Bonne année 2024 - Propriétés du nombre 2024

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Bonne année 2024

2024 quelques propriétés de cet entier naturel

Le site Math93.com vous souhaite une heureuse, chaleureuse et studieuse année 2024. Profitons-en pour revenir sur quelques caractéristiques de ce nombre pair.

 

  1. Ecriture de 2024
  2. L'année 2024
  3. Diviseurs de 2024
  4. Quelques décompositions de 2024
  5. 2024 nombres abondant
  6. Autres curiosités de 2024
  7. 2024 nombre tétraédrique
  8. Décomposition de 2024 en somme de carrés et cubes
  9. Décomposition de 2024 en sommes de carrés

 

1) Écriture du nombre 2024

Cet entier s'écrit ainsi, en tenant compte de l'orthographe réformée par les recommandations de l'Académie Française publiées en 1990. Il faut savoir que cette orthographe révisée est la référence dorénavant !

Français : Deux-mille-vingt-quatre
Anglais : Two thousand twenty-four
Allemand : zwei­tausend­vier­und­zwanzig
Espagnol : Dos mil veinticuatro
Italien : duemilaventiquatro
Portugais : Dois mil vinte e quatro

 Autres écriture de 2024

En chiffre romain MMXXIV
En binaire 11111101000
En octal 3750
En hexadécimal 7e8

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2) L'année 2024

2024 est une année bissextile qui commence un lundi et compte 52 semaines et 2 jours soit \(52\times7+2=366\) jours.

On rappelle que les années sont bissextiles une fois tous les quatre ans. Sauf l'année du siècle et cela trois fois sur quatre.
Soit toutes les années divisibles par 4, sauf les siècles à l'exception des siècles divisibles par 4 (années divisibles par 400).
Avec 2024 on a :
$$2024 = 4\times 506$$

L'année 2024 comporte 2 vendredis 13 : les 13 septembre et 13 décembre.

2024 est une année à 52 dimanches. On a eu 53 dimanches en : 2000, 2006, 2012, 2017 et 2023.

 

3) Diviseurs de 2 024 et nombres premiers

  • Diviseurs
    L'entier pair 2024 admet 16 diviseurs (2023 n'en avait que 6) qui sont : $$1, 2, 4, 8, 11, 22, 23, 44, 46, 88, 92, 184, 253, 506, 1012, 2024$$
      
  • Nombre premiers : 2024 n'est pas un nombre premier.
    On rappelle qu'un nombre premier est un entier naturel qui admet exactement deux diviseurs distincts entiers et positifs.
    La précédente année première était 2017 et la prochaine année première sera 2027. 
       
  • Décomposition en facteurs premiers
    $$2 024 = 2^3\times11\times 23$$ 
  • Le 2024e nombre premier est  17 599

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4) 2024 et quelques décompositions

Par exemple :

2000, 2001, 2004, 2007, 2010, 2016, 2020, 2022, 2023, 2024, 2025, 2028, 2030, 2034, 2040, 2043, 2052, 2061, 2064, 2070, 2080, 2085, 2088, 2090, 2100 …

 

Prochaines plages de quatre années de Harshad consécutives:

[2022, 2023, 2024, 2025], [3030, 3031, 3032, 3033], [10307, 10308, 10309, 10310], …

Précédentes: [510, 511, 512, 513], [1014, 1015, 1016, 1017]

 

  • Nombre 3-polis : trois fois somme d'entiers consécutifs.
    Un nombre poli ou escalier est un nombre qui peut s'écrire sous la forme de une ou plusieurs sommes de deux ou plusieurs nombres consécutifs. Le degré de politesse indique combien de fois un nombre est sommes de nombres consécutifs.
    2024 est un nombre 3-poli.
      
    • \(2024= 77+78 +\cdots + 99\)
    • \(2024= 119+120+\cdots+134\)
    • \(2024= 179+180+\cdots+189\)
        

Pour en savoir plus sur ces nombres : nombres polis
Remarque : Tous les nombres peuvent se mettent sous la forme de somme de consécutifs sauf les puissances de 2. 

