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Bonne année 2022

2022 quelques propriétés de cet entier naturel

Le site Math93.com vous souhaite une heureuse, chaleureuse et studieuse année 2022. Profitons-en pour revenir sur quelques caractéristiques de ce nombre pair.

Écriture du nombre 2022

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-deux
Anglais : Two thousand twenty-two
Allemand : zweitausendzweiundzwanzig
Espagnol : dos mil veintidós
Italien : duemilaventidue / Portugais : dois mil e vinte e dois

Autres écriture de 2022

En chiffre romain	MMXXII
En binaire	11111100110
En octal	3746
En hexadécimal	7e6
En dollars américains	USD 2,022.00 ($)
En euros	2 022,00 EUR (€)

Opérations mathématiques

L'année 2022

2022 n'est pas une année bissextile et compte donc 365 jours (52 x 7 + 1) donc 52 semaines plus 1 jours.

La dernière semaine de l'année 2021 prendra fin le dimanche 2 janvier 2022 (semaine N°52 2021). La 1ère semaine de l'année 2022 (semaine N°1 2022) débutera quant à elle le lundi 3 janvier 2022 pour se terminer le dimanche 9 janvier 2022.

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).

Diviseurs de 2 022 et nombres premiers

Diviseurs
L'entier pair 2022 admet 8 diviseurs qui sont : $1, 2, 3, 6, 337, 674, 1011, 2022$
On verra que c'est une propriété des nombres sphéniques.
Nombre premiers : 2022 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.
Moyenne
2022 est la moyenne de deux nombres premiers consécutifs (le dernière et la prochaine année première) :
$2022=\dfrac{2017+2027}{2}$
Décomposition en facteurs premiers
$2 022 = 2\times3\times337$
Le 2022^e nombre premier est 17 581

Ecritures de 2022

Avec les premiers nombres premiers :
$2022=2\times3\times\dfrac{5\times7!}{11+13+17+19}-23-29-31$
Avec les 10 chiffres :
$2022=9\times8\times(7+6)+543\times2\times1$
$2022=9-876+(5+4)\times321$
et on aura
$2023=\left(987+6-5\right)\times\dfrac{43}{21}$
Avec des factorielles et doubles factorielles et tous les chiffres
The factorial of n is the product of all integers from 1 to n :
$n!=1\times2\times\cdots\times n$
In mathematics, the double factorial or semifactorial of a number n, denoted by n‼, is the product of all the integers from 1 up to n that have the same parity (odd or even) as n.
For exemple $7!!=1\times3\times5\times7$ and $6!!=2\times4\times6$ . On a :
$2022=0!-1!+2!-3!-4!!-5!+6!+7!!+8!!+9!!$

2022 et quelques décompositions

Nombre de Harshad ou de Niven
C'est un nombre divisible par la somme de ses chiffres soit ici : $2+0+2+2=6$
$\dfrac{2022}{6}=337$
Nombre 3-polis : trois fois somme d'entiers consécutifs.
- $2022= 163 + 164 +\cdots + 174$
- $2022= 504+505+506+507$
- $2022= 673+674+675$
Nombre Sphénique
Un nombre sphénique est un entier strictement positif qui est le produit de trois facteurs premiers distincts. La définition exige que chacun des trois facteurs premiers ne soit exprimé qu'une seule fois.
$2 022 = 2\times3\times337$
Tous les nombres sphéniques ont exactement huit diviseurs.
Somme de deux premiers consécutifs
$2022=1009+1013$
Somme de deux nombres premiers
$2 022 = 5 + 2017 = 11+2011=\cdots=1009+1013$
2022 est 59 fois somme de deux premiers.
Nombre semi-parfait : somme de certains de ses diviseurs.
$2 022 = 1011 + 674 + 337$

2022 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 :

$2022<1+ 2+ 3+ 6+ 337+674+ 1011=2034$

2022 Somme de carrés

Le nombre 2022 est un entier qui peut s'écrire comme la somme de trois carrés (tous non nuls) 7 fois, comme la somme de quatre carrés 90 fois, comme somme de cinq carrés 379 fois, comme somme de six carrés 1 837 fois mais pas comme somme de deux carrés.

2022 somme de deux carrés ?

2022 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 2022 sont : $2 022 = 2\times3\times337$

Et l'on a :

$2=4\times0+2$ .
$3=4\times0+3$ , mais il n'est pas à une puissance paire dans la décomposition.
$337=4\times84+1$ .

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.

2022 somme de 3 carrés parfaits

Le nombre 2022 est un entier qui peut s'écrire comme la somme de trois carrés 7 fois :

$2022=2^2+13^2+43^2$
$2022= 5^2+29^2+34^2$
$2022=7^2+23^2+38^2$
$2022=10^2+31^2+31^2$
$2022=11^2+26^2+35^2$
$2022=13^2+22^2+37^2$
$2022= 17^2+17^2+38^2$

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 $4^i \left(8 j + 7\right)$ avec i et j entiers positifs ou nuls.

2022 somme de 4 carrés parfaits

Le nombre 2022 est un entier qui peut s'écrire comme la somme de quatre carrés 90 fois :

$2022=1^2+1^2+16^2 +42^2$
$2022 = 1^2+1^2+24^2 +38^2=\cdots$
$2022=19^2+19^2+20^2 +30^2$

Voici la liste des 90 quadruplets [a, b, c, d] tels que

$2022 = a^2 + b^2 + c^2 + d^2$ [1, 1, 16, 42], [1, 1, 24, 38], [1, 2, 9, 44], [1, 4, 18, 41], [1, 4, 22, 39], [1, 6, 7, 44], [1, 6, 31, 32], [1, 7, 26, 36], [1, 9, 28, 34], [1, 10, 20, 39], [1, 10, 25, 36], [1, 12, 14, 41], [1, 14, 15, 40], [1, 14, 23, 36], [1, 16, 26, 33], [1, 17, 24, 34], [1, 22, 24, 31], [2, 3, 28, 35], [2, 5, 12, 43], [2, 8, 27, 35], [2, 9, 16, 41], [2, 19, 19, 36], [2, 20, 23, 33], [3, 4, 29, 34], [3, 8, 10, 43], [3, 13, 20, 38], [4, 6, 11, 43], [4, 6, 17, 41], [4, 10, 15, 41], [4, 11, 11, 42], [4, 11, 21, 38], [4, 11, 27, 34], [4, 14, 17, 39], [4, 14, 21, 37], [4, 15, 25, 34], [4, 18, 29, 29], [5, 5, 6, 44], [5, 5, 26, 36], [5, 6, 19, 40], [5, 8, 13, 42], [5, 12, 22, 37], [5, 14, 24, 35], [5, 16, 29, 30], [5, 20, 21, 34], [5, 22, 27, 28], [6, 7, 16, 41], [6, 8, 31, 31], [6, 11, 29, 32], [6, 16, 19, 37], [6, 19, 20, 35], [6, 19, 28, 29], [6, 20, 25, 31], [7, 7, 18, 40], [7, 7, 30, 32], [7, 10, 28, 33], [7, 14, 16, 39], [7, 17, 28, 30], [7, 18, 25, 32], [7, 20, 22, 33], [8, 9, 14, 41], [8, 15, 17, 38], [8, 19, 21, 34], [8, 21, 26, 29], [8, 23, 23, 30], [9, 14, 28, 31], [9, 16, 23, 34], [10, 11, 24, 35], [10, 13, 27, 32], [10, 16, 21, 35], [11, 11, 22, 36], [11, 13, 24, 34], [11, 21, 26, 28], [11, 22, 24, 29], [12, 13, 22, 35], [12, 14, 29, 29], [12, 19, 19, 34], [12, 19, 26, 29], [13, 13, 28, 30], [13, 14, 19, 36], [13, 16, 21, 34], [14, 16, 27, 29], [14, 17, 24, 31], [14, 19, 21, 32], [14, 24, 25, 25], [15, 17, 22, 32], [15, 22, 23, 28], [16, 19, 26, 27], [16, 21, 22, 29], [17, 18, 25, 28], [19, 19, 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.

