Bonne année 2023
2023 quelques propriétés de cet entier naturel
Le site Math93.com vous souhaite une heureuse, chaleureuse et studieuse année 2023. Profitons-en pour revenir sur quelques caractéristiques de ce nombre impair.
Écriture du nombre 2023
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-trois
Anglais : Two thousand twenty-three
Allemand : zweitausenddreiundzwanzig
Espagnol : dos mil veintitrés
Italien : duemilaventitré / Portugais : dois mil e vinte e três
Autres écriture de 2023
| En chiffre romain | MMXXIII |
| En binaire | 11111100111 |
| En octal | 3747 |
| En hexadécimal | 7e7 (c'est un palyndrome) |
| En dollars américains | USD 2,023.00 ($) |
| En euros | 2 023,00 EUR (€) |
Opérations mathématiques
L'année 2023
2023 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 2022 prendra fin le dimanche 1 janvier 2023 (semaine N°52 2022). La 1ère semaine de l'année 2023 (semaine N°1 2023) débutera quant à elle le lundi 2 janvier 2023.
2023 est une année à 53 dimanches et pas 52. On a eu 53 dimanches en: 2000, 2006, 2012 et 2017.
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 2 023 et nombres premiers
- Diviseurs
L'entier impair 2023 admet seulement 6 diviseurs qui sont : $$1, 7, 17, 119, 289, 2023$$
- Nombre premiers : 2023 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
$$2 023 = 7\times17^2$$ - Le 2023e nombre premier est 17 597
2023 et quelques décompositions
- Nombre de Harshad ou de Niven
C'est un nombre divisible par la somme de ses chiffres soit ici : \(2+0+2+3=7\)
$$\dfrac{2023}{7}=17^2=289$$ - Nombre 5-polis : cinq fois somme d'entiers consécutifs.
- \(2023= 43+44 +\cdots + 76\)
- \(2023= 111+\cdots+127\)
- \(2023= 138+\cdots+151\)
- \(2023= 286+\cdots+292\)
- \(2023= 1011+1012\)
- Somme de deux premiers consécutifs
2023 n'est pas somme de 2 premiers consécutifs mais 2022 l'était : $$2022=1009+1013$$ - Somme de deux nombres premiers
2023 n'est pas somme de 2 premiers mais 2022 l'était : $$2 022 = 5 + 2017 = 11+2011=\cdots=1009+1013$$
2022 est 59 fois somme de deux premiers.
- Nombre semi-parfait : somme de certains de ses diviseurs.
2023 n'est pas semi parfait contrairement à 2022 $$2 022 = 1011 + 674 + 337$$
- Si on cherche à écrire 2 023 sous la forme d’une opération impliquant seulement avec les chiffres : $$2023 = (2+0+2+3)(2^2+0^2+2^2+3^2)^2$$
2023 nombre déficient
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 2023 est donc déficient :
$$2023>1+7+17+119+289=433$$
2023 Somme de carrés
2023 somme de deux carrés ?
2023 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 2023 sont : $$2 023 = 7\times17^2$$
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.
2023 somme de 3 carrés parfaits
Le nombre 2023 ne peut pas s'écrire comme somme de 3 carrés.
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.