 

  • Somme de deux premiers consécutifs
    2024 n'est pas somme de 2 premiers consécutifs mais 2022 l'était : $$2022=1009+1013$$

 

  • Somme de deux nombres premiers
    2024 est somme de 2 nombres premiers 32 fois.
    • \(2024= 7+2017\)
    • \(2024= 13+2011\)
    • ...
    • \(2024= 991+1033\)
     
  • Nombre parfait : c'est un entier somme de ses diviseurs propres.
    2024 n'est pas parfait.
    La somme des diviseurs propres de 2024 est :
    $$1+2+ 4+8+ 11+ 22+ 23+44+ 46+ 88+ 92+ 184+ 253+ 506+ 1012 =4320$$

  • Nombre semi-parfait : somme de certains de ses diviseurs.
    2024 n'est pas semi parfait contrairement à 2022 : $$2 022 = 1011 + 674 + 337$$
     

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5) 2024 nombre abondant

Un nombre abondant est un nombre qui est inférieur à la somme de ses diviseurs propres, c'est à dire ses diviseurs autre que lui-même et il est déficient dans le cas contraire et parfait si il y a égalité. L'entier 2024 est donc abondant:

$$1+2+ 4+8+ 11+ 22+ 23+44+ 46+ 88+ 92+ 184+ 253+ 506+ 1012 =4320 > 2024$$

 

6) Autres Curiosités de 2024

Avec 20 et 24

$$2024 = (20+24)+(20+24)+(20+24)(20+24)$$

 

Coefficient binomiaux et 2024

$$2024 = \frac{24!}{3!(21)!} = \binom{24}{3} =\dfrac{22\times23\times24}{1\times2\times3}$$

 

Ecrire 2024 avec tous les chiffres dans l'ordre

\begin{align*}
2024&= 1234 - 5 + 6 + 789\\ 2024&= 123 + 4(5 + 6 + 78) + 9 \\ 2024&= 9 \times 8 + 7 + 6 \times 54 × 3 \times 2 + 1
\end{align*}

 

A special countdown for 2024

$$\begin{align*}
2024&=10+(9\times8\times7+6-5)\times4-3-2-1\\
2024&=\left(10+(9+8\times7)\times6\right)\times 5 +4\times3\times2\times1\\
2024&=10+9\times8\times7\times(6+5-4-3)-2\times1\\
2024&=10+9\times8\times7\times(6-5)\times4-3+2-1\\
2024&=10+\left(9\times8\times7-6+5\right)\times4+3-2+1
\end{align*}$$

 

Avec seulement des 2

$$2024=2222 - 222 + 22 + 2$$

 

Avec les années précédentes et suivantes

$$2024=\dfrac{2023^2-1}{2022}=\dfrac{(2023-1)(2023+1)}{2022}=\dfrac{2022\times2024}{2022}$$

 

Avec quelques symboles

Expliquons un peut cette égalité :

  • \(\phi\) est le nombre d'or et vaut :$$\phi=\dfrac{1+\sqrt{5}}{2}\approx 1,618$$
  • Le nombre e est la base des logarithmes naturels, c'est-à-dire le nombre défini par \(\ln(e) = 1\). Cette constante mathématique, également appelée nombre d'Euler ou constante de Néper en référence aux mathématiciens Leonhard Euler et John Napier, vaut environ 2,71828.
  • \(\lceil x\rceil\) correspond à la partie entière supérieure (appelée en anglais ceiling, « plafond ») définie par : $$\lceil x  \rceil-1 < x \leq \lceil x \rceil$$
  • i nombre complexe dont le carré vaut \(-1\).
  • On a donc : $$\phi^2+i^0-\left(\pi^3\right)^2+\left(e^2\right)^4=\phi^2+1- \pi^6+ e^8\approx 2023,19$$
    Et la partie entière supérieure vaut bien 2024.

