Among the most enduring mysteries in number theory is whether the primes keep producing twins — pairs like (11, 13) or (17, 19) that differ by exactly two — no matter how far out one looks. The general form was set down by Alphonse de Polignac in 1849, and the first deep theorem came from Viggo Brun in 1915, who proved that the reciprocals of the twin primes converge to a finite value, now called Brun's constant; in doing so he invented modern sieve theory and showed that twins must thin out even if there are infinitely many. Hardy and Littlewood went further, conjecturing a precise density of about 2C₂·x/(ln x)² for the count of twins below x. For nearly a century the infinitude itself stood untouched, until Yitang Zhang's stunning announcement on 17 April 2013 that some gap below 70 million recurs infinitely often — the first finite bound ever proved. A Polymath collaboration led by Terence Tao, together with James Maynard's independent multidimensional sieve, soon drove that bound down to 246, where it still stands. Closing the gap all the way to 2 — the twin prime conjecture itself — remains open. This mission states it cleanly: the set of primes p for which p + 2 is also prime is infinite.
Congruent Numbers — Tunnell's Criterion (Even Case)Open Problem
Which whole numbers are the area of a right triangle with rational sides? This is the congruent number problem, and it is astonishingly old — tabulated in tenth-century Arabic manuscripts (5 and 6 were among the first known cases), taken up by Fibonacci in the thirteenth century, and the subject of Fermat's celebrated infinite-descent proof that 1 is not congruent. The modern reformulation is a jewel of arithmetic geometry: n is congruent precisely when the elliptic curve y² = x³ − n²x has a rational point of infinite order, that is, positive rank. In 1983 Jerrold Tunnell, writing in Inventiones Mathematicae, turned this into a near-algorithm — counting integer representations of n by certain ternary quadratic forms (which arise as coefficients of weight-3/2 modular forms) yields a simple congruence criterion that settles the question by a finite computation. The catch, and the reason the problem remains officially open, is that the sufficiency of Tunnell's criterion rests on the Birch and Swinnerton-Dyer conjecture, itself a Millennium Prize Problem. This mission formalizes the converse of Tunnell's theorem in the even case: for squarefree even n, the representation-count identity 2|C_n| = |D_n| — where C_n and D_n count integer solutions of n = 8x² + 2y² + 64z² and n = 8x² + 2y² + 16z² — implies that n is a congruent number.
Congruent Numbers — Tunnell's Criterion (Odd Case)Open Problem
Which whole numbers are the area of a right triangle with rational sides? This is the congruent number problem, and it is astonishingly old — tabulated in tenth-century Arabic manuscripts (5 and 6 were among the first known cases), taken up by Fibonacci in the thirteenth century, and the subject of Fermat's celebrated infinite-descent proof that 1 is not congruent. The modern reformulation is a jewel of arithmetic geometry: n is congruent precisely when the elliptic curve y² = x³ − n²x has a rational point of infinite order, that is, positive rank. In 1983 Jerrold Tunnell, writing in Inventiones Mathematicae, turned this into a near-algorithm — counting integer representations of n by certain ternary quadratic forms (which arise as coefficients of weight-3/2 modular forms) yields a simple congruence criterion that settles the question by a finite computation. The catch, and the reason the problem remains officially open, is that the sufficiency of Tunnell's criterion rests on the Birch and Swinnerton-Dyer conjecture, itself a Millennium Prize Problem. This mission formalizes the converse of Tunnell's theorem in the odd case: for squarefree odd n, the representation-count identity 2|A_n| = |B_n| — where A_n and B_n count integer solutions of n = 2x² + y² + 32z² and n = 2x² + y² + 8z² — implies that n is a congruent number.