Or ici avec \(i=0\) et \(j=253\) on remarque que : $$2023=4^0 \times (8\times 253-1)$$
2023 somme de 4 carrés parfaits
Le nombre 2023 est un entier qui peut s'écrire comme la somme de quatre carrés 61 fois :
- \( 2023=1^2+2^2+13^2 +43^2\)
- \( 2023 = 1^2+5^2+29^2 +34^2=\cdots\)
- \( 2023=19^2+19^2+25^2 +26^2\)
Voici la liste des 90 quadruplets [a, b, c, d] tels que $$2023 = a^2 + b^2 + c^2 + d^2$$ [1, 2, 13, 43], [1, 5, 29, 34], [1, 7, 23, 38], [1, 10, 31, 31], [1, 11, 26, 35], [1, 13, 22, 37], [1, 17, 17, 38], [2, 5, 25, 37], [2, 7, 11, 43], [2, 7, 17, 41], [2, 11, 23, 37], [2, 13, 13, 41], [2, 13, 25, 35], [2, 17, 19, 37], [2, 23, 23, 31], [3, 3, 18, 41], [3, 3, 22, 39], [3, 5, 15, 42], [3, 5, 30, 33], [3, 9, 13, 42], [3, 13, 18, 39], [3, 14, 27, 33], [3, 18, 27, 31], [3, 21, 22, 33], [5, 5, 23, 38], [5, 6, 21, 39], [5, 7, 10, 43], [5, 10, 23, 37], [5, 11, 14, 41], [5, 14, 29, 31], [5, 17, 22, 35], [5, 19, 26, 31], [6, 9, 15, 41], [6, 13, 27, 33], [6, 23, 27, 27], [7, 11, 22, 37], [7, 13, 19, 38], [7, 17, 23, 34], [7, 22, 23, 31], [9, 9, 30, 31], [9, 14, 15, 39], [9, 18, 23, 33], [9, 22, 27, 27], [10, 11, 11, 41], [10, 11, 29, 31], [10, 13, 23, 35], [11, 11, 25, 34], [11, 13, 17, 38], [13, 13, 23, 34], [13, 14, 17, 37], [13, 15, 27, 30], [13, 18, 21, 33], [13, 22, 23, 29], [14, 19, 25, 29], [15, 15, 22, 33], [17, 17, 17, 34], [17, 17, 22, 31], [17, 22, 25, 25], [17, 23, 23, 26], [18, 21, 23, 27], [19, 19, 25, 26]
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.
2023 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.
2023 somme de 5 carrés parfaits
Le nombre 2023 est un entier qui peut s'écrire comme la somme de cinq carrés 379 fois :
- \( 2023=1^2+1^2+18^2 +20^2+36^2\)
- \( 2023 = 1^2+2^2+12^2 +28^2+33^2=\cdots\)
- \( 2023=18^2+19^2+19^2+21^2 +24^2\)
Voici la liste des 379 quintuplets [a, b, c, d, e] tels que $$2023 = a^2 + b^2 + c^2 + d^2+ e^2$$ [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]
2023 somme de 6 carrés parfaits
Pour information, il y a 2 058 décompositions de 2023 en somme de 6 carrés :