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7) 2024 nombre tétraédrique

2024 est un nombre tétraédrique c'est à dire qu'il peut se représenter sous la forme d'un tétraèdre (pyramide dont toutes les faces sont des triangles équilatéraux) à 22 étages !

tetraedre

 2024 est le 22e nombre tétraédrique (voir la liste ici https://oeis.org/A000292).

Pour tout entier naturel n non nul, le n-ième nombre pyramidal triangulaire, somme des n premiers nombres triangulaires, est donc la formule générale est : $$T(n)=\dfrac{n(n+1)(n+2)}{6}$$

Et on a :

$$T(22)= \dfrac{22\times23\times24}{6}=2024$$

 

8) 2024 en somme de cubes

$$2024=2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3$$

 $$2024=2^2+4^2+6^2+8^2+10^2+12^2+14^2+16^2+18^2+20^2+22^2$$

 

9) 2024 Somme de carrés

2024 somme de deux carrés ?

2024 n'est pas décomposable en somme de 2 carrés.

 

Théorème des deux carrés (cas général) —
Un entier naturel est somme de deux carrés si et seulement si chacun de ses facteurs premiers de la forme \(4k + 3\) intervient à une puissance paire.
En particulier, la décomposition est unique lorsque l'entier ne possède aucun facteur premier de la forme \(4k + 1\), ou alors un seul et avec exposant 1.

Or les facteurs premiers de 2024 sont : $$2 024 = 7\times17^2$$

Et l'on a :

  • \(7=4\times1+3\) , mais il n'est pas à une puissance paire dans la décomposition.
  • \(17=4\times4+1\)

Une autre expression du nombre de décompositions d'un entier premier impair a été donnée par le mathématicien français Pierre de Fermat (1601-1655). On dispose d'un autre théorème dont  une preuve exposée sur ce poly (niveau supérieur) :

Théorème des deux carrés de Fermat (cas des nombres premiers) —
Un nombre premier impair (c'est-à-dire tous les nombres premiers sauf 2) est une somme de deux carrés parfaits si et seulement si le reste de sa division euclidienne par 4 est 1 ; dans ce cas, les carrés sont déterminés de manière unique.

 

2024 somme de 3 carrés parfaits

Le nombre 2024 est somme de 3 carrés 7 fois.

Le théorème des trois carrés démontré par Carl Friedrich GAUSS (1777-1855) en 1801, et s'exprime par:

Théorème des trois carrés —
Un entier naturel est somme de trois carrés si, et seulement si, il n'est pas de la forme \(4^i \left(8 j -1\right)\) avec i et j entiers positifs ou nuls.

 

Voici les 7 décompositions de 2024 en somme de 3 carrés :

$$[2, 16, 42], [2, 24, 38], [8, 14, 42], [10, 18, 40], [10, 30, 32], [16, 18, 38], [18, 26, 32]$$

  • \( 2024=2^2+16^2+42^2\)
  • ...
  • \( 2024=18^2+26^2 +32^2\)

 

2024 somme de 4 carrés parfaits

Le nombre 2024 est un entier qui peut s'écrire comme la somme de quatre carrés 16 fois :

  • \( 2024=2^2+18^2+20^2 +36^2\)
  • \( 2024  = 4^2+6^2+6^2 +44^2=\cdots\)
  • ...
  • \( 2024=18^2+20^2+20^2 +30^2\)

 

Voici la liste des 90 quadruplets [a, b, c, d] tels que $$2024 = a^2 + b^2 + c^2 + d^2$$ [[2, 18, 20, 36], [4, 6, 6, 44], [4, 6, 26, 36], [4, 10, 12, 42], [4, 18, 28, 30], [6, 8, 18, 40], [6, 8, 30, 32], [6, 12, 20, 38], [6, 16, 24, 34], [8, 22, 24, 30], [10, 12, 22, 36], [10, 18, 24, 32], [12, 14, 28, 30], [12, 18, 20, 34], [14, 24, 24, 26], [18, 20, 20, 30]]

 

Le théorème des quatre carrés de Lagrange, également connu sous le nom de conjecture de Bachet, s'énonce de la façon suivante :

Théorème des quatre carrés de Lagrange—
Tout entier positif peut s'exprimer comme la somme de quatre carrés (dont certains peuvent être nuls).
Plus formellement, pour tout entier positif n, il existe des entiers a, b, c, d tels que :
$$n = a^2 + b^2 + c^2 + d^2$$

 

On peut se demander si il est possible d'exiger que les carrés soient non nuls.

