List of incomplete proofs

Results later proved rigorously

• Euclid's Elements. Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 David Hilbert gave a complete set of axioms for Euclidean geometry, called Hilbert's axioms, and between 1926 and 1959 Tarski gave some complete sets of first order axioms, called Tarski's axioms.
• Isoperimetric inequality. For three dimensions it states that the shape enclosing the maximum volume for its surface area is the sphere. It was formulated by Archimedes but not proved rigorously until the 19th century, by Hermann Schwarz.
• Infinitesimals. In the 18th century there was widespread use of infinitesimals in calculus, though these were not really well defined. Calculus was put on firm foundations in the 19th century, and Robinson put infinitesimals in a rigorous basis with the introduction of nonstandard analysis in the 20th century.
• Fundamental theorem of algebra. Many incomplete or incorrect attempts were made at proving this theorem in the 18th century, including by d'Alembert, Euler, de Foncenex, Lagrange, Laplace, Wood, and Gauss. The first rigorous proof was published by Argand in 1806.
• Dirichlet's theorem on arithmetic progressions. In 1808 Legendre published an attempt at a proof of Dirichlet's theorem, but as Dupré pointed out in 1859 one of the lemmas used by Legendre is false. Dirichlet gave a complete proof in 1837.
• The proofs of the Kronecker–Weber theorem by Kronecker and Weber both had gaps. The first complete proof was given by Hilbert in 1896.
• In 1879, Alfred Kempe published a purported proof of the four color theorem, whose validity as a proof was accepted for eleven years before it was refuted by Percy Heawood. Peter Guthrie Tait gave another incorrect proof in 1880 which was shown to be incorrect by Julius Petersen in 1891. Kempe's proof did, however, suffice to show the weaker five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976.
• Schröder–Bernstein theorem. In 1896 Schröder published a proof sketch which, however, was shown to be faulty by Alwin Reinhold Korselt in 1911.
• Fermat's Last Theorem. An initial proof was released by Andrew Wiles in June 1993 but was found to contain an error in September of that year. Wiles would go on to publish a corrected proof in 1995.
• Jordan curve theorem. There has been some controversy about whether Jordan's original proof of this in 1887 contains gaps. Oswald Veblen in 1905 claimed that Jordan's proof is incomplete, but in 2007 Hales said that the gaps are minor and that Jordan's proof is essentially complete.
• In 1905 Lebesgue tried to prove the result that a function implicitly defined by a Baire function is Baire, but his proof incorrectly assumed that the projection of a Borel set is Borel. Suslin pointed out the error and was inspired by it to define analytic sets as continuous images of Borel sets.
• Dehn's lemma. Dehn published an attempted proof in 1910, but Kneser found a gap in 1929. It was finally proved in 1956 by Christos Papakyriakopoulos.
• Hilbert's sixteenth problem about the finiteness of the number of limit cycles of a plane polynomial vector field. Henri Dulac published a partial solution to this problem in 1923, but in about 1980 Écalle and Ilyashenko independently found a serious gap, and fixed it in about 1991.
• In 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the theorem of the three geodesics, which was later found to be flawed. The proof was completed by Werner Ballmann about 50 years later.
• Littlewood–Richardson rule. Robinson published an incomplete proof in 1938, though the gaps were not noticed for many years. The first complete proofs were given by Marcel-Paul Schützenberger in 1977 and Thomas in 1974.
• Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete proof as it contained a gap, and the first complete proofs were given in about 1967 by Baker and Stark. In 1969 Stark showed how to fill the gap in Heegner's paper.
• In 1954 Igor Shafarevich published a proof that every finite solvable group is a Galois group over the rationals. However Schmidt pointed out a gap in the argument at the prime 2, which Shafarevich fixed in 1989.
• Nielsen realization problem. Kravetz claimed to solve this in 1959 by first showing that Teichmüller space is negatively curved, but in 1974 Masur showed that it is not negatively curved. The Nielsen realization problem was finally solved in 1980 by Kerckhoff.
• Yamabe problem. Yamabe claimed a solution in 1960, but Trudinger discovered a gap in 1968, and a complete proof was not given until 1984.
• Mordell conjecture over function fields. Manin published a proof in 1963, but found and corrected a gap in the proof.
• In 1973 Britton published a 282-page attempted solution of Burnside's problem. In his proof he assumed the existence of a set of parameters satisfying some inequalities, but Adian pointed out that these inequalities were inconsistent. Novikov and Adian had previously found a correct solution around 1968.
• Classification of finite simple groups. In 1983, Gorenstein announced that the proof of the classification had been completed, but he had been misinformed about the status of the proof of classification of quasithin groups, which had a serious gap in it. A complete proof for this case was published by Aschbacher and Smith in 2004.
• In 1986, Spencer Bloch published the paper "Algebraic Cycles and Higher K-theory" which introduced a higher Chow group, a precursor to motivic cohomology. The paper used an incorrect moving lemma; the lemma was later replaced by 30 pages of complex arguments that "took many years to be accepted as correct."
• Kepler conjecture. Hsiang published an incomplete proof of this in 1993. In 1998 Hales published a proof depending on long computer calculations.