2022 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.

2022 somme de 5 carrés parfaits

Le nombre 2022 est un entier qui peut s'écrire comme la somme de cinq carrés 379 fois :

$2022=1^2+1^2+18^2 +20^2+36^2$
$2022 = 1^2+2^2+12^2 +28^2+33^2=\cdots$
$2022=18^2+19^2+19^2+21^2 +24^2$

Voici la liste des 379 quintuplets [a, b, c, d, e] tels que

$2022 = a^2 + b^2 + c^2 + d^2+ e^2$ [1, 1, 18, 20, 36], [1, 2, 12, 28, 33], [1, 2, 18, 18, 37], [1, 2, 21, 26, 30], [1, 4, 4, 15, 42], [1, 4, 4, 30, 33], [1, 4, 9, 18, 40], [1, 4, 9, 30, 32], [1, 4, 12, 30, 31], [1, 4, 15, 22, 36], [1, 4, 23, 24, 30], [1, 5, 6, 14, 42], [1, 5, 14, 30, 30], [1, 6, 6, 10, 43], [1, 6, 8, 20, 39], [1, 6, 8, 25, 36], [1, 6, 10, 11, 42], [1, 6, 10, 21, 38], [1, 6, 10, 27, 34], [1, 6, 17, 20, 36], [1, 6, 24, 25, 28], [1, 7, 8, 12, 42], [1, 7, 10, 24, 36], [1, 8, 15, 24, 34], [1, 9, 12, 14, 40], [1, 9, 16, 28, 30], [1, 10, 11, 30, 30], [1, 10, 12, 16, 39], [1, 10, 15, 20, 36], [1, 10, 16, 24, 33], [1, 10, 18, 21, 34], [1, 11, 18, 26, 30], [1, 12, 12, 17, 38], [1, 12, 18, 23, 32], [1, 12, 24, 25, 26], [1, 14, 14, 27, 30], [1, 14, 15, 24, 32], [1, 14, 21, 22, 30], [1, 16, 17, 24, 30], [1, 18, 22, 22, 27], [2, 2, 3, 18, 41], [2, 2, 3, 22, 39], [2, 2, 5, 15, 42], [2, 2, 5, 30, 33], [2, 2, 9, 13, 42], [2, 2, 13, 18, 39], [2, 2, 14, 27, 33], [2, 2, 18, 27, 31], [2, 2, 21, 22, 33], [2, 3, 3, 8, 44], [2, 3, 3, 20, 40], [2, 3, 4, 12, 43], [2, 3, 6, 23, 38], [2, 3, 7, 14, 42], [2, 3, 8, 24, 37], [2, 3, 9, 22, 38], [2, 3, 12, 29, 32], [2, 3, 16, 27, 32], [2, 3, 18, 23, 34], [2, 3, 21, 28, 28], [2, 3, 22, 25, 30], [2, 4, 9, 20, 39], [2, 4, 9, 25, 36], [2, 4, 15, 16, 39], [2, 5, 11, 24, 36], [2, 5, 15, 18, 38], [2, 5, 16, 21, 36], [2, 5, 24, 24, 29], [2, 6, 7, 13, 42], [2, 6, 9, 26, 35], [2, 6, 10, 19, 39], [2, 6, 11, 30, 31], [2, 6, 17, 18, 37], [2, 7, 12, 12, 41], [2, 7, 12, 15, 40], [2, 7, 12, 23, 36], [2, 7, 13, 30, 30], [2, 8, 9, 28, 33], [2, 8, 12, 17, 39], [2, 8, 12, 21, 37], [2, 8, 17, 24, 33], [2, 8, 21, 27, 28], [2, 9, 9, 16, 40], [2, 9, 13, 18, 38], [2, 9, 14, 29, 30], [2, 9, 19, 26, 30], [2, 9, 20, 24, 31], [2, 10, 10, 27, 33], [2, 10, 15, 18, 37], [2, 10, 17, 27, 30], [2, 11, 12, 27, 32], [2, 11, 18, 22, 33], [2, 12, 12, 19, 37], [2, 12, 15, 25, 32], [2, 12, 16, 23, 33], [2, 12, 17, 17, 36], [2, 12, 19, 27, 28], [2, 13, 18, 25, 30], [2, 14, 15, 21, 34], [2, 15, 15, 28, 28], [2, 15, 16, 24, 31], [2, 15, 21, 26, 26], [2, 17, 18, 26, 27], [2, 18, 18, 23, 29], [3, 3, 8, 28, 34], [3, 3, 14, 28, 32], [3, 4, 5, 6, 44], [3, 4, 5, 26, 36], [3, 4, 6, 19, 40], [3, 4, 8, 13, 42], [3, 4, 12, 22, 37], [3, 4, 14, 24, 35], [3, 4, 16, 29, 30], [3, 4, 20, 21, 34], [3, 4, 22, 27, 28], [3, 5, 8, 18, 40], [3, 5, 8, 30, 32], [3, 5, 12, 20, 38], [3, 5, 16, 24, 34], [3, 6, 7, 22, 38], [3, 6, 8, 8, 43], [3, 6, 10, 14, 41], [3, 6, 11, 16, 40], [3, 6, 13, 28, 32], [3, 6, 14, 25, 34], [3, 6, 25, 26, 26], [3, 7, 10, 10, 42], [3, 7, 14, 18, 38], [3, 7, 18, 22, 34], [3, 8, 8, 11, 42], [3, 8, 8, 21, 38], [3, 8, 8, 27, 34], [3, 8, 12, 19, 38], [3, 8, 13, 22, 36], [3, 8, 14, 27, 32], [3, 8, 16, 18, 37], [3, 8, 18, 20, 35], [3, 8, 18, 28, 29], [3, 8, 21, 22, 32], [3, 9, 10, 26, 34], [3, 10, 12, 13, 40], [3, 10, 12, 20, 37], [3, 10, 14, 14, 39], [3, 10, 16, 19, 36], [3, 10, 20, 27, 28], [3, 10, 22, 23, 30], [3, 11, 14, 20, 36], [3, 11, 18, 28, 28], [3, 12, 13, 16, 38], [3, 12, 13, 26, 32], [3, 12, 19, 22, 32], [3, 13, 22, 24, 28], [3, 14, 20, 24, 29], [3, 18, 22, 23, 26], [3, 19, 20, 24, 26], [3, 20, 20, 22, 27], [4, 4, 6, 27, 35], [4, 4, 15, 26, 33], [4, 4, 18, 21, 35], [4, 4, 19, 27, 30], [4, 5, 6, 24, 37], [4, 5, 18, 19, 36], [4, 5, 24, 26, 27], [4, 6, 7, 20, 39], [4, 6, 7, 25, 36], [4, 6, 8, 15, 41], [4, 6, 9, 17, 40], [4, 6, 13, 24, 35], [4, 6, 15, 28, 31], [4, 6, 16, 25, 33], [4, 6, 20, 27, 29], [4, 7, 7, 12, 42], [4, 7, 15, 24, 34], [4, 8, 9, 30, 31], [4, 8, 14, 15, 39], [4, 8, 18, 23, 33], [4, 8, 22, 27, 27], [4, 9, 9, 20, 38], [4, 9, 10, 12, 41], [4, 9, 10, 15, 40], [4, 9, 10, 23, 36], [4, 9, 12, 25, 34], [4, 9, 15, 16, 38], [4, 9, 15, 26, 32], [4, 9, 20, 25, 