbach's conjecture

\[ \text{Infinitely many primes } p \text{ with } p+2 \text{ also prime} \]

What it asks: Are there infinitely many primes p such that p + 2 is also prime? Examples: 3 and 5, 11 and 13, 17 and 19.

Status: Unsolved. Zhang (2013) proved infinitely many prime pairs within a gap of 70 million; Maynard / Polymath reduced the unconditional bound to 246. That proves some gap occurs infinitely often, not gap 2.

Gatekeeper / prize holder: Research community peer review. No prize.

Steps to count as solved: Prove infinitely many pairs at exactly gap 2, published and accepted.

Verification link: Twin prime conjecture

\[ \forall n \ge 1, \; \exists \text{ prime } p: \; n^2 < p < (n+1)^2 \]

What it asks: For every positive integer n there is at least one prime between every pair of consecutive squares.

Status: Unsolved. Non-peer-reviewed "proof" claims on sites such as Academia.edu are not proofs.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Legendre's conjecture

\[ \text{Infinitely many primes of the form } n^2 + 1 \]

What it asks: Are there infinitely many primes of the form n² + 1? Examples: 2, 5, 17, 37, 101.

Status: Unsolved. Best partial result: Iwaniec proved n² + 1 has at most two prime factors infinitely often — that is not primality. This is the fourth of Landau's 1912 problems; the other three are Goldbach, twin primes, and Legendre, so Landau's set as a whole is still fully open.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Landau's problems

\[ \forall \text{ even } k > 0, \; \text{infinitely many consecutive prime pairs differ by } k \]

What it asks: For every positive even number k there are infinitely many pairs of consecutive primes differing by exactly k. Twin primes are k = 2; k = 4 gives cousin primes, k = 6 gives sexy primes.

Status: Unsolved for every value of k. Bounded-gap results (at most 246) prove some gap occurs infinitely often, not that each even gap does.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Polignac's conjecture

\[ p \text{ prime and } 2p+1 \text{ prime} \quad \text{— infinitely many?} \]

What it asks: A Sophie Germain prime is a prime p such that 2p + 1 is also prime (2, 3, 5, 11, 23, 29, and so on). Are there infinitely many?

Status: Unsolved — conjectured infinite, not proven. Some wiki and paper claims of a proof are not accepted by the community.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Safe and Sophie Germain primes

\[ \text{Primes of the form } 2^p - 1, \; p \text{ prime} \quad \text{— infinitely many?} \]

What it asks: A Mersenne prime is a prime of the form 2^p − 1 with p prime (3, 7, 31, 127, and so on). Are there infinitely many?

Status: Unsolved — it is not even known whether the set is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture predicts infinitely many, at roughly 5.92 per decimal-digit length, but that is a heuristic, not a proof.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Mersenne prime

\[ \sqrt{p_{n+1}} - \sqrt{p_n} < 1, \qquad \text{i.e. } g_n < 2\sqrt{p_n} + 1 \]

What it asks: For consecutive primes p_n and p_n+1, the square roots are always less than 1 apart.

Status: Unsolved. Verified empirically for primes up to 2 × 10¹⁹, but verification is not proof. A 2025 journal paper claiming a breakthrough proof is not accepted as a resolution.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Andrica's conjecture

What it asks: Write the primes, then repeatedly replace each row by the absolute differences of consecutive terms. Gilbreath (1958) claimed every row after the first starts with 1.

Status: Unsolved. Computationally verified for primes up to 1.5 × 10¹⁵. An "elementary proof" found on rxiv is not accepted.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Gilbreath's conjecture

\[ \text{Composite } n \text{ with } \varphi(n) \mid (n-1) \quad \text{— does one exist?} \]

What it asks: Is there a composite number n such that Euler's totient φ(n) divides n − 1? For primes φ(n) = n − 1 trivially, so the question is only interesting for composites.

Status: Unsolved; no composite example is known. Lehmer (1932) conjectured none exists. Any composite solution would have to be odd, square-free, a Carmichael number, with at least 14 distinct prime factors (Cohen–Hagis 1980).

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Lehmer's totient problem

\[ a + b = c, \; \gcd(a,b)=1 \implies \text{only finitely many triples with } c > \operatorname{rad}(abc)^{1+\varepsilon} \text{ for each } \varepsilon > 0 \]

What it asks: For coprime positive integers a + b = c, and every ε > 0, only finitely many triples have c greater than rad(abc)^1+ε, where rad(n) is the product of its distinct prime factors.

Status: Claimed but unverified; regarded as unproven by the mainstream community as of March 2026. Shinichi Mochizuki claimed a proof in 2012 via Inter-universal Teichmüller theory; Scholze–Stix identified a gap at Corollary 3.12; publication by RIMS did not settle mainstream acceptance; supporting work in 2024–2025 is likewise not accepted in the sources checked.

What is missing: A proof the wider community accepts as closing the Scholze–Stix gap — not another publication of the same argument.

