Lagrangen neljän neliön lause

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Lukuteoriassa Lagrangen lauseen mukaan jokainen positiivinen kokonaisluku on esitettävissä neljän neliön summana. Lauseen otaksui Claude Gaspard Bachet de Méziriac ja sen todisti 1770 Joseph Louis Lagrange. Lause esiintyi jo Diofantoksen Arithmeticassa, josta Bachet käänsi sen latinaksi vuonna 1621. Esimerkkeinä lauseesta on

3 = 12 + 12 + 12 + 02
31 = 52 + 22 + 12 + 12
310 = 172 + 42 + 22 + 12.

Adrien-Marie Legendre täydensi lausetta vuonna 1798 väittämällä, että positiivinen kokonaisluku voidaan esittää kolmen neliön summana jos ja vain jos se ei ole muotoa 4k(8m + 7). Legendren todistus oli puutteellinen ja lauseen todisti ensimmäisenä Carl Friedrich Gauss.

Vuonna 1834 Carl Gustav Jakob Jacobi löysi täsmällisen kaavan sille, monellako tavalla annettu positiivinen kokonaisluku voidaan esittää neljän neliön summana. Tämä luku on kahdeksan kertaa n:n tekijöiden lukumäärä jos n on pariton ja 24 kertaa n:n parittomien tekijöiden lukumäärä jos n on parillinen.

Lagrangen neljän neliön lause on erikoistapaus Fermat'n monikulmiolauseesta ja Waringin probleemasta. Myös seuraava tulos on yleisempi kuin Lagrangen lause: Olkoon a, b, c ja d annettuja luonnollisia lukuja. Voidaanko löytää kokonaisluvut luvut x1, x2, x3, x4 siten, että yhtälöllä

n = ax12 + bx22 + cx32 + dx42

on ratkaisu n:n kaikilla positiivisilla kokonaislukuarvoilla. Tapaus a=b=c=d=1 on Lagrangen lause. Yleisen tapauksen todisti Srinivasa Ramanujan. Hän todisti, että jos a ≤ b ≤ c ≤ d, on olemassa täsmälleen 54 eri tapaa valita a, b, c ja d siten, että yhtälöllä on ratkaisu kaikilla n arvoilla, kun x1, x2, x3, x4 ovat kokonaislukuja. (Ramanujan listasi 55:nnen mahdollisuuden a=1, b=2, c=5, d=5, mutta tässä tapauksessa yhtälöllä ei ole ratkaisuja, jos n=15. [1]

Lähteet

  • Ireland ja Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1990, ISBN 0-387-97329-X

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