30], [4, 10, 13, 21, 36], [4, 10, 21, 21, 32], [4, 11, 12, 29, 30], [4, 11, 16, 27, 30], [4, 12, 13, 18, 37], [4, 12, 14, 21, 35], [4, 12, 15, 26, 31], [4, 12, 17, 22, 33], [4, 13, 18, 27, 28], [4, 13, 19, 24, 30], [4, 14, 15, 17, 36], [4, 15, 15, 20, 34], [4, 15, 16, 25, 30], [4, 15, 23, 24, 26], [4, 18, 20, 21, 29], [5, 5, 8, 12, 42], [5, 5, 10, 24, 36], [5, 6, 10, 30, 31], [5, 6, 14, 26, 33], [5, 6, 18, 26, 31], [5, 6, 19, 24, 32], [5, 8, 14, 21, 36], [5, 10, 12, 27, 32], [5, 10, 18, 22, 33], [5, 11, 12, 24, 34], [5, 11, 16, 18, 36], [5, 11, 20, 24, 30], [5, 12, 12, 22, 35], [5, 12, 13, 28, 30], [5, 12, 14, 19, 36], [5, 12, 16, 21, 34], [5, 13, 24, 24, 26], [5, 14, 15, 26, 30], [5, 14, 21, 24, 28], [5, 16, 18, 24, 29], [5, 16, 20, 21, 30], [5, 17, 18, 22, 30], [5, 19, 22, 24, 24], [6, 6, 7, 26, 35], [6, 6, 10, 13, 41], [6, 6, 10, 25, 35], [6, 6, 13, 25, 34], [6, 6, 14, 23, 35], [6, 6, 22, 25, 29], [6, 7, 8, 28, 33], [6, 7, 9, 16, 40], [6, 7, 13, 18, 38], [6, 7, 14, 29, 30], [6, 7, 19, 26, 30], [6, 7, 20, 24, 31], [6, 8, 11, 24, 35], [6, 8, 13, 27, 32], [6, 8, 16, 21, 35], [6, 9, 10, 19, 38], [6, 9, 14, 22, 35], [6, 9, 16, 25, 32], [6, 10, 11, 26, 33], [6, 10, 13, 14, 39], [6, 10, 14, 27, 31], [6, 10, 17, 21, 34], [6, 10, 19, 25, 30], [6, 10, 21, 22, 31], [6, 11, 11, 12, 40], [6, 11, 13, 20, 36], [6, 11, 14, 15, 38], [6, 11, 15, 22, 34], [6, 11, 17, 26, 30], [6, 11, 20, 21, 32], [6, 12, 16, 19, 35], [6, 12, 16, 25, 31], [6, 12, 17, 23, 32], [6, 12, 23, 23, 28], [6, 13, 20, 24, 29], [6, 14, 14, 15, 37], [6, 14, 18, 25, 29], [6, 14, 19, 23, 30], [6, 15, 20, 20, 31], [6, 16, 23, 24, 25], [6, 17, 22, 22, 27], [6, 18, 19, 25, 26], [6, 19, 20, 21, 28], [7, 7, 12, 22, 36], [7, 7, 18, 24, 32], [7, 8, 8, 9, 42], [7, 8, 8, 18, 39], [7, 8, 12, 26, 33], [7, 8, 15, 28, 30], [7, 8, 17, 18, 36], [7, 9, 14, 20, 36], [7, 9, 18, 28, 28], [7, 10, 18, 18, 35], [7, 12, 12, 23, 34], [7, 12, 16, 22, 33], [7, 12, 20, 23, 30], [7, 13, 18, 18, 34], [7, 14, 15, 16, 36], [7, 14, 24, 24, 25], [7, 15, 18, 20, 32], [7, 17, 18, 24, 28], [7, 18, 18, 22, 29], [8, 8, 15, 15, 38], [8, 8, 18, 27, 29], [8, 9, 9, 14, 40], [8, 9, 10, 16, 39], [8, 9, 12, 17, 38], [8, 9, 18, 23, 32], [8, 9, 24, 25, 26], [8, 10, 11, 21, 36], [8, 10, 12, 25, 33], [8, 10, 20, 27, 27], [8, 10, 21, 24, 29], [8, 11, 12, 18, 37], [8, 11, 18, 27, 28], [8, 11, 19, 24, 30], [8, 12, 14, 23, 33], [8, 12, 17, 25, 30], [8, 12, 18, 23, 31], [8, 13, 13, 18, 36], [8, 13, 18, 21, 32], [8, 13, 22, 24, 27], [8, 14, 15, 24, 31], [8, 15, 15, 22, 32], [8, 15, 18, 25, 28], [8, 16, 17, 18, 33], [8, 16, 19, 21, 30], [8, 18, 23, 23, 24], [8, 20, 21, 21, 26], [9, 9, 20, 26, 28], [9, 10, 10, 29, 30], [9, 10, 16, 17, 36], [9, 10, 18, 19, 34], [9, 10, 18, 26, 29], [9, 12, 17, 22, 32], [9, 12, 22, 23, 28], [9, 13, 14, 26, 30], [9, 14, 14, 18, 35], [9, 14, 16, 20, 33], [9, 14, 19, 22, 30], [9, 15, 16, 26, 28], [9, 16, 16, 23, 30], [9, 16, 18, 20, 31], [9, 16, 22, 24, 25], [9, 17, 20, 24, 26], [10, 10, 15, 21, 34], [10, 11, 12, 19, 36], [10, 11, 15, 26, 30], [10, 11, 21, 24, 28], [10, 12, 15, 23, 32], [10, 12, 17, 20, 33], [10, 12, 19, 24, 29], [10, 13, 18, 23, 30], [10, 14, 14, 21, 33], [10, 14, 18, 21, 31], [10, 15, 22, 22, 27], [10, 16, 19, 24, 27], [10, 16, 21, 21, 28], [11, 12, 12, 13, 38], [11, 13, 16, 24, 30], [11, 14, 15, 18, 34], [11, 14, 20, 24, 27], [11, 15, 18, 26, 26], [11, 20, 21, 22, 24], [12, 12, 13, 14, 37], [12, 12, 17, 17, 34], [12, 12, 17, 22, 31], [12, 12, 22, 25, 25], [12, 12, 23, 23, 26], [12, 13, 14, 27, 28], [12, 13, 18, 19, 32], [12, 13, 21, 22, 28], [12, 14, 20, 21, 29], [12, 16, 19, 19, 30], [12, 17, 17, 20, 30], [12, 17, 22, 23, 24], [12, 18, 20, 23, 25], [12, 19, 20, 21, 26], [13, 13, 18, 24, 28], [13, 16, 16, 21, 30], [13, 18, 18, 23, 26], [13, 18, 20, 20, 27], [14, 14, 15, 26, 27], [14, 14, 17, 21, 30], [14, 15, 16, 16, 33], [14, 15, 20, 24, 25], [14, 15, 21, 22, 26], [14, 16, 20, 21, 27], [14, 17, 18, 22, 27], [15, 16, 16, 18, 31], [15, 16, 17, 24, 26], [15, 17, 18, 20, 28], [16, 20, 21, 21, 22], [17, 18, 21, 22, 22], [18, 18, 19, 22, 23], [18, 19, 19, 20, 24]

