Some conjectures are disproved by a single counter-example and garner little or no further interest or study, such as (to my knowledge) Euler's conjecture in number theory that at least $n$ $n^{th}$ powers are required to sum to an $n^{th}$ power, for $n>2$ (disproved by counter-example by L. J. Lander and T. R. Parkin in 1966). But other conjectures retain interest, even after being disproved by counter-example, and lead to analytic disproofs and insights.

**Question**

What are some conjectures, first known to be false through counter-example, but whose subsequent analytic disproofs shed novel insights into the problem? More generally, what are the properties of known false conjectures that nevertheless garner significant interest and efforts of mathematicians to produce analytic disproofs?

analyticdisproofs. Was that intentional? $\endgroup$ – KConrad Jul 12 '18 at 1:59particularcounter-example contains information, but in an extreme case a counter-example merely answers the single yes-no question as to whether the conjecture is true. "Analytic" or other proofs generally provide much more information and insight. $\endgroup$ – David G. Stork Jul 12 '18 at 2:24