That the prime numbers, thinning out as they climb yet never quite vanishing, should nonetheless contain arithmetic progressions of every finite length is one of the most celebrated discoveries of twenty-first-century mathematics. Ben Green and Terence Tao proved it in 2004 (published in the Annals of Mathematics in 2008), resolving a question whose roots reach back to Lagrange and Waring around 1770 and which had crystallized in the Erdős–Turán conjecture. The primes have density zero, so Szemerédi's theorem — which guarantees long progressions only in positive-density sets — does not apply directly; the genius of the proof was a transference principle extending Szemerédi's theorem to sets sitting densely inside a 'pseudorandom' host, built from the sieve ideas of Goldston, Pintz, and Yıldırım. The result was a centerpiece of the citation for Tao's 2006 Fields Medal and opened a whole industry, including the Tao–Ziegler extension to polynomial progressions. Unusually for a headline problem, this theorem is already proved — which makes it an ideal flagship formalization mission: a deep, decomposable argument whose pieces, from Szemerédi's theorem to the transference principle, the community can rebuild and verify in Lean.
In a 1999 Vienna doctoral thesis, Gerhard Zauner conjectured that in every finite dimension d one can find d² unit vectors in complex d-space that are mutually as spread out as possible — any two sharing the same squared overlap 1/(d+1). Such a configuration, a symmetric informationally complete positive operator-valued measure (SIC-POVM), is the optimal minimal measurement for reconstructing an unknown quantum state, which is why the idea was rediscovered and named by Renes, Blume-Kohout, Scott, and Caves in 2004 and became central to quantum tomography, quantum cryptography, and the QBist reading of quantum mechanics. Geometrically these are maximal sets of complex equiangular lines; physically they are the most efficient quantum measurements; and, remarkably, they appear to be governed by deep number theory — recent work by Appleby, Flammia, Kopp, and others ties exact SICs to Stark units and Hilbert's twelfth problem on explicit class field theory. Exact solutions have been hand-built in scores of dimensions and numerical ones found in every dimension checked, yet a general existence proof remains out of reach. Formalizing Zauner's conjecture gives this problem — straddling quantum information, geometry, and algebraic number theory — a precise shared target.
First raised for two variables by Ludwig Kraus in 1884 and stated in full generality by Ott-Heinrich Keller in 1939, the Jacobian conjecture asks something that sounds almost like freshman calculus: if a polynomial map from complex n-space to itself has a Jacobian determinant equal to a nonzero constant, must it be invertible by another polynomial map? That constant-Jacobian condition is precisely the algebraic shadow of the inverse function theorem, yet producing a polynomial — not merely analytic — inverse has resisted every attack for over eighty years. Shreeram Abhyankar championed the problem because it can be stated 'using little beyond a knowledge of calculus,' and Stephen Smale placed it sixteenth on his 1998 list of problems for the new century. Its notoriety is sharpened by a graveyard of published 'proofs' that later collapsed. Deep reductions exist — Bass, Connell, and Wright showed in 1982 that the general case reduces to maps of degree three — and the problem is equivalent, through work of Tsuchimoto, Belov-Kanel, and Kontsevich, to the Dixmier conjecture on the Weyl algebra. A formal statement anchors this famously slippery problem so that progress can be verified rather than merely believed.
A Hadamard matrix is a square array of +1s and −1s whose rows are mutually orthogonal — equivalently, one whose determinant attains the absolute maximum that Jacques Hadamard proved in 1893 any ±1 matrix can reach. The story opens earlier, with James Joseph Sylvester's 1867 doubling construction producing such matrices in every power-of-two order; Hadamard himself added orders 12 and 20. The conjecture bearing his name asserts that a Hadamard matrix exists for every order divisible by four. Raymond Paley's 1933 construction from finite fields settled vast new families, and computer searches filled stubborn gaps — beginning with order 92 at JPL in 1962 and reaching order 428 only in 2005, after which 668 became the smallest order whose existence is still unknown. Far from a curiosity, these matrices are workhorses of applied mathematics, underpinning error-correcting codes (the Reed–Muller code that sharpened Mariner spacecraft imagery), spread-spectrum and CDMA signal design, optimal statistical designs of experiments, and coded-aperture spectroscopy. Settling the conjecture would close a 130-year-old gap where combinatorics, number theory, and design theory meet.