Si on exige de plus qu'aucun des carrés de la somme ne soit nul (autrement dit que la décomposition soit en quatre carrés exactement, et non en quatre carrés ou moins), on a le résultat suivant

Théorème —
Les seuls entiers non décomposables en somme de 4 carrés tous non nuls sont :

    • 0, 1, 3, 5, 9, 11, 17, 29, 41,
    • et pour \(m\) entier positif ou nul, les nombres de la forme :
    • \(\displaystyle 2\times 4^{m}\), \(\displaystyle 6\times 4^{m}\) et \(\displaystyle 14\times 4^{m}\)

Une preuve ici.

 

2024 somme de 5 carrés parfaits ... et plus

On plus généralement on a le théorème de Hasse-Minkowski :

Théorème de Hasse-Minkowski —
Pour \(k\geq5\), tout entier positif \(n\) peut s'exprimer comme la somme de \(k\) carrés.

Preuve : seminaire JL Lagrange.

 

2024 somme de 5 carrés parfaits

Le nombre 2023 est un entier qui peut s'écrire comme la somme de cinq carrés 314 fois :

  • \( 2024=1^2+1^2+2^2 +13^2+43^2\)
  • \( 2024  = 1^2+1^2+5^2 +29^2+34^2=\cdots\)
  • ...
  • \( 2024=18^2+18^2+20^2+20^2 +24^2\)