Gatekeeper / prize holder: Mainstream mathematical community peer review. No prize for abc itself.

Verification link: abc conjecture

\[ A^x + B^y = C^z, \; x,y,z > 2 \implies A, B, C \text{ share a common prime factor} \]

What it asks: If A, B, C, x, y, z are positive integers with x, y, z greater than 2 and A^x + B^y = C^z, then A, B, and C share a common prime factor. Fermat's Last Theorem is the special case x = y = z.

Status: Unsolved.

Gatekeeper and prize holder: $1,000,000, held by the American Mathematical Society in trust, supervised by the Beal Prize Committee appointed by the AMS president, funded by D. Andrew Beal.

Steps to collect: A proof or a counterexample, published in a refereed and respected mathematics publication, satisfying the AMS prize rules. Check the AMS rules as AMS states them before relying on any paraphrase.

Verification links: Beal conjecture · AMS Beal Prize listing

\[ n! + 1 = m^2 \quad \text{— for which } n? \]

What it asks: For which n is n! + 1 a perfect square? Known solutions only: (n, m) = (4, 5), (5, 11), (7, 71).

Status: Unsolved. No further solutions up to n ≤ 10¹⁵ in computational searches; Erdős conjectured there are no more. "Resolutions" on vixra / rxiv are not accepted.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Brocard's problem

\[ \forall n \ge 2, \; \exists x,y,z > 0: \quad \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]

What it asks: For every integer n ≥ 2 there exist positive integers x, y, z with 4/n = 1/x + 1/y + 1/z, a sum of three unit fractions.

Status: Unsolved. Verified computationally to n ≤ 10¹⁷; it suffices to check prime n; constructive results for some prime classes do not prove the whole claim.

Do not merge with: the Sierpiński variant (5/n as three unit fractions) — a related but separate conjecture.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Erdős–Straus conjecture

\[ a^2+b^2=d^2, \; a^2+c^2=e^2, \; b^2+c^2=f^2, \; a^2+b^2+c^2=g^2 \]

What it asks: A rectangular box with integer edges a, b, c, integer face diagonals, and integer space diagonal g. Does one exist?

Status: Unsolved — no example found, no proof that none exists. Older exhaustive-search bounds (odd edge greater than 2.5 × 10¹³, smallest edge greater than 5 × 10¹¹, space diagonal greater than 9 × 10¹⁵) should be treated as dated, not current limits. Preprint claims of non-existence are not accepted as proofs.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Euler brick / perfect cuboid

\[ \sigma(N) = 2N, \; N \text{ odd} \quad \text{— does one exist?} \]

What it asks: A perfect number equals the sum of its proper divisors. Even perfect numbers are known; does an odd perfect number exist?

Status: Unsolved; no example known. Any odd perfect N must satisfy N > 10¹⁵⁰⁰, have at least 101 prime factors counting multiplicity, and satisfy Euler's form constraints. Sources disagree on the distinct-prime-factor count (at least 10 in one bound, at least 9 or 12 in others, depending on divisibility by 3) — do not smooth that into a single figure. Preprint claims that odd perfect numbers do not exist are not accepted.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Perfect number

\[ \operatorname{trdeg}_{\mathbb{Q}} \mathbb{Q}(z_1,\dots,z_n, e^{z_1},\dots,e^{z_n}) \ge n \]

What it asks: If z₁, …, z_n are complex numbers linearly independent over the rationals, then the field obtained by adjoining them and their exponentials has transcendence degree at least n over the rationals. In plain terms it would settle, among other things, the algebraic independence of e and π — even the irrationality of e + π is currently unknown.

Status: Unsolved. Special cases are proved (Lindemann–Weierstrass, Baker's theorem, Ax's functional analogue), but the full numerical conjecture is not. Unaccepted 2025–2026 "proof" preprints are not proofs.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Schanuel's conjecture

What it asks: Over a field of characteristic zero, if a polynomial map f: k² → k² has Jacobian determinant a nonzero constant, then f has a polynomial inverse. The n = 1 case is trivial.

Status — read carefully: The general conjecture is FALSE; only the planar case is open. On July 19–20, 2026, Levent Alpöge announced an explicit 3-variable counterexample (degrees 7, 6, 4; Jacobian determinant identically −2; three distinct rational points mapping to the same output, so no inverse exists), extended to every dimension ≥ 3 by padding with identity coordinates. An independent Isabelle/HOL formal verification was published October 29, 2026. List this as "planar Jacobian conjecture, open; general Jacobian conjecture, disproved in dimensions ≥ 3 in 2026" — not as a fully open conjecture.

Gatekeeper / prize holder: Research community peer review / formal verification. No prize.

Verification links: Jacobian conjecture · Isabelle/HOL counterexample verification

What it asks: Is there a smooth 4-manifold homeomorphic to the 4-sphere but not diffeomorphic to it — does an exotic smooth structure on S⁴ exist?

