errors in some of these incomplete proofs. A strengthening of Hilbert's sixteenth problem asking whether there exists a uniform finite upper bound for the number of limit cycles of planar polynomial vector fields of given degree n. In the 1950s, Evgenii Landis and Ivan Petrovsky published a purported solution, but it was shown wrong in the early 1960s. In 1954 Zarankiewicz claimed to have solved Turán's brick factory problem about the crossing number of complete bipartite graphs, but Kainen and Ringel later noticed a gap in his proof. Complex structures on the 6-sphere. In 1969 Alfred Adler published a paper in the American Journal of Mathematics claiming that the 6-sphere has no complex structure. His argument was incomplete, and this is (as of 2016) still a major open problem. Closed geodesics. In 1978 Wilhelm Klingenberg published a proof that smooth compact manifolds without boundary have infinitely many closed geodesics. His proof was controversial, and there is currently (as of 2016) no consensus on whether his proof is complete. Telescope conjecture. Ravenel announced a refutation of this in 1992, but later withdrew it, and the conjecture is still open. Matroid bundles. In 2003 Daniel Biss published a paper in the Annals of Mathematics claiming to show that matroid bundles are equivalent to real vector bundles, but in 2009 published a correction pointing out a serious gap in the proof. His correction was based on a 2007 paper by Mnëv. In 2012, the Japanese mathematician Shinichi Mochizuki released online a series of papers in which he claims to prove the abc conjecture. Despite the publication in a peer-reviewed journal later, his proof has not been accepted as correct in the mainstream mathematical community. See also List of long mathematical proofs List of disproved mathematical ideas Notes References Further reading — Lists over a hundred pages of (mostly trivial) published errors made by mathematicians. External links David Mumford email about the errors of the Italian algebraic geometry school under Severi The first 9 pages of mention some examples of incorrect results in homotopy theory. MathOverflow questions Ilya Nikokoshev, Most interesting mathematics mistake? Kevin Buzzard what mistakes did the Italian algebraic geometers actually make? Will Jagy, Widely accepted mathematical results that were later shown wrong? John Stillwell, What are some correct results discovered with incorrect (or no) proofs? Moritz. Theorems demoted back to conjectures Mei Zhang, Proofs shown to be wrong after formalization with