Editing Édouard Lucas (section)

==Biography==
Lucas was born in [[Amiens]] and educated at the [[École Normale Supérieure]].<ref name=":0">{{Cite web|url=http://mathshistory.st-andrews.ac.uk/Biographies/Lucas.html|title=Édouard Lucas|last=O'Connor|first=John|website=MacTutor History of Mathematics archive, University of St Andrews.}}</ref> He worked in the [[Paris Observatory]] and later became a professor of mathematics at the Lycée Saint Louis and the  Lycée Charlemagne in  [[Paris]].<ref name=":0" />

Lucas served as an artillery officer in the French Army during the Franco-Prussian War of 1870–1871.<ref name=":0" />

In 1875, Lucas posed a challenge to [[mathematical proof|prove]] that the only solution of the [[Diophantine equation]]

:<math>\sum_{n=1}^{N} n^2 = M^2\;</math>

with ''N'' > 1 is when ''N'' = 24 and ''M'' = 70. This is known as the [[cannonball problem]], since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a [[square pyramid]] out of them. It was not until 1918 that a proof (using [[elliptic function]]s) was found for this remarkable fact, which has relevance to the [[bosonic string theory]] in 26 dimensions.<ref>{{cite web|url=http://math.ucr.edu/home/baez/week95.html |title=week95 |publisher=Math.ucr.edu |date=1996-11-26 |access-date=2012-01-04}}</ref> More recently, [[elementary proof]]s have been published.<ref>{{Cite journal|author=Ma, D. G. |title=An Elementary Proof of the Solutions to the Diophantine Equation <math>6y^2=x(x+1)(2x+1)</math> |journal=Sichuan Daxue Xuebao |volume=4 |pages=107–116 |year=1985}}</ref><ref>{{Cite journal|author=Anglin, W. S. |title=The Square Pyramid Puzzle|jstor=2323911 |journal=[[American Mathematical Monthly]] |volume=97 |issue=2 |pages=120–124 |year=1990 |doi=10.2307/2323911}}</ref>

He devised methods for testing the [[prime number|primality]] of numbers. In 1857, at age 15, Lucas began testing the primality of 2<sup>127</sup>&nbsp;&minus;&nbsp;1, a number with 39 decimal digits, by hand, using [[Lucas sequence]]s. In 1876, after 19 years of testing,<ref>{{cite web|url=http://primes.utm.edu/curios/page.php?number_id=135 |title=Prime Curios!: 17014...05727 (39-digits) |publisher=Primes.utm.edu |access-date=2012-01-04}}</ref> he finally proved that 2<sup>127</sup>&nbsp;&minus;&nbsp;1 was prime; this would remain the largest known [[Mersenne prime]] for three-quarters of a century. This may stand forever as the largest prime number proven by hand. Later [[Derrick Henry Lehmer]] refined Lucas's [[Lucas primality test|primality tests]] and obtained the [[Lucas–Lehmer primality test]].

He worked on the development of the [[umbral calculus]].

Lucas is credited as the first to publish the [[Kempner function]].<ref>{{cite web|author=Sondow, Jonathan|author2=Weisstein, Eric W.|author-link2=Eric W. Weisstein|title=Smarandache Function|website=MathWorld—A Wolfram Web Resource|url=https://mathworld.wolfram.com/SmarandacheFunction.html}}</ref>

Lucas was also interested in [[recreational mathematics]]. He found an elegant [[Binary numeral system|binary]] solution to the [[Baguenaudier]] puzzle.<ref>{{cite journal | last = Lucas | first = Édouard | date = 1880 | title = Récréations scientifiques sur l'arithmétique et sur la géométrie de situation | publisher = G. Baillière | url = https://gallica.bnf.fr/ark:/12148/bpt6k2150950 | journal = La Revue scientifique de la France et de l'étranger: Revue des cours scientifiques | language = fr | volume = 10 | issue = 1 | pages = 36–42 | access-date = 2019-05-13}}</ref> He also invented the [[Tower of Hanoi]] puzzle in 1883, which he marketed under the nickname ''N. Claus de Siam'', an [[anagram]] of ''Lucas d'Amiens'', and published for the first time a description of the [[dots and boxes]] game in 1889.

Lucas died in unusual circumstances. At the banquet of the annual congress of the ''Association française pour l'avancement des sciences'', a waiter dropped some [[crockery]] and a piece of broken plate cut Lucas on the cheek. He died a few days later of a severe skin inflammation, probably caused by [[sepsis]], at 49 years old.