Voici la liste des 2058 sextuplets [a, b, c, d, e, f] tels que $$2023 = a^2 + b^2 + c^2 + d^2+ e^2+ f^2$$ ([[1, 1, 1, 18, 20, 36], [1, 1, 2, 12, 28, 33], [1, 1, 2, 18, 18, 37], [1, 1, 2, 21, 26, 30], [1, 1, 4, 4, 15, 42], [1, 1, 4, 4, 30, 33], [1, 1, 4, 9, 18, 40], [1, 1, 4, 9, 30, 32], [1, 1, 4, 12, 30, 31], [1, 1, 4, 15, 22, 36], [1, 1, 4, 23, 24, 30], [1, 1, 5, 6, 14, 42], [1, 1, 5, 14, 30, 30], [1, 1, 6, 6, 10, 43], [1, 1, 6, 8, 20, 39], [1, 1, 6, 8, 25, 36], [1, 1, 6, 10, 11, 42], [1, 1, 6, 10, 21, 38], [1, 1, 6, 10, 27, 34], [1, 1, 6, 17, 20, 36], [1, 1, 6, 24, 25, 28], [1, 1, 7, 8, 12, 42], [1, 1, 7, 10, 24, 36], [1, 1, 8, 15, 24, 34], [1, 1, 9, 12, 14, 40], [1, 1, 9, 16, 28, 30], [1, 1, 10, 11, 30, 30], [1, 1, 10, 12, 16, 39], [1, 1, 10, 15, 20, 36], [1, 1, 10, 16, 24, 33], [1, 1, 10, 18, 21, 34], [1, 1, 11, 18, 26, 30], [1, 1, 12, 12, 17, 38], [1, 1, 12, 18, 23, 32], [1, 1, 12, 24, 25, 26], [1, 1, 14, 14, 27, 30], [1, 1, 14, 15, 24, 32], [1, 1, 14, 21, 22, 30], [1, 1, 16, 17, 24, 30], [1, 1, 18, 22, 22, 27], [1, 2, 2, 3, 18, 41], [1, 2, 2, 3, 22, 39], [1, 2, 2, 5, 15, 42], [1, 2, 2, 5, 30, 33], [1, 2, 2, 9, 13, 42], [1, 2, 2, 13, 18, 39], [1, 2, 2, 14, 27, 33], [1, 2, 2, 18, 27, 31], [1, 2, 2, 21, 22, 33], [1, 2, 3, 3, 8, 44], [1, 2, 3, 3, 20, 40], [1, 2, 3, 4, 12, 43], [1, 2, 3, 6, 23, 38], [1, 2, 3, 7, 14, 42], [1, 2, 3, 8, 24, 37], [1, 2, 3, 9, 22, 38], [1, 2, 3, 12, 29, 32], [1, 2, 3, 16, 27, 32], [1, 2, 3, 18, 23, 34], [1, 2, 3, 21, 28, 28], [1, 2, 3, 22, 25, 30], [1, 2, 4, 9, 20, 39], [1, 2, 4, 9, 25, 36], [1, 2, 4, 15, 16, 39], [1, 2, 5, 11, 24, 36], [1, 2, 5, 15, 18, 38], [1, 2, 5, 16, 21, 36], [1, 2, 5, 24, 24, 29], [1, 2, 6, 7, 13, 42], [1, 2, 6, 9, 26, 35], [1, 2, 6, 10, 19, 39], [1, 2, 6, 11, 30, 31], [1, 2, 6, 17, 18, 37], [1, 2, 7, 12, 12, 41], [1, 2, 7, 12, 15, 40], [1, 2, 7, 12, 23, 36], [1, 2, 7, 13, 30, 30], [1, 2, 8, 9, 28, 33], [1, 2, 8, 12, 17, 39], [1, 2, 8, 12, 21, 37], [1, 2, 8, 17, 24, 33], [1, 2, 8, 21, 27, 28], [1, 2, 9, 9, 16, 40], [1, 2, 9, 13, 18, 38], [1, 2, 9, 14, 29, 30], [1, 2, 9, 19, 26, 30], [1, 2, 9, 20, 24, 31], [1, 2, 10, 10, 27, 33], [1, 2, 10, 15, 18, 37], [1, 2, 10, 17, 27, 30], [1, 2, 11, 12, 27, 32], [1, 2, 11, 18, 22, 33], [1, 2, 12, 12, 19, 37], [1, 2, 12, 15, 25, 32], [1, 2, 12, 16, 23, 33], [1, 2, 12, 17, 17, 36], [1, 2, 12, 19, 27, 28], [1, 2, 13, 18, 25, 30], [1, 2, 14, 15, 21, 34], [1, 2, 15, 15, 28, 28], [1, 2, 15, 16, 24, 31], [1, 2, 15, 21, 26, 26], [1, 2, 17, 18, 26, 27], [1, 2, 18, 18, 23, 29], [1, 3, 3, 8, 28, 34], [1, 3, 3, 14, 28, 32], [1, 3, 4, 5, 6, 44], [1, 3, 4, 5, 26, 36], [1, 3, 4, 6, 19, 