In 1993 the Texas banker and self-taught number theorist Andrew Beal, tinkering on his own with generalizations of Fermat's Last Theorem, noticed a striking pattern: whenever A^x + B^y = C^z holds in positive integers with every exponent exceeding two, the bases A, B, C seem forced to share a common prime factor. Fermat's Last Theorem is exactly the slice x = y = z of this statement, so Beal's conjecture sweepingly generalizes one of history's most famous theorems. Beal backed his question with money, raising the prize from $5,000 in 1997 to $1,000,000, now held in trust by the American Mathematical Society. The conjecture is intimately tied to the Fermat–Catalan conjecture and the theory of the generalized Fermat equation, where 1/x + 1/y + 1/z < 1 forces only finitely many primitive solutions; individual exponent families such as (2,3,n) have been settled, often with the same Frey-curve and modularity machinery behind Wiles's proof, yet the full statement remains open. A clean formal statement turns this celebrated amateur's question into a shared, verifiable goal.
Formulated in 1985 by Joseph Oesterlé and David Masser as an arithmetic distillation of Szpiro's conjecture on elliptic curves, the abc conjecture makes a deceptively simple claim about coprime triples with a + b = c: the three numbers cannot all be built from many repeated small primes at once, so c can only rarely exceed rad(abc)^(1+ε). Dorian Goldfeld called it 'the most important unsolved problem in Diophantine analysis,' and for good reason — a single proof would cascade through number theory, delivering Fermat's Last Theorem for all large exponents almost for free, along with Roth's theorem, the Mordell–Faltings theorem, the Fermat–Catalan conjecture, infinitely many non-Wieferich primes, and all but finitely many counterexamples to Beal's conjecture. Since 2012 Shinichi Mochizuki has claimed a proof via inter-universal Teichmüller theory, published in 2021, but the community has not accepted it: in 2018 Peter Scholze and Jakob Stix identified a gap they regarded as fatal. A precise formal statement gives everyone a shared, machine-checkable target around which to organize verified progress.
Every time a streaming service guesses what you would rate a film you have never seen, it is solving a matrix completion problem: fill in the missing entries of a vast user-by-item table from the few that are observed. The question became famous during the Netflix Prize (2006-2009), and it looks hopeless - infinitely many matrices fit the observed entries - until one assumes the structure that makes recommendation possible: the table is essentially low rank, because tastes are governed by a few latent factors. In their landmark 2009 paper 'Exact Matrix Completion via Convex Optimization' (Foundations of Computational Mathematics), Emmanuel Candes and Benjamin Recht proved that an n-by-n matrix of rank r can be recovered exactly, with high probability, from only about n^1.2 * r * log n randomly observed entries - not by the NP-hard route of minimizing rank, but by minimizing the nuclear norm, a convex surrogate (the sum of the singular values) solvable efficiently. The proof, in the lineage of Candes-Romberg-Tao compressed sensing, turns on two ideas: an incoherence condition ensuring the singular vectors are spread out rather than spiky, and a dual certificate witnessing optimality, whose existence rests on delicate random-matrix concentration. It transformed a practical engineering puzzle into rigorous theory and seeded a decade of work across machine learning, signal processing, computer vision, and sensor localization. This mission formalizes the Candes-Recht exact-recovery theorem in Lean, decomposed into its dual-certificate construction and the probabilistic concentration reductions at its core.
A complete formal proof of Fermats Last Theorem for exponent 5: for all positive natural numbers a,b,c, a^5 + b^5 != c^5. The proof follows the classical Legendre-Dirichlet approach (1825-1830): Case 1 (5 does not divide a,b,c) is dispatched by congruences, and Case 2 (5 divides one of them) uses infinite descent through the ring Z[zeta_5]. The open hard leaf is the Z[zeta_5] PID step (flt5_zeta5_ring_witnesses).