Many famous results were discovered through non-rigorous proofs, with correct proofs being found only later and with greater difficulty. One that is well known is Euler's 1737 proof that

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\frac{\pi^2}{6}$

in which he pretends that the power series for $\frac{\sin\sqrt{x}}{\sqrt{x}}$ is an infinite polynomial and factorizes it from knowledge of its roots.

Another example, of a different type, is the Jordan curve theorem. In this case, the theorem seems obvious, and Jordan gets credit for realizing that it requires proof. However, the proof was harder than he thought, and the first rigorous proof was found some decades later than Jordan's attempt. Many of the basic theorems of topology are like this.

Then of course there is Ramanujan, who is in a class of his own when it comes to discovering theorems without proving them.

I'd be interested to see other examples, and in your thoughts on what the examples reveal about the connection between discovery and proof.

**Clarification**. When I posed the question I was hoping for some explanations
for the gap between discovery and proof to emerge, without any hinting from me. Since this hasn't happened much yet, let me suggest some possible
explanations that I had in mind:

*Physical intuition*. This lies behind results such as the Jordan curve theorem,
Riemann mapping theorem, Fourier analysis.

*Lack of foundations*. This accounts for the late arrival of rigor in calculus, topology,
and (?) algebraic geometry.

*Complexity*. Hard results cannot proved correctly the first time, only via a
series of partially correct, or incomplete, proofs. Example: Fermat's last theorem.

I hope this gives a better idea of what I was looking for. Feel free to edit your answers if you have anything to add.

et al.) the dawn of modern statistical and quantum physics had a great deal to do with theconsolidation ofrigor throughout mathematics. Indeed, ergodic theory and functional analysis owe a great deal to these disciplines, and neither could have existed in the time of (say) Euler because the approach to mathematics was different. $\endgroup$10more comments