2022 somme de 6 carrés parfaits

Pour information, il y a 1 837 décompositions de 2022 en somme de 6 carrés :

Voici la liste des 1837 sextuplets [a, b, c, d, e, f] tels que

$2022 = a^2 + b^2 + c^2 + d^2+ e^2+ f^2$ [1, 1, 1, 1, 13, 43], [1, 1, 1, 5, 25, 37], [1, 1, 1, 7, 11, 43], [1, 1, 1, 7, 17, 41], [1, 1, 1, 11, 23, 37], [1, 1, 1, 13, 13, 41], [1, 1, 1, 13, 25, 35], [1, 1, 1, 17, 19, 37], [1, 1, 1, 23, 23, 31], [1, 1, 2, 4, 8, 44], [1, 1, 2, 4, 20, 40], [1, 1, 2, 12, 24, 36], [1, 1, 3, 7, 21, 39], [1, 1, 3, 9, 9, 43], [1, 1, 3, 9, 29, 33], [1, 1, 3, 21, 27, 29], [1, 1, 4, 8, 28, 34], [1, 1, 4, 14, 28, 32], [1, 1, 5, 5, 11, 43], [1, 1, 5, 5, 17, 41], [1, 1, 5, 23, 25, 29], [1, 1, 7, 7, 31, 31], [1, 1, 7, 11, 13, 41], [1, 1, 7, 11, 25, 35], [1, 1, 7, 13, 29, 31], [1, 1, 7, 15, 15, 39], [1, 1, 7, 17, 29, 29], [1, 1, 7, 21, 21, 33], [1, 1, 8, 10, 16, 40], [1, 1, 8, 16, 16, 38], [1, 1, 8, 16, 26, 32], [1, 1, 8, 20, 20, 34], [1, 1, 9, 11, 27, 33], [1, 1, 9, 15, 25, 33], [1, 1, 11, 13, 19, 37], [1, 1, 11, 21, 27, 27], [1, 1, 11, 23, 23, 29], [1, 1, 12, 12, 24, 34], [1, 1, 12, 16, 18, 36], [1, 1, 12, 20, 24, 30], [1, 1, 13, 13, 29, 29], [1, 1, 13, 19, 23, 31], [1, 1, 14, 16, 28, 28], [1, 1, 14, 20, 20, 32], [1, 1, 15, 15, 27, 29], [1, 1, 15, 21, 25, 27], [1, 1, 16, 16, 22, 32], [1, 1, 17, 19, 23, 29], [1, 2, 2, 2, 28, 35], [1, 2, 2, 4, 29, 34], [1, 2, 2, 8, 10, 43], [1, 2, 2, 13, 20, 38], [1, 2, 3, 6, 6, 44], [1, 2, 3, 6, 26, 36], [1, 2, 3, 10, 12, 42], [1, 2, 3, 18, 28, 30], [1, 2, 4, 4, 7, 44], [1, 2, 4, 4, 31, 32], [1, 2, 4, 8, 16, 41], [1, 2, 4, 10, 26, 35], [1, 2, 4, 13, 26, 34], [1, 2, 4, 14, 19, 38], [1, 2, 4, 16, 28, 31], [1, 2, 4, 19, 22, 34], [1, 2, 4, 22, 26, 29], [1, 2, 5, 8, 22, 38], [1, 2, 5, 14, 14, 40], [1, 2, 5, 22, 22, 32], [1, 2, 6, 6, 24, 37], [1, 2, 6, 18, 19, 36], [1, 2, 6, 24, 26, 27], [1, 2, 7, 20, 28, 28], [1, 2, 8, 8, 17, 40], [1, 2, 8, 9, 24, 36], [1, 2, 8, 10, 22, 37], [1, 2, 8, 11, 26, 34], [1, 2, 8, 18, 27, 30], [1, 2, 8, 20, 23, 32], [1, 2, 9, 24, 24, 28], [1, 2, 10, 11, 14, 40], [1, 2, 10, 19, 20, 34], [1, 2, 10, 20, 26, 29], [1, 2, 10, 21, 24, 30], [1, 2, 11, 14, 16, 38], [1, 2, 11, 14, 26, 32], [1, 2, 11, 16, 22, 34], [1, 2, 12, 18, 18, 35], [1, 2, 14, 14, 16, 37], [1, 2, 14, 14, 20, 35], [1, 2, 14, 14, 28, 29], [1, 2, 14, 19, 26, 28], [1, 2, 16, 20, 20, 31], [1, 2, 17, 24, 24, 24], [1, 2, 18, 21, 24, 26], [1, 3, 3, 3, 25, 37], [1, 3, 3, 7, 27, 35], [1, 3, 3, 9, 31, 31], [1, 3, 3, 11, 19, 39], [1, 3, 3, 17, 25, 33], [1, 3, 4, 6, 14, 42], [1, 3, 4, 14, 30, 30], [1, 3, 5, 5, 21, 39], [1, 3, 5, 9, 15, 41], [1, 3, 5, 13, 27, 33], [1, 3, 5, 23, 27, 27], [1, 3, 6, 6, 28, 34], [1, 3, 6, 12, 26, 34], [1, 3, 6, 14, 22, 36], [1, 3, 6, 20, 26, 30], [1, 3, 7, 9, 19, 39], [1, 3, 9, 9, 13, 41], [1, 3, 9, 9, 25, 35], [1, 3, 9, 11, 17, 39], [1, 3, 9, 11, 21, 37], [1, 3, 9, 19, 27, 29], [1, 3, 9, 21, 23, 31], [1, 3, 10, 12, 18, 38], [1, 3, 11, 15, 21, 35], [1, 3, 11, 19, 21, 33], [1, 3, 12, 22, 22, 30], [1, 3, 13, 15, 23, 33], [1, 3, 13, 21, 21, 31], [1, 3, 14, 14, 18, 36], [1, 3, 15, 23, 23, 27], [1, 3, 17, 21, 21, 29], [1, 4, 4, 4, 23, 38], [1, 4, 4, 7, 28, 34], [1, 4, 4, 9, 12, 42], [1, 4, 4, 10, 17, 40], [1, 4, 4, 12, 18, 39], [1, 4, 4, 16, 17, 38], [1, 4, 4, 17, 26, 32], [1, 4, 4, 18, 24, 33], [1, 4, 4, 23, 26, 28], [1, 4, 6, 6, 13, 42], [1, 4, 6, 12, 12, 41], [1, 4, 6, 12, 15, 40], [1, 4, 6, 12, 23, 36], [1, 4, 6, 13, 30, 30], [1, 4, 7, 10, 16, 40], [1, 4, 7, 16, 16, 38], [1, 4, 7, 16, 26, 32], [1, 4, 7, 20, 20, 34], [1, 4, 8, 8, 14, 41], [1, 4, 8, 14, 28, 31], [1, 4, 8, 16, 23, 34], [1, 4, 9, 12, 22, 36], [1, 4, 9, 18, 24, 32], [1, 4, 10, 10, 19, 38], [1, 4, 10, 14, 22, 35], [1, 4, 10, 16, 25, 32], [1, 4, 12, 12, 14, 39], [1, 4, 12, 14, 24, 33], [1, 4, 12, 18, 24, 31], [1, 4, 13, 14, 14, 38], [1, 4, 13, 14, 22, 34], [1, 4, 13, 22, 26, 26], [1, 4, 14, 16, 23, 32], [1, 4, 14, 20, 25, 28], [1, 4, 14, 22, 22, 29], [1, 4, 16, 17, 26, 28], [1, 4, 18, 23, 24, 24], [1, 4, 19, 22, 22, 26], [1, 4, 20, 20, 23, 26], [1, 5, 5, 7, 31, 31], [1, 5, 5, 11, 13, 41], [1, 5, 5, 11, 25, 35], [1, 5, 5, 13, 29, 31], [1, 5, 5, 15, 15, 39], [1, 5, 5, 17, 29, 29], [1, 5, 5, 21, 21, 33], [1, 5, 6, 6, 18, 40], [1, 5, 6, 6, 30, 32], [1, 5, 6, 22, 24, 30], [1, 5, 7, 7, 7, 43], [1, 5, 7, 7, 23, 37], [1, 5, 7, 17, 17, 37], [1, 5, 7, 19, 19, 35], [1, 5, 7, 