Voici la liste des 379 quintuplets [a, b, c, d, e] tels que $$2024 = a^2 + b^2 + c^2 + d^2+ e^2$$ [[1, 1, 2, 13, 43], [1, 1, 5, 29, 34], [1, 1, 7, 23, 38], [1, 1, 10, 31, 31], [1, 1, 11, 26, 35], [1, 1, 13, 22, 37], [1, 1, 17, 17, 38], [1, 2, 5, 25, 37], [1, 2, 7, 11, 43], [1, 2, 7, 17, 41], [1, 2, 11, 23, 37], [1, 2, 13, 13, 41], [1, 2, 13, 25, 35], [1, 2, 17, 19, 37], [1, 2, 23, 23, 31], [1, 3, 3, 18, 41], [1, 3, 3, 22, 39], [1, 3, 5, 15, 42], [1, 3, 5, 30, 33], [1, 3, 9, 13, 42], [1, 3, 13, 18, 39], [1, 3, 14, 27, 33], [1, 3, 18, 27, 31], [1, 3, 21, 22, 33], [1, 5, 5, 23, 38], [1, 5, 6, 21, 39], [1, 5, 7, 10, 43], [1, 5, 10, 23, 37], [1, 5, 11, 14, 41], [1, 5, 14, 29, 31], [1, 5, 17, 22, 35], [1, 5, 19, 26, 31], [1, 6, 9, 15, 41], [1, 6, 13, 27, 33], [1, 6, 23, 27, 27], [1, 7, 11, 22, 37], [1, 7, 13, 19, 38], [1, 7, 17, 23, 34], [1, 7, 22, 23, 31], [1, 9, 9, 30, 31], [1, 9, 14, 15, 39], [1, 9, 18, 23, 33], [1, 9, 22, 27, 27], [1, 10, 11, 11, 41], [1, 10, 11, 29, 31], [1, 10, 13, 23, 35], [1, 11, 11, 25, 34], [1, 11, 13, 17, 38], [1, 13, 13, 23, 34], [1, 13, 14, 17, 37], [1, 13, 15, 27, 30], [1, 13, 18, 21, 33], [1, 13, 22, 23, 29], [1, 14, 19, 25, 29], [1, 15, 15, 22, 33], [1, 17, 17, 17, 34], [1, 17, 17, 22, 31], [1, 17, 22, 25, 25], [1, 17, 23, 23, 26], [1, 18, 21, 23, 27], [1, 19, 19, 25, 26], [2, 2, 4, 8, 44], [2, 2, 4, 20, 40], [2, 2, 12, 24, 36], [2, 3, 7, 21, 39], [2, 3, 9, 9, 43], [2, 3, 9, 29, 33], [2, 3, 21, 27, 29], [2, 4, 8, 28, 34], [2, 4, 14, 28, 32], [2, 5, 5, 11, 43], [2, 5, 5, 17, 41], [2, 5, 23, 25, 29], [2, 7, 7, 31, 31], [2, 7, 11, 13, 41], [2, 7, 11, 25, 35], [2, 7, 13, 29, 31], [2, 7, 15, 15, 39], [2, 7, 17, 29, 29], [2, 7, 21, 21, 33], [2, 8, 10, 16, 40], [2, 8, 16, 16, 38], [2, 8, 16, 26, 32], [2, 8, 20, 20, 34], [2, 9, 11, 27, 33], [2, 9, 15, 25, 33], [2, 11, 13, 19, 37], [2, 11, 21, 27, 27], [2, 11, 23, 23, 29], [2, 12, 12, 24, 34], [2, 12, 16, 18, 36], [2, 12, 20, 24, 30], [2, 13, 13, 29, 29], [2, 13, 19, 23, 31], [2, 14, 16, 28, 28], [2, 14, 20, 20, 32], [2, 15, 15, 27, 29], [2, 15, 21, 25, 27], [2, 16, 16, 22, 32], [2, 17, 19, 23, 29], [3, 3, 3, 29, 34], [3, 3, 6, 11, 43], [3, 3, 6, 17, 41], [3, 3, 10, 15, 