# Invalid mathematical proof that was considered valid for the longest period of time?

Sometimes, mathematical proofs accepted in the scientific community are found to be wrong. What is the disproved proof that was believed to be correct for the longest period of time?

clarification: Note that I'm asking only about proofs where the result itself turned out to be false.

• The longest should be among the subtractions you can do in the (currently 39) replies at mathoverflow.net/questions/35468/…. Either that, or you’ll become convinced that the question is meaningless. Dec 21, 2017 at 23:28

## 2 Answers

In 1923 Henri Dulac published a solution of Hilbert problem 16, second part. This was accepted, he received a prize of the French academy, and the problem was considered solved until a gap in the proof was found by Ilyashenko in 1982. It took about 10 years to fill the gap (Ilyashenko and Ecalle, independently, 1992). 1982-1923=59, so for 59 years the fact was believed to be proven while it was not. This is the longest example that I know.

Are you asking about proofs of results where the result itself turns out to be false, or do you allow proofs with one part that was incomplete (a proof not covering all cases is strictly speaking not a valid proof of the claimed theorem) and the gap was filled in later?

The history of the Kronecker-Weber theorem provides an example of an incomplete proof that went unquestioned for a long time. Weber published a proof in 1886 and a gap was found in it and filled in 95 years later by Neumann. See the Wikipedia page about the Kronecker-Weber theorem. Arguably the reason nobody noticed the gap for so long is that, in the intervening years (starting with Hilbert ten years after Weber) other proofs and further conceptual generalizations of the Kronecker-Weber theorem were found, so there was not a strong incentive to recheck Weber's proof in order to have a proof of the result.

Again, if what you are really after is proofs that were fundamentally mistaken rather than being incomplete, this example I give does not fit. Please clarify your intentions.

Arguably, until a rigorous definition of real numbers and a proof of their completeness was found in the 1800s, lots of earlier work involving the real numbers was not fully justified. For example, Euclidean geometry needs further conditions related to continuity or completeness of the reals, which the ancient Greeks did not realize. Thus you could say there are 2000-year old examples in Euclidean geometry, and you'd be hard-pressed to find examples over a longer time period than that.

• According to Wikipedia, the first complete proof was given by Hilbert in 1896. Dec 22, 2017 at 15:33