19, 25, 31], [1, 5, 8, 10, 26, 34], [1, 5, 9, 13, 15, 39], [1, 5, 9, 15, 27, 31], [1, 5, 10, 10, 14, 40], [1, 5, 10, 14, 16, 38], [1, 5, 10, 14, 26, 32], [1, 5, 10, 16, 22, 34], [1, 5, 11, 11, 23, 35], [1, 5, 11, 17, 19, 35], [1, 5, 11, 17, 25, 31], [1, 5, 11, 25, 25, 25], [1, 5, 13, 13, 17, 37], [1, 5, 13, 19, 25, 29], [1, 5, 14, 18, 24, 30], [1, 5, 18, 18, 18, 32], [1, 6, 6, 6, 8, 43], [1, 6, 6, 8, 11, 42], [1, 6, 6, 8, 21, 38], [1, 6, 6, 8, 27, 34], [1, 6, 6, 12, 19, 38], [1, 6, 6, 13, 22, 36], [1, 6, 6, 14, 27, 32], [1, 6, 6, 16, 18, 37], [1, 6, 6, 18, 20, 35], [1, 6, 6, 18, 28, 29], [1, 6, 6, 21, 22, 32], [1, 6, 7, 8, 24, 36], [1, 6, 7, 24, 24, 28], [1, 6, 8, 11, 30, 30], [1, 6, 8, 12, 16, 39], [1, 6, 8, 15, 20, 36], [1, 6, 8, 16, 24, 33], [1, 6, 8, 18, 21, 34], [1, 6, 10, 12, 29, 30], [1, 6, 10, 16, 27, 30], [1, 6, 12, 16, 17, 36], [1, 6, 12, 18, 19, 34], [1, 6, 12, 18, 26, 29], [1, 6, 13, 14, 18, 36], [1, 6, 14, 18, 21, 32], [1, 6, 14, 22, 24, 27], [1, 6, 15, 20, 24, 28], [1, 6, 16, 18, 26, 27], [1, 6, 18, 19, 20, 30], [1, 6, 21, 22, 22, 24], [1, 7, 7, 11, 11, 41], [1, 7, 7, 11, 29, 31], [1, 7, 7, 13, 23, 35], [1, 7, 8, 8, 20, 38], [1, 7, 8, 10, 28, 32], [1, 7, 8, 20, 22, 32], [1, 7, 9, 9, 17, 39], [1, 7, 9, 9, 21, 37], [1, 7, 9, 15, 21, 35], [1, 7, 9, 19, 21, 33], [1, 7, 11, 11, 19, 37], [1, 7, 11, 13, 29, 29], [1, 7, 11, 19, 23, 31], [1, 7, 12, 12, 28, 30], [1, 7, 12, 24, 24, 26], [1, 7, 13, 17, 17, 35], [1, 7, 15, 17, 27, 27], [1, 7, 16, 16, 26, 28], [1, 7, 17, 19, 19, 31], [1, 7, 17, 23, 23, 25], [1, 7, 19, 19, 25, 25], [1, 7, 19, 21, 21, 27], [1, 8, 8, 16, 26, 31], [1, 8, 8, 22, 25, 28], [1, 8, 9, 12, 24, 34], [1, 8, 9, 16, 18, 36], [1, 8, 9, 20, 24, 30], [1, 8, 10, 17, 28, 28], [1, 8, 11, 14, 14, 38], [1, 8, 11, 14, 22, 34], [1, 8, 11, 22, 26, 26], [1, 8, 12, 12, 15, 38], [1, 8, 12, 18, 20, 33], [1, 8, 14, 14, 14, 37], [1, 8, 14, 20, 20, 31], [1, 8, 15, 16, 24, 30], [1, 8, 16, 16, 17, 34], [1, 8, 16, 16, 22, 31], [1, 8, 16, 20, 25, 26], [1, 8, 17, 20, 22, 28], [1, 9, 9, 13, 13, 39], [1, 9, 9, 13, 27, 31], [1, 9, 9, 17, 27, 29], [1, 9, 10, 12, 20, 36], [1, 9, 11, 15, 15, 37], [1, 9, 11, 17, 21, 33], [1, 9, 11, 19, 27, 27], [1, 9, 12, 14, 24, 32], [1, 9, 15, 15, 23, 31], [1, 9, 15, 19, 25, 27], [1, 9, 16, 18, 24, 28], [1, 9, 21, 21, 23, 23], [1, 10, 10, 10, 11, 40], [1, 10, 10, 11, 16, 38], [1, 10, 10, 11, 26, 32], [1, 10, 10, 14, 16, 37], [1, 10, 10, 14, 20, 35], [1, 10, 10, 14, 28, 29], [1, 10, 10, 19, 26, 28], [1, 10, 11, 18, 24, 30], [1, 10, 12, 15, 16, 36], [1, 10, 12, 24, 24, 25], [1, 10, 13, 14, 20, 34], [1, 10, 13, 20, 26, 26], [1, 10, 13, 22, 22, 28], [1, 10, 14, 20, 22, 29], [1, 10, 16, 16, 25, 28], [1, 10, 16, 18, 21, 30], [1, 10, 19, 20, 22, 26], [1, 11, 11, 11, 17, 37], [1, 11, 11, 17, 23, 31], [1, 11, 11, 23, 25, 25], [1, 11, 13, 19, 23, 29], [1, 11, 14, 14, 22, 32], [1, 11, 15, 15, 15, 35], [1, 11, 15, 15, 19, 33], [1, 11, 16, 22, 22, 26], [1, 11, 17, 17, 19, 31], [1, 11, 17, 19, 25, 25], [1, 11, 17, 21, 21, 27], [1, 11, 18, 18, 24, 26], [1, 12, 12, 14, 24, 31], [1, 12, 12, 15, 22, 32], [1, 12, 12, 18, 25, 28], [1, 12, 13, 18, 22, 30], [1, 12, 14, 23, 24, 24], [1, 12, 15, 20, 24, 26], [1, 13, 13, 13, 17, 35], [1, 13, 13, 15, 27, 27], [1, 13, 13, 19, 19, 31], [1, 13, 13, 23, 23, 25], [1, 13, 14, 14, 26, 28], [1, 13, 15, 15, 21, 31], [1, 13, 17, 19, 19, 29], [1, 13, 20, 22, 22, 22], [1, 13, 21, 21, 21, 23], [1, 14, 14, 18, 24, 27], [1, 14, 14, 19, 22, 28], [1, 14, 16, 16, 17, 32], [1, 14, 16, 16, 23, 28], [1, 14, 18, 21, 22, 24], [1, 14, 20, 20, 20, 25], [1, 15, 15, 17, 21, 29], [1, 15, 17, 21, 21, 25], [1, 16, 16, 20, 22, 25], [1, 16, 17, 18, 24, 24], [1, 16, 17, 20, 20, 26], [1, 16, 18, 18, 21, 26], [1, 17, 17, 17, 23, 25], [2, 2, 2, 4, 25, 37], [2, 2, 2, 5, 7, 44], [2, 2, 2, 5, 31, 32], [2, 2, 2, 7, 19, 40], [2, 2, 2, 11, 17, 40], [2, 2, 2, 16, 23, 35], [2, 2, 2, 19, 25, 32], [2, 2, 3, 4, 15, 42], [2, 2, 3, 4, 30, 33], [2, 2, 3, 9, 18, 40], [2, 2, 3, 9, 30, 32], [2, 2, 3, 12, 30, 31], [2, 2, 3, 15, 22, 36], [2, 2, 3, 23, 24, 30], [2, 2, 4, 5, 23, 38], [2, 2, 4, 6, 21, 39], [2, 2, 4, 7, 10, 43], [2, 2, 4, 10, 23, 37], [2, 2, 4, 11, 14, 41], [2, 2, 4, 14, 29, 31], [2, 2, 4, 17, 22, 35], [2, 2, 4, 19, 26, 31], [2, 2, 5, 7, 28, 34], [2, 2, 5, 9, 12, 42], [2, 2, 5, 10, 17, 40], [2, 2, 5, 12, 18, 39], [2, 2, 5, 16, 17, 38], [2, 