Status: Open. Distinguish carefully from the solved cases: Smale proved dimensions greater than 4 (1961), Freedman proved the topological 4-dimensional case (1982), Perelman proved dimension 3 (2003). Belief by some mathematicians that the smooth statement is false is a belief, not a proof or disproof.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification links: Poincaré conjecture · Exotic sphere

What it asks: What is the smallest number of colors needed to color the plane so that no two points at unit distance share a color?

Status: Unsolved; known to be 5, 6, or 7 as of August 2026. De Grey (2018) proved at least 5; the smallest known 5-chromatic unit-distance graph was the 509-vertex Parts graph; the 7-color hexagonal tiling gives the upper bound. A low-quality 2026 report claiming an AI proof of at least 6 is not a basis to upgrade the range to 6–7.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Hadwiger–Nelson problem

What it asks: What is the largest area of a rigid two-dimensional shape that can be maneuvered through a unit-width L-shaped corridor? Gerver's sofa (area approximately 2.2195, 18 curve sections) is the leading candidate.

Status: Claimed but unverified; still listed as open in Wikipedia in the sources checked. Jineon Baek posted a 119-page arXiv preprint in November 2024 claiming Gerver's value is optimal; sources say it was undergoing peer review with optimistic initial reactions, but that is not general acceptance. Do not list it as solved.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification links: Moving sofa problem · Baek preprint (PDF)

\[ 43 \le R(5,5) \le 48 \]

What it asks: The smallest n such that any red/blue coloring of the edges of the complete graph on n vertices contains a monochromatic 5-clique. In party terms: among n people there are always 5 mutual acquaintances or 5 mutual strangers. Known: R(3,3) = 6, R(4,4) = 18.

Status: Unsolved; 43 ≤ R(5,5) ≤ 48. The leading guess is 43, unproven. Google DeepMind's AlphaEvolve (March 2026) improved lower bounds for some Ramsey numbers but did not close R(5,5). A low-credibility web article claiming a proof that R(5,5) = 43 is not accepted; reputable sources as of 2026 still list it as unknown 43–48.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Ramsey theory

\[ HH^T = nI, \quad H \text{ a } \pm 1 \text{ matrix} \]

What it asks: A Hadamard matrix is a square ±1 matrix with mutually orthogonal rows. Such matrices can only exist for n = 1, 2, or a multiple of 4. Conjecture: one exists for every multiple of 4.

Status: Unsolved. 2026 update: in August 2026 Alpöge, Voinov, and Reynolds-Haertle announced Hadamard matrices at the twelve orders below 2000 at which none was known, verified and surveyed in a subsequent arXiv paper, which lists 95 orders up to 10,000 still unknown. The conjecture is still open — do not repeat the old "668 is the smallest open case" line; that is outdated.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification links: Hadamard matrix · 2026 Hadamard survey

What it asks: Is there a set of points dense in the whole plane such that every pairwise distance between points of the set is rational?

Status: Unsolved. Dense rational-distance sets on a circle exist (when the radius squared is rational), but those are dense only on their circle, not in the plane. Assuming the Bombieri–Lang conjecture, Tao and Shaffaf independently showed no dense planar set exists — but that is conditional, not a proof.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Erdős–Ulam problem

\[ \gamma = \lim_{n \to \infty} \left(1 + \frac{1}{2} + \cdots + \frac{1}{n} - \log n\right) \approx 0.5772156649 \]

What it asks: Is γ rational, irrational, or transcendental?

Status: Unsolved — not even known to be irrational, let alone transcendental. Partial results: Shidlovsky (1959) proved at least one of γ and the Gompertz constant δ is irrational; Rivoal (2012) proved at least one of them is transcendental. A vixra paper claiming γ's irrationality is not accepted.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Euler's constant

\[ n \to \begin{cases} n/2 & n \text{ even} \\ 3n+1 & n \text{ odd} \end{cases} \quad \text{— does it always reach 1?} \]

What it asks: Start with any positive integer n; if even, halve it; if odd, replace by 3n + 1; repeat. Does it always reach 1?

Status: Unsolved since 1937. Verified computationally to about 2⁶⁸; Tao's 2019 / 2022 "almost all" result is partial, not a proof for every start. A 2017 paper arguing Collatz may be unprovable is not the mainstream status; do not present Collatz as proven unprovable.

Gatekeeper / prize holder: Research community peer review. No prize.

Verification link: Collatz conjecture

What it asks: Take a number, add it to its digit-reversal, repeat. Most numbers reach a palindrome quickly (89 takes 24 steps, reaching 8813200023188). A Lychrel number would never reach a palindrome. Is 196 one?

Status: Unsolved in base 10 — no base-10 number has been proven Lychrel; 196 is merely the smallest candidate, tested through hundreds of millions of iterations (over a billion in one account, reaching a 413-million-digit number) without a palindrome. Note the honest asymmetry: in bases that are powers of 2 (binary, hexadecimal), some numbers can be proven never to palindromise — that proof does not exist for base 10.

Gatekeeper / prize holder: Research community peer review / computation. No prize.

Verification link: Lychrel number