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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=2017+20272    
  • Décomposition en facteurs premiers
    2022=2×3×337 
  • Le 2022e nombre premier est 17 581

 

 Ecritures de 2022

  • Avec les premiers nombres premiers :
    2022=2×3×5×7!11+13+17+19232931
  • Avec les 10 chiffres :
    2022=9×8×(7+6)+543×2×1
    2022=9876+(5+4)×321
    et on aura
    2023=(987+65)×4321
  • 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!=1×2××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!!=1×3×5×7 and 6!!=2×4×6. 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
    20226=337
  • Nombre 3-polis : trois fois somme d'entiers consécutifs.
    • 2022=163+164++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.

    2022=2×3×337
    Tous les nombres sphéniques ont exactement huit diviseurs.
      
  • Somme de deux premiers consécutifs
    2022=1009+1013
  • Somme de deux nombres premiers
    2022=5+2017=11+2011==1009+1013
    2022 est 59 fois somme de deux premiers.
     
  • Nombre semi-parfait : somme de certains de ses diviseurs.
    2022=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 : 2022=2×3×337

Et l'on a :

  • 2=4×0+2.
  • 3=4×0+3, mais il n'est pas à une puissance paire dans la décomposition.
  • 337=4×84+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=22+132+432
  • 2022=52+292+342
  • 2022=72+232+382
  • 2022=102+312+312
  • 2022=112+262+352
  • 2022=132+222+372
  • 2022=172+172+382

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 4i(8j+7) 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=12+12+162+422
  • 2022=12+12+242+382=
  • 2022=192+192+202+302


Voici la liste des 90 quadruplets [a, b, c, d] tels que 2022=a2+b2+c2+d2 [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=a2+b2+c2+d2

 

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 :
    • 2×4m, 6×4m et 14×4m

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 k5, 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=12+12+182+202+362
  • 2022=12+22+122+282+332=
  • 2022=182+192+192+212+242


Voici la liste des 379 quintuplets [a, b, c, d, e] tels que 2022=a2+b2+c2+d2+e2 [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=a2+b2+c2+d2+e2+f2 [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, 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14, 14, 19, 23, 26], [8, 14, 15, 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 2e 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 : a2+b2=c2
Les triplets pythagoriciens les plus connus sont (3, 4, 5) et (5, 12, 13) mais l'on a aussi :

20222=17282+10502

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 divise 2+3+3+3+7=18=6×3
    Elle divise aussi la somme de ses facteurs premiers puisque :
    6 divise 2+3+337=342=6×57
  • Avec π, e et la partie entière (floor in english)
    2022=π6+e6+π5+e5+π4+e4+π3+e3
    En effet on a :
    π6+e6+π5+e5+π4+e4+π3+e32022,35
  • Exponentielle
    e2022>10878
    On estime que le nombre de particules dans l'univers est de l'ordre de 1080.
     
    Remarque : pour calculer le nombre de chiffres d'un grand ombre on utilise le logarithme décimale, voir cette page.
    Et ici on a : Log(e2022)=2022×Log(e)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
    • 20222=4 088 484
    • 22022=4 848 804

 

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