2, 5, 17, 26, 32], [2, 2, 5, 18, 24, 33], [2, 2, 5, 23, 26, 28], [2, 2, 6, 15, 27, 32], [2, 2, 6, 21, 24, 31], [2, 2, 7, 8, 26, 35], [2, 2, 7, 10, 29, 32], [2, 2, 7, 11, 20, 38], [2, 2, 7, 13, 14, 40], [2, 2, 7, 14, 20, 37], [2, 2, 7, 16, 22, 35], [2, 2, 8, 10, 13, 41], [2, 2, 8, 10, 25, 35], [2, 2, 8, 13, 25, 34], [2, 2, 8, 14, 23, 35], [2, 2, 8, 22, 25, 29], [2, 2, 9, 14, 21, 36], [2, 2, 10, 13, 28, 31], [2, 2, 10, 16, 17, 37], [2, 2, 10, 17, 20, 35], [2, 2, 10, 17, 28, 29], [2, 2, 10, 19, 23, 32], [2, 2, 11, 16, 26, 31], [2, 2, 11, 22, 25, 28], [2, 2, 12, 21, 23, 30], [2, 2, 13, 14, 25, 32], [2, 2, 13, 15, 18, 36], [2, 2, 13, 17, 20, 34], [2, 2, 13, 20, 22, 31], [2, 2, 15, 18, 21, 32], [2, 2, 15, 22, 24, 27], [2, 2, 17, 20, 22, 29], [2, 3, 3, 12, 16, 40], [2, 3, 3, 20, 24, 32], [2, 3, 4, 11, 24, 36], [2, 3, 4, 15, 18, 38], [2, 3, 4, 16, 21, 36], [2, 3, 4, 24, 24, 29], [2, 3, 6, 6, 16, 41], [2, 3, 6, 7, 18, 40], [2, 3, 6, 7, 30, 32], [2, 3, 6, 10, 28, 33], [2, 3, 6, 14, 16, 39], [2, 3, 6, 17, 28, 30], [2, 3, 6, 18, 25, 32], [2, 3, 6, 20, 22, 33], [2, 3, 7, 22, 24, 30], [2, 3, 8, 9, 10, 42], [2, 3, 8, 10, 18, 39], [2, 3, 8, 12, 24, 35], [2, 3, 9, 14, 24, 34], [2, 3, 9, 24, 26, 26], [2, 3, 10, 12, 26, 33], [2, 3, 10, 15, 28, 30], [2, 3, 10, 17, 18, 36], [2, 3, 11, 12, 12, 40], [2, 3, 12, 13, 20, 36], [2, 3, 12, 14, 15, 38], [2, 3, 12, 15, 22, 34], [2, 3, 12, 17, 26, 30], [2, 3, 12, 20, 21, 32], [2, 3, 14, 18, 20, 33], [2, 3, 15, 18, 26, 28], [2, 3, 15, 20, 22, 30], [2, 3, 16, 18, 23, 30], [2, 3, 18, 18, 20, 31], [2, 3, 18, 22, 24, 25], [2, 4, 4, 4, 11, 43], [2, 4, 4, 4, 17, 41], [2, 4, 4, 5, 5, 44], [2, 4, 4, 5, 19, 40], [2, 4, 4, 7, 16, 41], [2, 4, 4, 8, 31, 31], [2, 4, 4, 11, 29, 32], [2, 4, 4, 16, 19, 37], [2, 4, 4, 19, 20, 35], [2, 4, 4, 19, 28, 29], [2, 4, 4, 20, 25, 31], [2, 4, 5, 7, 22, 38], [2, 4, 5, 8, 8, 43], [2, 4, 5, 10, 14, 41], [2, 4, 5, 11, 16, 40], [2, 4, 5, 13, 28, 32], [2, 4, 5, 14, 25, 34], [2, 4, 5, 25, 26, 26], [2, 4, 6, 6, 9, 43], [2, 4, 6, 6, 29, 33], [2, 4, 6, 9, 11, 42], [2, 4, 6, 9, 21, 38], [2, 4, 6, 9, 27, 34], [2, 4, 6, 11, 18, 39], [2, 4, 6, 15, 29, 30], [2, 4, 6, 21, 25, 30], [2, 4, 7, 8, 17, 40], [2, 4, 7, 9, 24, 36], [2, 4, 7, 10, 22, 37], [2, 4, 7, 11, 26, 34], [2, 4, 7, 18, 27, 30], [2, 4, 7, 20, 23, 32], [2, 4, 8, 13, 13, 40], [2, 4, 8, 13, 20, 37], [2, 4, 8, 16, 29, 29], [2, 4, 8, 17, 25, 32], [2, 4, 8, 23, 25, 28], [2, 4, 9, 11, 30, 30], [2, 4, 9, 12, 16, 39], [2, 4, 9, 15, 20, 36], [2, 4, 9, 16, 24, 33], [2, 4, 9, 18, 21, 34], [2, 4, 10, 10, 11, 41], [2, 4, 10, 10, 29, 31], [2, 4, 10, 11, 25, 34], [2, 4, 10, 13, 17, 38], [2, 4, 11, 12, 21, 36], [2, 4, 11, 14, 23, 34], [2, 4, 11, 16, 16, 37], [2, 4, 11, 16, 20, 35], [2, 4, 11, 16, 28, 29], [2, 4, 11, 23, 26, 26], [2, 4, 11, 24, 24, 27], [2, 4, 12, 12, 25, 33], [2, 4, 12, 20, 27, 27], [2, 4, 12, 21, 24, 29], [2, 4, 13, 14, 26, 31], [2, 4, 14, 17, 19, 34], [2, 4, 14, 17, 26, 29], [2, 4, 14, 19, 22, 31], [2, 4, 15, 15, 16, 36], [2, 4, 15, 24, 24, 25], [2, 4, 16, 16, 23, 31], [2, 4, 16, 19, 19, 32], [2, 4, 16, 21, 24, 27], [2, 4, 17, 17, 20, 32], [2, 4, 17, 19, 26, 26], [2, 4, 17, 20, 23, 28], [2, 4, 18, 18, 25, 27], [2, 4, 19, 20, 20, 29], [2, 5, 5, 20, 28, 28], [2, 5, 6, 6, 20, 39], [2, 5, 6, 6, 25, 36], [2, 5, 6, 7, 12, 42], [2, 5, 6, 15, 24, 34], [2, 5, 7, 10, 20, 38], [2, 5, 7, 12, 30, 30], [2, 5, 7, 18, 18, 36], [2, 5, 7, 22, 26, 28], [2, 5, 8, 8, 29, 32], [2, 5, 8, 11, 28, 32], [2, 5, 8, 14, 17, 38], [2, 5, 8, 17, 22, 34], [2, 5, 8, 19, 28, 28], [2, 5, 8, 22, 22, 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18, 22, 27], [8, 14, 16, 16, 17, 31], [8, 14, 16, 16, 25, 25], [8, 14, 16, 19, 19, 28], [8, 14, 17, 17, 20, 28], [8, 15, 15, 16, 24, 26], [8, 15, 15, 18, 20, 28], [8, 15, 18, 21, 22, 22], [8, 16, 17, 20, 22, 23], [8, 16, 18, 19, 21, 24], [8, 18, 18, 18, 19, 25], [9, 9, 9, 11, 17, 37], [9, 9, 9, 17, 23, 31], [9, 9, 9, 23, 25, 25], [9, 9, 10, 20, 24, 28], [9, 9, 11, 11, 23, 33], [9, 9, 11, 13, 27, 29], [9, 9, 11, 15, 17, 35], [9, 9, 11, 17, 19, 33], [9, 9, 12, 16, 26, 28], [9, 9, 13, 15, 25, 29], [9, 9, 13, 17, 21, 31], [9, 9, 13, 21, 25, 25], [9, 9, 16, 16, 18, 32], [9, 9, 17, 17, 21, 29], [9, 9, 19, 21, 23, 23], [9, 9, 20, 20, 22, 24], [9, 10, 10, 11, 18, 36], [9, 10, 10, 12, 21, 34], [9, 10, 10, 18, 24, 29], [9, 10, 10, 20, 21, 30], [9, 10, 