The Basel problem, $\sum_{n=1}^\infty{\frac{1}{n^2}}$, took 90 years for a closed form, $\frac{\pi^2}{6}$, to be found.

I'm curious to know what other mathematical expressions, especially those involving limits, took a long time for a closed form to be found. (By "found", I mean discovered and verified.)

  • 4
    $\begingroup$ Cardano's formula for the roots of a cubic. It is hard to say when the idea of a closed form algebraic solution was first conceived, but Omar Khayyam already talked of its desirability c. 1100. The formula was only derived c. 1540. $\endgroup$
    – Conifold
    Jun 19, 2022 at 8:55
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    $\begingroup$ In more recent times, one of Ramanujan's closed form expressions for a series from his lost notebook (written in 1919-20) was only proved in 2019 by Berndt, Li, and Zaharescu, see (1.10). In fairness, the notebook was only found in 1976. $\endgroup$
    – Conifold
    Jun 19, 2022 at 9:35

1 Answer 1


Euler's Pentagonal number theorem was discovered by Euler in 1775, published in 1783. In his paper Euler gave only heuristic arguments. The first
rigorous proof was obtained by Jacobi in 1829.

The theorem is an identity for formal series: $$\prod_{n=1}^\infty(1-x^n)=\sum_{-\infty}^\infty(-1)^nx^{n(3n-1)/2}.$$

  • $\begingroup$ I am guessing that this was in his Fundamenta Nova. Please give the page number? $\endgroup$
    – Somos
    Jun 25, 2022 at 2:46

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