  Incorrect results

• In 1759 Euler claimed that there were no closed knight tours on a chess board with 3 rows, but in 1917 Ernest Bergholt found tours on 3 × 10 and 3 × 12 boards.
• Euler's conjecture on Graeco-Latin squares. In the 1780s Euler conjectured that no such squares exist for any oddly even number n ≡ 2. In 1959, R. C. Bose and S. S. Shrikhande constructed counterexamples of order 22. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search. Finally Parker, Bose, and Shrikhande showed this conjecture to be false for all n ≥ 10.
• In 1798 A. M. Legendre claimed that 6 is not the sum of 2 rational cubes, which as Lamé pointed out in 1865 is false as 6 = ³ + ³.
• In 1803, Gian Francesco Malfatti claimed to prove that a certain arrangement of three circles would cover the maximum possible area inside a right triangle. However, to do so he made certain unwarranted assumptions about the configuration of the circles. It was shown in 1930 that circles in a different configuration could cover a greater area, and in 1967 that Malfatti's configuration was never optimal. See Malfatti circles.
• In 1806 André-Marie Ampère claimed to prove that a continuous function is differentiable at most points. However, in 1872 Weierstrass gave an example of a continuous function that was not differentiable anywhere: The Weierstrass function.
• Intersection theory. In 1848 Steiner claimed that the number of conics tangent to 5 given conics is 7776 = 6⁵, but later realized this was wrong. The correct number 3264 was found by Berner in 1865 and by Ernest de Jonquieres around 1859 and by Chasles in 1864 using his theory of characteristics. However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and Macpherson in about 1978.
• Dirichlet's principle. This was used by Riemann in 1851, but Weierstrass found a counterexample to one version of this principle in 1870, and Hilbert stated and proved a correct version in 1900.
• incorrectly claimed that there are three different groups of order 6. This mistake is strange because in an earlier 1854 paper he correctly stated that there are just two such groups.
• In 1885, Evgraf Fedorov classified the convex polyhedra with congruent rhombic faces, but missed a case. Stanko Bilinski in 1960 rediscovered the Bilinski dodecahedron and proved that, with the addition of this shape, the classification was complete.
• Wronskians. In 1887 Mansion claimed in his textbook that if a Wronskian of some functions vanishes everywhere then the functions are linearly dependent. In 1889 Peano pointed out the counterexample x² and x|''x|. The result is correct if the functions are analytic.
• published a purported example of an algebraic curve in 3-dimensional projective space that could not be defined as the zeros of 3 polynomials, but in 1941 Perron found 3 equations defining Vahlen's curve. In 1961 Kneser showed that any algebraic curve in projective 3-space can be given as the zeros of 3 polynomials.
• In 1898 Miller published a paper incorrectly claiming to prove that the Mathieu group M₂₄ does not exist, though in 1900 he pointed out that his proof was wrong.
• Little claimed in 1900 that the writhe of a reduced knot diagram is an invariant. However, in 1974 Perko discovered a counterexample called the Perko pair, a pair of knots listed as distinct in tables for many years that are in fact the same.
• Hilbert's twenty-first problem. In 1908 Plemelj claimed to have shown the existence of Fuchsian differential equations with any given monodromy group, but in 1989 Bolibruch discovered a counterexample.
• In 1925 Ackermann published a proof that a weak system can prove the consistency of a version of analysis, but von Neumann found an explicit mistake in it a few years later. Gödel's incompleteness theorems showed that it is not possible to prove the consistency of analysis using weaker systems.
• Groups of order 64. In 1930 Miller published a paper claiming that there are 294 groups of order 64. Hall and Senior showed in 1964 that the correct number is 267.
• Kurt Gödel proved in 1933 that the truth of a certain class of sentences of first-order arithmetic, known in the literature as , was decidable. That is, there was a method for deciding correctly whether any statement of that form was true. In the final sentence of that paper, he asserted that the same proof would work for the decidability of the larger class ₌, which also includes formulas that contain an equality predicate. However, in the mid-1960s, Stål Aanderaa showed that Gödel's proof would not go through for the larger class, and in 1982 Warren Goldfarb showed that validity of formulas from the larger class was in fact undecidable.
• Grunwald–Wang theorem. Wilhelm Grunwald published an incorrect proof in 1933 of an incorrect theorem, and George Whaples later published another incorrect proof. Shianghao Wang found a counterexample in 1948 and published a corrected version of the theorem in 1950.
• In 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but showed that this is false for surfaces of positive geometric genus.
• One of many examples from algebraic geometry in the first half of the 20th century: claimed that a degree-n'' surface in 3-dimensional projective space has at most −4 nodes, B. Segre pointed out that this was wrong; for example, for degree 6 the maximum number of nodes is 65, achieved by the Barth sextic, which is more than the maximum of 52 claimed by Severi.
• Rokhlin invariant. Rokhlin incorrectly claimed in 1951 that the third stable stem of the homotopy groups of spheres is of order 12. In 1952 he discovered his error: it is in fact cyclic of order 24. The difference is crucial as it results in the existence of the Rokhlin invariant, a fundamental tool in the theory of 3- and 4-dimensional manifolds.
• In 1961, Jan-Erik Roos published an incorrect theorem about the vanishing of the first derived functor of the inverse limit functor under certain general conditions. However, in 2002, Amnon Neeman constructed a counterexample. Roos showed in 2006 that the theorem holds if one adds the assumption that the category has a set of generators.
• The Schur multiplier of the Mathieu group M₂₂ is particularly notorious as it was miscalculated more than once: first claimed it had order 3, then in a 1968 correction claimed it had order 6; its order is in fact 12. This caused an error in the title of Janko's paper A new finite simple group of order 86,775,570,046,077,562,880 which possesses M₂₄ and the full covering group of M₂₂ as subgroup on J4: it does not have the full covering group as a subgroup, as the full covering group is larger than was realized at the time.
• The original statement of the classification of N-groups by Thompson in 1968 accidentally omitted the Tits group, though he soon fixed this.
• In 1975, Leitzel, Madan, and Queen incorrectly claimed that there are only 7 function fields over finite fields with genus > 0 and class number 1, but in 2013 Stirpe found another; there are in fact exactly 8.
• Busemann–Petty problem. Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R⁴ has a negative solution, and in the second of which he proved that it has a positive solution.
• Algebraic stacks. The book on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by.