40], [1, 3, 4, 8, 13, 42], [1, 3, 4, 12, 22, 37], [1, 3, 4, 14, 24, 35], [1, 3, 4, 16, 29, 30], [1, 3, 4, 20, 21, 34], [1, 3, 4, 22, 27, 28], [1, 3, 5, 8, 18, 40], [1, 3, 5, 8, 30, 32], [1, 3, 5, 12, 20, 38], [1, 3, 5, 16, 24, 34], [1, 3, 6, 7, 22, 38], [1, 3, 6, 8, 8, 43], [1, 3, 6, 10, 14, 41], [1, 3, 6, 11, 16, 40], [1, 3, 6, 13, 28, 32], [1, 3, 6, 14, 25, 34], [1, 3, 6, 25, 26, 26], [1, 3, 7, 10, 10, 42], [1, 3, 7, 14, 18, 38], [1, 3, 7, 18, 22, 34], [1, 3, 8, 8, 11, 42], [1, 3, 8, 8, 21, 38], [1, 3, 8, 8, 27, 34], [1, 3, 8, 12, 19, 38], [1, 3, 8, 13, 22, 36], [1, 3, 8, 14, 27, 32], [1, 3, 8, 16, 18, 37], [1, 3, 8, 18, 20, 35], [1, 3, 8, 18, 28, 29], [1, 3, 8, 21, 22, 32], [1, 3, 9, 10, 26, 34], [1, 3, 10, 12, 13, 40], [1, 3, 10, 12, 20, 37], [1, 3, 10, 14, 14, 39], [1, 3, 10, 16, 19, 36], [1, 3, 10, 20, 27, 28], [1, 3, 10, 22, 23, 30], [1, 3, 11, 14, 20, 36], [1, 3, 11, 18, 28, 28], [1, 3, 12, 13, 16, 38], [1, 3, 12, 13, 26, 32], [1, 3, 12, 19, 22, 32], [1, 3, 13, 22, 24, 28], [1, 3, 14, 20, 24, 29], [1, 3, 18, 22, 23, 26], [1, 3, 19, 20, 24, 26], [1, 3, 20, 20, 22, 27], [1, 4, 4, 6, 27, 35], [1, 4, 4, 15, 26, 33], [1, 4, 4, 18, 21, 35], [1, 4, 4, 19, 27, 30], [1, 4, 5, 6, 24, 37], [1, 4, 5, 18, 19, 36], [1, 4, 5, 24, 26, 27], [1, 4, 6, 7, 20, 39], [1, 4, 6, 7, 25, 36], [1, 4, 6, 8, 15, 41], [1, 4, 6, 9, 17, 40], [1, 4, 6, 13, 24, 35], [1, 4, 6, 15, 28, 31], [1, 4, 6, 16, 25, 33], [1, 4, 6, 20, 27, 29], [1, 4, 7, 7, 12, 42], [1, 4, 7, 15, 24, 34], [1, 4, 8, 9, 30, 31], [1, 4, 8, 14, 15, 39], [1, 4, 8, 18, 23, 33], [1, 4, 8, 22, 27, 27], [1, 4, 9, 9, 20, 38], [1, 4, 9, 10, 12, 41], [1, 4, 9, 10, 15, 40], [1, 4, 9, 10, 23, 36], [1, 4, 9, 12, 25, 34], [1, 4, 9, 15, 16, 38], [1, 4, 9, 15, 26, 32], [1, 4, 9, 20, 25, 30], [1, 4, 10, 13, 21, 36], [1, 4, 10, 21, 21, 32], [1, 4, 11, 12, 29, 30], [1, 4, 11, 16, 27, 30], [1, 4, 12, 13, 18, 37], [1, 4, 12, 14, 21, 35], [1, 4, 12, 15, 26, 31], [1, 4, 12, 17, 22, 33], [1, 4, 13, 18, 27, 28], [1, 4, 13, 19, 24, 30], [1, 4, 14, 15, 17, 36], [1, 4, 15, 15, 20, 34], [1, 4, 15, 16, 25, 30], [1, 4, 15, 23, 24, 26], [1, 4, 18, 20, 21, 29], [1, 5, 5, 8, 12, 42], [1, 5, 5, 10, 24, 36], [1, 5, 6, 10, 30, 31], [1, 5, 6, 14, 26, 33], [1, 5, 6, 18, 26, 31], [1, 5, 6, 19, 24, 32], [1, 5, 8, 14, 21, 36], [1, 5, 10, 12, 27, 32], [1, 5, 10, 18, 22, 33], [1, 5, 11, 