41], [3, 3, 11, 11, 42], [3, 3, 11, 21, 38], [3, 3, 11, 27, 34], [3, 3, 14, 17, 39], [3, 3, 14, 21, 37], [3, 3, 15, 25, 34], [3, 3, 18, 29, 29], [3, 5, 6, 27, 35], [3, 5, 15, 26, 33], [3, 5, 18, 21, 35], [3, 5, 19, 27, 30], [3, 6, 7, 9, 43], [3, 6, 7, 29, 33], [3, 6, 9, 23, 37], [3, 6, 13, 17, 39], [3, 6, 13, 21, 37], [3, 6, 15, 23, 35], [3, 6, 17, 27, 31], [3, 6, 19, 23, 33], [3, 6, 25, 25, 27], [3, 7, 9, 11, 42], [3, 7, 9, 21, 38], [3, 7, 9, 27, 34], [3, 7, 11, 18, 39], [3, 7, 15, 29, 30], [3, 7, 21, 25, 30], [3, 9, 9, 22, 37], [3, 9, 13, 26, 33], [3, 9, 15, 22, 35], [3, 9, 19, 22, 33], [3, 9, 23, 26, 27], [3, 10, 13, 15, 39], [3, 10, 15, 27, 31], [3, 11, 15, 15, 38], [3, 11, 18, 27, 29], [3, 13, 21, 26, 27], [3, 14, 15, 15, 37], [3, 14, 17, 21, 33], [3, 14, 19, 27, 27], [3, 15, 18, 25, 29], [3, 15, 19, 23, 30], [3, 17, 18, 21, 31], [3, 18, 21, 25, 25], [3, 19, 21, 22, 27], [4, 4, 8, 22, 38], [4, 4, 14, 14, 40], [4, 4, 22, 22, 32], [4, 6, 8, 12, 42], [4, 6, 10, 24, 36], [4, 8, 10, 20, 38], [4, 8, 12, 30, 30], [4, 8, 18, 18, 36], [4, 8, 22, 26, 28], [4, 10, 10, 28, 32], [4, 10, 20, 22, 32], [4, 14, 16, 20, 34], [4, 16, 20, 26, 26], [4, 16, 22, 22, 28], [4, 18, 18, 24, 28], [5, 5, 5, 10, 43], [5, 5, 11, 22, 37], [5, 5, 13, 19, 38], [5, 5, 17, 23, 34], [5, 5, 22, 23, 31], [5, 6, 9, 19, 39], [5, 7, 7, 26, 35], [5, 7, 10, 13, 41], [5, 7, 10, 25, 35], [5, 7, 13, 25, 34], [5, 7, 14, 23, 35], [5, 7, 22, 25, 29], [5, 9, 10, 27, 33], [5, 9, 15, 18, 37], [5, 9, 17, 27, 30], [5, 10, 13, 19, 37], [5, 10, 21, 27, 27], [5, 10, 23, 23, 29], [5, 11, 13, 22, 35], [5, 11, 14, 29, 29], [5, 11, 19, 19, 34], [5, 11, 19, 26, 29], [5, 13, 23, 25, 26], [5, 14, 17, 17, 35], [5, 15, 15, 18, 35], [5, 15, 18, 19, 33], [5, 19, 22, 23, 25], [5, 21, 21, 21, 26], [6, 6, 12, 28, 32], [6, 6, 16, 20, 36], [6, 7, 11, 27, 33], [6, 7, 15, 25, 33], [6, 8, 12, 22, 36], [6, 8, 18, 24, 32], [6, 9, 15, 29, 29], [6, 9, 17, 23, 33], [6, 9, 21, 25, 29], [6, 10, 12, 12, 40], [6, 11, 11, 15, 39], [6, 12, 12, 16, 38], [6, 12, 12, 26, 32], [6, 12, 22, 24, 28], [6, 13, 15, 15, 37], [6, 13, 17, 21, 33], [6, 13, 19, 27, 27], [6, 15, 19, 21, 31], [6, 16, 16, 24, 30], [6, 17, 21, 23, 27], [7, 7, 