Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered.
I am interested in more such examples, especially, in former (famous or interesting) "theorems" where no remedy could be found to the present day.
I just discovered that Wikipedia maintains a page entitled "List of incomplete proofs." Each of the more than $60$ entries is marked with these symbols:
For over $130$ years people have been steadily looking for a resolution to the following problem: what is the maximum number of limit cycles for the system of differential equations $x'=f(x,y), y'=g(x,y)$ where $f$ and $g$ are any real quadratic polynomials, in $x$ and $y$. In the 1950's, two Russian mathematicians (Petrovskii and Landis) wrote a paper claiming that the maximum is $3$. People tried to understand the proof but found holes in it, and attempted to fix it up. The famous Arnold was skeptical about the possibility. In the 1970's, two Chinese mathematicians (Chen and Wang) discovered a specific set of coefficients for the polynomials $f$ and $g$ and showed that this system has $4$ limit cycles. Of course, nobody tried to patch up the previous claim of max=$3$.
I thought this to be an amusing story, perhaps slightly in the direction of your question/request.
Here is one survey article on Hilbert's 16th Problem, where this issue is on p 5, section 4, problem 3.
Let $k$ be a field, and let $K=k(t)$ be the field of rational functions over $k$. Tate and Shafarevich [1] proved that if $k$ is a finite field, then for every $r$ there exists an elliptic curve $E/K$ with $\text{rank}~E(K)\ge r$. Their construction uses ideas from an earlier paper of Lapin [2], and indeed, the Tate-Shafarevich paper says that "analogous examples have been constructed over the field $k(t)$, when $k$ is an algebraically closed field of characteristic zero, by A. I. Lapin." However, various problems with Lapin's construction were discovered, so although the Tate-Shafarevich proof is fine for $k$ finite, the characteristic $0$ case, e.g., for $K=\mathbb C(t)$, is regarded as an open problem. (Technical note: In the case that $k$ is infinite, one requires that $E$ not be isotrivial, which means that $E$ is not isomorphic over $K$ to an elliptic curve defined over $k$.)
Tate, J., Shafarevich, I.R., The rank of elliptic curves. Dokl. Akad. Nauk SSSR 175 (1967), 770–773.
Lapin, A.I., Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 953–988.
Not unresolved today, but...
The Lusternik-Schnirelmann "Theorem of the three geodesics" claims that on any Riemannian manifold topologically a sphere there are three simple (non-self-intersecting) closed geodesics (and for the ellipsoid, exactly three). This was "proved" in 1929, but the proof was soon recognized to be flawed. It was not entirely settled until ~50 years later, by Ballmann in 1978, followed by another proof and generalization by Klingenberg in 1985:
Werner Ballmann, "Der Satz von Lusternik und Schnirelmann" Math. Shriften 102, pp. 1-25, 1978.
Wilhelm Klingenberg, "The existence of three short closed geodesics," Differential Geometry and Complex Analysis, Springer, Berlin, pp. 169–179, 1985.
Carmichael's totient function conjecture:
For every positive integer $m$, there is at least one integer $n \neq m$ such that $\varphi(m)=\varphi(n)$.
Carmichael stated this as a theorem in 1907, but he retracted his proof in 1922 and stated this conjecture as an open problem.
As mentioned in Kevin Casto's comment, this question is almost a duplicate of Widely accepted mathematical results that were later shown wrong. As I mentioned in an answer to that question, R. B. Kershner's paper "On Paving the Plane," Amer. Math. Monthly 75 (1968), 839–844, announced the classification of all convex pentagons that tile the plane. Kershner said that "The proof…is extremely laborious and will be given elsewhere." The proof was wrong; Richard James III and Marjorie Rice found counterexamples in the 1970s, Stein found another counterexample in 1985, and Mann–McLoud–Von Derau found yet another counterexample in 2015. The classification of all such tilings remains an open problem to this day.
The third "theorem" of Fermat, formulated in 1637, stood as a conjecture for a long time: although Fermat claimed to have proved it, he did not share his proof and nobody was able to offer any proof for many centuries. Many "proofs" were discovered and later refuted, so the assertion kept oscillating between theorem and conjecture status, until Andrew Wiles finally proved the conjecture in 1994 (having himself previously offered an unsuccessful proof). See https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