11, 12, 26, 30], [9, 10, 12, 12, 23, 32], [9, 10, 12, 22, 22, 27], [9, 10, 13, 14, 24, 30], [9, 10, 13, 18, 18, 32], [9, 10, 15, 16, 24, 28], [9, 10, 16, 18, 19, 30], [9, 10, 17, 20, 24, 24], [9, 10, 18, 20, 21, 26], [9, 11, 11, 13, 21, 33], [9, 11, 11, 21, 23, 27], [9, 11, 12, 14, 18, 34], [9, 11, 12, 18, 26, 26], [9, 11, 14, 18, 20, 30], [9, 11, 15, 15, 23, 29], [9, 11, 15, 21, 23, 25], [9, 11, 17, 19, 21, 27], [9, 12, 14, 14, 26, 27], [9, 12, 14, 16, 16, 33], [9, 12, 14, 20, 24, 25], [9, 12, 14, 21, 22, 26], [9, 12, 16, 16, 18, 31], [9, 12, 16, 17, 24, 26], [9, 12, 17, 18, 20, 28], [9, 13, 13, 15, 17, 33], [9, 13, 14, 18, 24, 26], [9, 13, 15, 15, 19, 31], [9, 13, 15, 17, 23, 27], [9, 13, 19, 21, 21, 23], [9, 14, 14, 18, 21, 28], [9, 14, 16, 18, 18, 29], [9, 14, 18, 19, 22, 24], [9, 15, 15, 17, 19, 29], [9, 15, 16, 20, 22, 24], [9, 15, 17, 19, 21, 25], [9, 16, 16, 18, 23, 24], [9, 16, 18, 18, 19, 26], [10, 10, 10, 13, 23, 32], [10, 10, 10, 16, 25, 29], [10, 10, 10, 19, 20, 31], [10, 10, 11, 15, 24, 30], [10, 10, 11, 16, 17, 34], [10, 10, 11, 16, 22, 31], [10, 10, 11, 20, 25, 26], [10, 10, 12, 18, 25, 27], [10, 10, 14, 16, 23, 29], [10, 10, 15, 16, 21, 30], [10, 10, 16, 19, 23, 26], [10, 11, 11, 16, 20, 32], [10, 11, 12, 12, 12, 37], [10, 11, 12, 12, 27, 28], [10, 11, 14, 14, 25, 28], [10, 11, 14, 20, 23, 26], [10, 11, 15, 18, 24, 26], [10, 11, 16, 19, 20, 28], [10, 12, 12, 12, 23, 31], [10, 12, 12, 13, 13, 36], [10, 12, 12, 13, 21, 32], [10, 12, 12, 15, 25, 28], [10, 12, 12, 16, 17, 33], [10, 12, 12, 23, 23, 24], [10, 12, 13, 14, 18, 33], [10, 12, 13, 15, 22, 30], [10, 12, 13, 18, 18, 31], [10, 12, 14, 18, 23, 27], [10, 12, 17, 18, 18, 29], [10, 12, 18, 21, 22, 23], [10, 12, 19, 20, 21, 24], [10, 13, 13, 20, 20, 28], [10, 13, 14, 14, 20, 31], [10, 13, 14, 16, 25, 26], [10, 13, 14, 17, 22, 28], [10, 13, 16, 16, 20, 29], [10, 13, 16, 22, 22, 23], [10, 13, 18, 18, 23, 24], [10, 14, 14, 15, 24, 27], [10, 14, 14, 17, 20, 29], [10, 14, 15, 21, 22, 24], [10, 14, 16, 19, 22, 25], [10, 14, 17, 19, 20, 26], [10, 15, 16, 17, 24, 24], [10, 15, 16, 18, 21, 26], [10, 16, 16, 16, 23, 25], [10, 16, 17, 18, 18, 27], [10, 19, 19, 20, 20, 20], [11, 11, 11, 11, 13, 37], [11, 11, 11, 13, 23, 31], [11, 11, 11, 17, 23, 29], [11, 11, 12, 12, 14, 36], [11, 11, 12, 22, 24, 24], [11, 11, 13, 17, 19, 31], [11, 11, 13, 19, 25, 25], [11, 11, 13, 21, 21, 27], [11, 11, 14, 20, 20, 28], [11, 11, 16, 16, 22, 28], [11, 11, 17, 17, 19, 29], [11, 11, 19, 19, 23, 23], [11, 12, 12, 14, 24, 29], [11, 12, 12, 19, 24, 26], [11, 12, 12, 20, 22, 27], [11, 12, 14, 16, 24, 27], [11, 12, 16, 21, 22, 24], [11, 12, 18, 18, 22, 25], [11, 13, 13, 13, 13, 35], [11, 13, 13, 19, 19, 29], [11, 13, 14, 16, 16, 32], [11, 13, 15, 15, 21, 29], [11, 13, 15, 21, 21, 25], [11, 13, 16, 18, 24, 24], [11, 13, 16, 20, 20, 26], [11, 13, 17, 17, 23, 25], [11, 14, 14, 14, 17, 32], [11, 14, 14, 14, 23, 28], [11, 14, 14, 20, 22, 25], [11, 14, 15, 16, 18, 30], [11, 14, 16, 17, 22, 26], [11, 15, 15, 19, 19, 27], [11, 16, 19, 20, 20, 22], [11, 17, 17, 21, 21, 21], [11, 17, 19, 19, 19, 23], [11, 18, 18, 18, 20, 23], [12, 12, 12, 13, 14, 35], [12, 12, 12, 17, 25, 26], [12, 12, 13, 18, 20, 29], [12, 12, 14, 15, 17, 32], [12, 12, 14, 15, 23, 28], [12, 12, 14, 16, 21, 29], [12, 12, 15, 20, 22, 25], [12, 12, 16, 17, 17, 30], [12, 12, 16, 18, 23, 25], [12, 12, 16, 19, 21, 26], [12, 13, 14, 17, 18, 30], [12, 13, 14, 19, 24, 24], [12, 13, 15, 18, 22, 26], [12, 13, 16, 18, 20, 27], [12, 14, 14, 15, 19, 30], [12, 14, 15, 16, 24, 25], [12, 14, 16, 16, 21, 27], [12, 14, 20, 20, 21, 21], [12, 15, 16, 17, 18, 28], [12, 15, 18, 19, 22, 22], [12, 15, 18, 20, 20, 23], [12, 16, 16, 21, 21, 22], [12, 16, 18, 19, 19, 24], [12, 17, 17, 18, 20, 24], [13, 13, 13, 19, 23, 25], [13, 13, 15, 17, 21, 27], [13, 13, 19, 21, 21, 21], [13, 13, 20, 20, 20, 22], [13, 14, 14, 16, 23, 26], [13, 14, 15, 18, 18, 28], [13, 14, 16, 16, 19, 28], [13, 14, 17, 20, 22, 22], [13, 16, 16, 18, 21, 24], [13, 16, 18, 18, 18, 25], [13, 17, 17, 17, 19, 25], [14, 14, 14, 17, 19, 28], [14, 14, 16, 19, 22, 23], [14, 14, 17, 18, 21, 24], [14, 15, 15, 20, 20, 24], [14, 16, 16, 16, 23, 23], [14, 16, 16, 17, 20, 25], [15, 15, 15, 17, 23, 23], [15, 15, 15, 19, 19, 25], [15, 15, 16, 16, 22, 24], [16, 16, 16, 17, 17, 26], [17, 17, 19, 19, 19, 19], [17, 18, 18, 18, 19, 20]