7, 14, 41], [7, 7, 9, 9, 42], [7, 7, 9, 18, 39], [7, 7, 11, 19, 38], [7, 7, 14, 19, 37], [7, 7, 17, 26, 31], [7, 7, 25, 25, 26], [7, 9, 15, 15, 38], [7, 9, 18, 27, 29], [7, 10, 11, 23, 35], [7, 10, 17, 19, 35], [7, 10, 17, 25, 31], [7, 10, 25, 25, 25], [7, 11, 11, 17, 38], [7, 11, 13, 23, 34], [7, 11, 14, 17, 37], [7, 11, 15, 27, 30], [7, 11, 18, 21, 33], [7, 11, 22, 23, 29], [7, 13, 13, 26, 31], [7, 13, 17, 19, 34], [7, 13, 17, 26, 29], [7, 13, 19, 22, 31], [7, 14, 17, 23, 31], [7, 14, 23, 25, 25], [7, 15, 15, 25, 30], [7, 17, 19, 22, 29], [8, 8, 8, 26, 34], [8, 8, 10, 14, 40], [8, 8, 14, 16, 38], [8, 8, 14, 26, 32], [8, 8, 16, 22, 34], [8, 10, 20, 26, 28], [8, 12, 14, 18, 36], [8, 14, 14, 28, 28], [8, 14, 16, 22, 32], [8, 18, 22, 24, 24], [8, 20, 20, 22, 26], [9, 9, 9, 10, 41], [9, 9, 9, 25, 34], [9, 9, 11, 29, 30], [9, 9, 13, 18, 37], [9, 9, 14, 21, 35], [9, 9, 15, 26, 31], [9, 9, 17, 22, 33], [9, 10, 15, 23, 33], [9, 10, 21, 21, 31], [9, 11, 15, 21, 34], [9, 13, 15, 18, 35], [9, 13, 18, 19, 33], [9, 14, 17, 27, 27], [9, 15, 17, 23, 30], [9, 17, 21, 22, 27], [9, 18, 19, 23, 27], [10, 10, 16, 28, 28], [10, 10, 20, 20, 32], [10, 11, 11, 29, 29], [10, 11, 17, 17, 35], [10, 13, 13, 19, 35], [10, 13, 13, 25, 31], [10, 13, 15, 21, 33], [10, 13, 17, 25, 29], [10, 14, 24, 24, 24], [10, 15, 21, 23, 27], [10, 16, 16, 16, 34], [10, 16, 20, 22, 28], [10, 19, 19, 19, 29], [11, 11, 13, 13, 38], [11, 11, 14, 19, 35], [11, 11, 14, 25, 31], [11, 11, 18, 27, 27], [11, 11, 21, 21, 30], [11, 13, 13, 14, 37], [11, 13, 17, 17, 34], [11, 13, 17, 22, 31], [11, 13, 22, 25, 25], [11, 13, 23, 23, 26], [11, 15, 18, 25, 27], [11, 17, 17, 22, 29], [11, 19, 22, 23, 23], [12, 12, 16, 18, 34], [12, 12, 22, 24, 26], [12, 14, 18, 24, 28], [12, 16, 18, 20, 30], [13, 13, 13, 19, 34], [13, 13, 13, 26, 29], [13, 13, 14, 23, 31], [13, 13, 19, 22, 29], [13, 14, 17, 23, 29], [13, 15, 15, 26, 27], [13, 15, 17, 21, 30], [13, 17, 19, 23, 26], [13, 18, 19, 21, 27], [14, 15, 15, 17, 33], [14, 17, 17, 17, 31], [14, 17, 17, 25, 25], [15, 15, 17, 18, 31], [15, 15, 18, 25, 25], [15, 15, 19, 22, 27], [16, 20, 20, 22, 22], [17, 18, 21, 21, 23], [17, 19, 19, 22, 23], [18, 18, 20, 20, 24]]

