# When did bounties and prize money for open mathematical problems start being a thing?

I'm a science/math journalist [ger] and currently I'm working on an article about the culture of prize money/bounties for solving open mathematical problems (Millennium Prize Problems and such). One section will be about the history of such bounties for which I'd like to ask you for some historical examples. I will certainly include the bounties in the Scottish Book from the Lwow School, as well as Erdös' faible for betting money on open problems. Thanks to a respond on the same question on MathOverflow I will also include the bounty for solving the three body problem, announced by king Oscar II of Sweden as an older example.

Now my question is: Does any of you know further examples for such rewards and maybe even know reliable sources for these?

Thank you very much for your help!

• Another example would be the Wolfskehl Prize. I don't have a reference handy, but I think prize money was involved when various mathematicians competed to solve various forms of cubic equations in the 16h century. Sep 14, 2022 at 20:24
• This claymath.org/library/monographs/MPPc.pdf purports to contain an overview (starting p.3). The first prize mentioned was won by Tartaglia in 1535.
– abo
Sep 14, 2022 at 20:43
• The reference @abo points to is: Jeremy Gray, "A History of Prizes in Mathematics", In: J. Carlson, A. Jaffe, and A. Wiles (eds.), The Millennium Prize Problems, Clay Mathematics Institute & American Mathematical Society, 2006, pp. 3-33 (online) Sep 14, 2022 at 20:44
• For details of two prizes, you may want to read the following: June Barrow-Green, "Oscar II's Prize Competition and the Error in Poincaré's Memoir on the Three Body Problem," Archive for history of exact sciences, June 1994, pp. 107-131 (scan online); Klaus Barner, "Paul Wolfskehl and the Wolfskehl prize," Notices of the AMS, Vol. 44, No. 10, November 1997, pp. 1294-1303 (scan online) Sep 14, 2022 at 23:02
• Donald Knuth offers money prizes for mistakes found in his book. Sep 15, 2022 at 13:34

## 3 Answers

I'm not sure which is the earliest. But one of the earliest I knew of is due to Mobius.

In 1846, Mobius invited Grassmann to paricipate in a competition to solve a problem first set by Liebniz: to derive a geometric calculus without coordinates, his famous analysis situs. His entry was the winning entry - it was also the only entry. Grassmann of course is responsible for the modern development of linear and exterior algebra.

I think I first read about this first in a biography of Grassmann.

On a much smaller scale and at very personal level consider Erdős problems

Erdős had a reputation for posing new problems as well as solving existing ones - Ernst Strauss called him "the absolute monarch of problem posers". Throughout his career, Erdős would offer payments for solutions to unresolved problems. These ranged from USD 25 for problems that he felt were just out of the reach of the current mathematical thinking (both his and others) up to USD 10,000 for problems that were both difficult to attack and mathematically significant. Some of these problems have since been solved, including the most lucrative - Erdős's conjecture on prime gaps was solved in 2014, and the USD 10,000 paid.

Surely, what I refer below it is not one of the earliest prizes, but it is important in the history of mathematics.$$^1$$

There is a famous story about heat diffusion, at the beginning of 19th century, that involved J. B. Fourier and his formulation of trigonometric series, and a prize which Fourier won, but with reserve and critics. The rather troubled story of this prize is intertwined with the history of the development of Fourier’s series and heat theory.

There was, at the beginning of the 19th century, in France, a sound interest, theorical and practical, about the subject of heat nature and diffusion, on which had worked mathematicians and physicists as Lagrange and Lavoisier. During these years, Fourier began to deal with the subject in an original manner.

His most important book, La Théorie Analytique de la Chaleur, published in 1822, in which Fourier presented his theory, was written in several stages.

It was preceded by a memoir that Fourier submitted to the Paris Academy of Sciences in 1807, but its publication was rejected because of the opposition of Lagrange, who considered not rigorous the Fourier's treatment of trigonometric series.

Fourier answered to the objections of Lagrange in an article, sent to Lagrange, and in another article sent to the Academia, in 1808. But his 1807 memoir remained unpublished, and has been published only recently.$$^2$$

But the question of heat diffusion remained open and, in 1810, the Academy of Sciences announced a prize competition on heat diffusion. Fourier submitted a revised version of his memoir, and won the competition, but was again criticized for lack of rigour and generality, as can be read in the report of the judges of the competition:

This work contains the true differential equations of heat diffusion [...] the novelty of the subject, together with its importance, convinced the Class of Mathematical, Physical and Natural Sciences of the Academy to award the prize to this work,[..] but observing that it isn’t without difficulties and that its analysis, to integrate them, is still unsatisfactory, with regard to generality and also with regard to rigour.$$^3$$

For this reason, probably, this memoir was not published until 1824, when Fourier had become secretary of the Academy of Sciences.

The Théorie Analytique de la Chaleur had been published in 1822, after Lagrange’s death and when Fourier’s reputation among the Academy of Sciences was rising.

$$^1$$ I don’t need to tell that Fourier’s work about heat diffusion and trigonometric series are one of the most important achievements in the first half of 19th analysis, and beyond: think that also Cantor’s set theory came from Cantor’s studies about Fourier’s series.

$$^2$$ In Grattan Guinnnes I. and Ravetz J.R., Joseph Fourier, 1768-1830, Cambridge, Mass., 1972.

$$^3$$ Quoted in U. Bottazzini, Il calcolo sublime, Boringhieri, 1981. p. 60 (en. transl. Bottazzini, U, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer/Verlag, 2012.