L'intérêt pour les sommes de carrés remonte à l’Antiquité : on trouve de telles sommes dans des tablettes en cunéiforme du début du 2^e millénaire avant notre ère et deux propositions dans les Éléments d'Euclide expliquent comment construire des carrés parfaits dont la somme ou la différence forment encore des carrés parfaits, ou au contraire comment ne pas obtenir un carré en sommant deux carrés.

Triplets pythagoriciens

En arithmétique, un triplet pythagoricien ou triplet de Pythagore est un triplet (a, b, c) d'entiers naturels non nuls vérifiant la relation de Pythagore : $\displaystyle a^{2}+b^{2}=c^{2}$
Les triplets pythagoriciens les plus connus sont (3, 4, 5) et (5, 12, 13) mais l'on a aussi :

$2022^2=1728^2+1050^2$

Donc (1050, 1728, 2022) est un triplet pythagoricien.

2022 pythagore

2022 - triplet pythagoricien

Autres curiosités du nombre 2022

Somme des chiffres et somme des diviseurs premiers
La somme des chiffres de 2022 soit $(2+0+2+2=6)$ divise la somme des chiffres de ses facteurs premiers 2, 3 et 337 puisque :
$6~ \text{divise}~2+3+3+3+7=18 =6\times3$
Elle divise aussi la somme de ses facteurs premiers puisque :
$6~ \text{divise}~2+3+337=342=6\times 57$
Avec $\pi$ , e et la partie entière (floor in english)
$2022=\lfloor \pi^6+e^6+ \pi^5+e^5+ \pi^4+e^4+ \pi^3+e^3\rfloor$
En effet on a :
$\pi^6+e^6+ \pi^5+e^5+ \pi^4+e^4+ \pi^3+e^3\approx 2022,35$
Exponentielle
$e^{2022} > 10^{878}$
On estime que le nombre de particules dans l'univers est de l'ordre de $10^{80}$ .

: pour calculer le nombre de chiffres d'un grand ombre on utilise le logarithme décimale, voir cette page.
Et ici on a : $Log\left(e^{2022}\right) = 2022\times Log(e) \approx 878,1434$
Image miroire d'un nombre
On appelle l'image miroir d'un nombre, le nombre lu à l'envers. Par exemple, l'image miroire de 324 est 423.
Le miroir de 2022 est 2202.
Une proprité amusante est que le carré de son miroir est égal au miroir de son carré
- $2022^{2}=4~088~484$
- $2202^{2}=4~848~804$