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Claimed proofs

Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.

Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the abc conjecture. He released a series of four preprints developing a new theory he called inter-universal Teichmüller theory (IUTT), which is then applied to prove the abc conjecture. The papers have not been widely accepted by the mathematical community as providing a proof of abc. This is not only because of their length and the difficulty of understanding them, but also because at least one specific point in the argument has been identified as a gap by some other experts. Although a few mathematicians have vouched for the correctness of the proof and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.

In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki. While they did not resolve the differences, they brought them into clearer focus.Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that... small modifications will not rescue the proof strategy"; Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.

On April 3, 2020, two mathematicians from the Kyoto research institute where Mochizuki works announced that his claimed proof would be published in Publications of the Research Institute for Mathematical Sciences, the institute's journal. Mochizuki is chief editor of the journal but recused himself from the review of the paper. The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp". In March 2021, Mochizuki's proof was published in RIMS.