With what kind of proof was the Binet formula derived for the first time?

Question

As everyone knows, there are dozens of proofs of Binet's Fibonacci Number Formula.

My question is, what was the first approach used by Binet? What let people to say this result (the Binet formula) was known to Euler and Bernoulli?

Answer 1

The original proof by Binet was published in 1843 in Paris.

Binet, J. P.: Mémoire sur l'intégration des équations linéaires aux différences finies, d'un ordre quelconque à coefficients variables, Comptes Rendus des Séances de l'Académie des Sciences 17 (1843), 559-567.

You could find the online version of the paper here or here, and a list of publications by Binet here.

cs.cas.cz says:

Like many results in mathematics it is often not the original discoverer who gets the glory of having their name attached to the result. J. P. M. Binet (1786-1856) published [Bin] this result now known as the Binet’s formula in 1843 although the result was known earlier. It seems that Daniel Bernoulli (1700-1782) discovered and proved this formula in 1726 ([Ber], §7). Two years later also Euler mentioned the formula in a letter to Bernoulli, but he published [Eul] it only in 1765. A. de Moivre (1667-1754) published it in the general form in the first systematic account of linear recurrences ([Mo1], pp. 26-42) and it seems that Binet’s formula was known to him already in 1718 [Mo2]. From the later authors let us mention for instance E.Lucas who may rediscovered it but did not report it before 1876.

References:

[Bin] Binet, J. P. (1843). Mémoire sur l’intégration des équations linéaires aux différences finies, d’un ordre quelconque, à coefficients variables. Comptes Rendus hebdomadaires des séances de l’Académie des Sciences (Paris), 17, 559-567.

[Ber] Bernoulli, D. (1728). Comment. Acad. Sci. Petrop., 3, 85-100.

[Eul] Euler, L. (1765). Observationes analyticae. Novi commentarii ascaemiae scientiarum imperialis Petropolotanae, 11, 124-143.

[Mo1] de Moivre, A. (1730). Miscellanea analytica de seriebus et quadraturis . London.

[Mo2] de Moivre, A. (1722). Philos. Trans., 32, 162-178.

maths.surrey.ac.uk says:

Many authors say that this formula was discovered by J. P. M. Binet (1786-1856) in 1843 and so call it Binet's Formula. Graham, Knuth and Patashnik in Concrete Mathematics (2nd edition, 1994) mention that Euler had already published this formula in 1765. But Don Knuth in The Art of Computer Programming, Volume 1 Fundamental Algorithms, section 1.2.8, traces it back even further, to A de Moivre (1667-1754). He had written about "Binet's" formula in 1730 and had indeed found a method for finding formulae for any general series of numbers formed in a similar way to the Fibonacci series. Like many results in Mathematics, it is often not the original discoverer who gets the glory of having their name attached to the result, but someone later!

References:

Concrete Mathematics (2nd edition, 1994) by Graham, Knuth and Patashnik, Addison-Wesley.

The Art of Computer Programming D E Knuth, Volume 1: Fundamental Algorithms (now in its Third Edition, 1997).

mathworld.wolfram.com says:

It was derived by Binet in 1843, although the result was known to Euler, Daniel Bernoulli, and de Moivre more than a century earlier.

References:

Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, p. 108, 2002.

Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 21, 2000.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 62, 1986.