 

 

2024 somme de 6 carrés parfaits

Pour information, il y a 2 058 décompositions de 2024 en somme de 6 carrés :

Voici la liste des 1850 sextuplets [a, b, c, d, e, f] tels que $$2024 = a^2 + b^2 + c^2 + d^2+ e^2+ f^2$$ ([[1, 1, 1, 1, 16, 42], [1, 1, 1, 1, 24, 38], [1, 1, 1, 2, 9, 44], [1, 1, 1, 4, 18, 41], [1, 1, 1, 4, 22, 39], [1, 1, 1, 6, 7, 44], [1, 1, 1, 6, 31, 32], [1, 1, 1, 7, 26, 36], [1, 1, 1, 9, 28, 34], [1, 1, 1, 10, 20, 39], [1, 1, 1, 10, 25, 36], [1, 1, 1, 12, 14, 41], [1, 1, 1, 14, 15, 40], [1, 1, 1, 14, 23, 36], [1, 1, 1, 16, 26, 33], [1, 1, 1, 17, 24, 34], [1, 1, 1, 22, 24, 31], [1, 1, 2, 3, 28, 35], [1, 1, 2, 5, 12, 43], [1, 1, 2, 8, 27, 35], [1, 1, 2, 9, 16, 41], [1, 1, 2, 19, 19, 36], [1, 1, 2, 20, 23, 33], [1, 1, 3, 4, 29, 34], [1, 1, 3, 8, 10, 43], [1, 1, 3, 13, 20, 38], [1, 1, 4, 6, 11, 43], [1, 1, 4, 6, 17, 41], [1, 1, 4, 10, 15, 41], [1, 1, 4, 11, 11, 42], [1, 1, 4, 11, 21, 38], [1, 1, 4, 11, 27, 34], [1, 1, 4, 14, 17, 39], [1, 1, 4, 14, 21, 37], [1, 1, 4, 15, 25, 34], [1, 1, 4, 18, 29, 29], [1, 1, 5, 5, 6, 44], [1, 1, 5, 5, 26, 36], [1, 1, 5, 6, 19, 40], [1, 1, 5, 8, 13, 42], [1, 1, 5, 12, 22, 37], [1, 1, 5, 14, 24, 35], [1, 1, 5, 16, 29, 30], [1, 1, 5, 20, 21, 34], [1, 1, 5, 22, 27, 28], [1, 1, 6, 7, 16, 41], [1, 1, 6, 8, 31, 31], [1, 1, 6, 11, 29, 32], [1, 1, 6, 16, 19, 37], [1, 1, 6, 19, 20, 35], [1, 1, 6, 19, 28, 29], [1, 1, 6, 20, 25, 31], [1, 1, 7, 7, 18, 40], [1, 1, 7, 7, 30, 32], [1, 1, 7, 10, 28, 33], [1, 1, 7, 14, 16, 39], [1, 1, 7, 17, 28, 30], [1, 1, 7, 18, 25, 32], [1, 1, 7, 20, 22, 33], [1, 1, 8, 9, 14, 41], [1, 1, 8, 15, 17, 38], [1, 1, 8, 19, 21, 34], [1, 1, 8, 21, 26, 29], [1, 1, 8, 23, 23, 30], [1, 1, 9, 14, 28, 31], [1, 1, 9, 16, 23, 34], [1, 1, 10, 11, 24, 35], [1, 1, 10, 13, 27, 32], [1, 1, 10, 16, 21, 35], [1, 1, 11, 11, 22, 36], [1, 1, 11, 13, 24, 34], [1, 1, 11, 21, 26, 28], [1, 1, 11, 22, 24, 29], [1, 1, 12, 13, 22, 35], [1, 1, 12, 14, 29, 29], [1, 1, 12, 19, 19, 34], [1, 1, 12, 19, 26, 29], [1, 1, 13, 13, 28, 30], [1, 1, 13, 14, 19, 36], [1, 1, 13, 16, 21, 34], [1, 1, 14, 16, 27, 29], [1, 1, 14, 17, 24, 31], [1, 1, 14, 19, 21, 32], [1, 1, 14, 24, 25, 25], [1, 1, 15, 17, 22, 32], [1, 1, 15, 22, 23, 28], [1, 1, 16, 19, 26, 27], [1, 1, 16, 21, 22, 29], [1, 1, 17, 18, 25, 28], [1, 1, 19, 19, 20, 30], [1, 2, 3, 4, 25, 37], [1, 2, 3, 5, 7, 44], [1, 2, 3, 5, 31, 32], [1, 2, 3, 7, 19, 40], [1, 2, 3, 11, 17, 40], [1, 2, 3, 16, 23, 35], [1, 2, 3, 19, 25, 32], [1, 2, 4, 7, 27, 35], [1, 2, 4, 9, 31, 31], [1, 2, 4, 11, 19, 39], [1, 2, 4, 17, 25, 33], [1, 2, 5, 7, 24, 37], [1, 2, 5, 8, 9, 43], [1, 2, 5, 8, 29, 33], [1, 2, 5, 11, 28, 33], [1, 2, 5, 12, 13, 41], [1, 2, 5, 12, 25, 35], [1, 2, 5, 13, 15, 40], [1, 2, 5, 13, 23, 36], [1, 2, 5, 15, 20, 37], [1, 2, 5, 21, 23, 32], [1, 2, 7, 7, 20, 39], [1, 2, 7, 7, 25, 36], [1, 2, 7, 8, 15, 41], [1, 2, 7, 9, 17, 40], [1, 2, 7, 13, 24, 35], [1, 2, 7, 15, 28, 31], [1, 2, 7, 16, 25, 33], [1, 2, 7, 20, 27, 29], [1, 2, 8, 15, 19, 37], [1, 2, 8, 17, 21, 35], [1, 2, 9, 13, 13, 40], [1, 2, 9, 13, 20, 37], [1, 2, 9, 16, 29, 29], [1, 2, 9, 17, 25, 32], [1, 2, 9, 23, 25, 28], [1, 2, 11, 11, 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