12, 24, 34], [1, 5, 11, 16, 18, 36], [1, 5, 11, 20, 24, 30], [1, 5, 12, 12, 22, 35], [1, 5, 12, 13, 28, 30], [1, 5, 12, 14, 19, 36], [1, 5, 12, 16, 21, 34], [1, 5, 13, 24, 24, 26], [1, 5, 14, 15, 26, 30], [1, 5, 14, 21, 24, 28], [1, 5, 16, 18, 24, 29], [1, 5, 16, 20, 21, 30], [1, 5, 17, 18, 22, 30], [1, 5, 19, 22, 24, 24], [1, 6, 6, 7, 26, 35], [1, 6, 6, 10, 13, 41], [1, 6, 6, 10, 25, 35], [1, 6, 6, 13, 25, 34], [1, 6, 6, 14, 23, 35], [1, 6, 6, 22, 25, 29], [1, 6, 7, 8, 28, 33], [1, 6, 7, 9, 16, 40], [1, 6, 7, 13, 18, 38], [1, 6, 7, 14, 29, 30], [1, 6, 7, 19, 26, 30], [1, 6, 7, 20, 24, 31], [1, 6, 8, 11, 24, 35], [1, 6, 8, 13, 27, 32], [1, 6, 8, 16, 21, 35], [1, 6, 9, 10, 19, 38], [1, 6, 9, 14, 22, 35], [1, 6, 9, 16, 25, 32], [1, 6, 10, 11, 26, 33], [1, 6, 10, 13, 14, 39], [1, 6, 10, 14, 27, 31], [1, 6, 10, 17, 21, 34], [1, 6, 10, 19, 25, 30], [1, 6, 10, 21, 22, 31], [1, 6, 11, 11, 12, 40], [1, 6, 11, 13, 20, 36], [1, 6, 11, 14, 15, 38], [1, 6, 11, 15, 22, 34], [1, 6, 11, 17, 26, 30], [1, 6, 11, 20, 21, 32], [1, 6, 12, 16, 19, 35], [1, 6, 12, 16, 25, 31], [1, 6, 12, 17, 23, 32], [1, 6, 12, 23, 23, 28], [1, 6, 13, 20, 24, 29], [1, 6, 14, 14, 15, 37], [1, 6, 14, 18, 25, 29], [1, 6, 14, 19, 23, 30], [1, 6, 15, 20, 20, 31], [1, 6, 16, 23, 24, 25], [1, 6, 17, 22, 22, 27], [1, 6, 18, 19, 25, 26], [1, 6, 19, 20, 21, 28], [1, 7, 7, 12, 22, 36], [1, 7, 7, 18, 24, 32], [1, 7, 8, 8, 9, 42], [1, 7, 8, 8, 18, 39], [1, 7, 8, 12, 26, 33], [1, 7, 8, 15, 28, 30], [1, 7, 8, 17, 18, 36], [1, 7, 9, 14, 20, 36], [1, 7, 9, 18, 28, 28], [1, 7, 10, 18, 18, 35], [1, 7, 12, 12, 23, 34], [1, 7, 12, 16, 22, 33], [1, 7, 12, 20, 23, 30], [1, 7, 13, 18, 18, 34], [1, 7, 14, 15, 16, 36], [1, 7, 14, 24, 24, 25], [1, 7, 15, 18, 20, 32], [1, 7, 17, 18, 24, 28], [1, 7, 18, 18, 22, 29], [1, 8, 8, 15, 15, 38], [1, 8, 8, 18, 27, 29], [1, 8, 9, 9, 14, 40], [1, 8, 9, 10, 16, 39], [1, 8, 9, 12, 17, 38], [1, 8, 9, 18, 23, 32], [1, 8, 9, 24, 25, 26], [1, 8, 10, 11, 21, 36], [1, 8, 10, 12, 25, 33], [1, 8, 10, 20, 27, 27], [1, 8, 10, 21, 24, 29], [1, 8, 11, 12, 18, 37], [1, 8, 11, 18, 27, 28], [1, 8, 11, 19, 24, 30], [1, 8, 12, 14, 23, 33], [1, 8, 12, 17, 25, 30], [1, 8, 12, 18, 23, 31], [1, 8, 13, 13, 18, 36], [1, 8, 13, 18, 21, 32], [1, 8, 13, 22, 24, 27], [1, 8, 14, 15, 24, 31], [1, 8, 15, 15, 22, 32], [1, 8, 15, 18, 25, 28], [1, 8, 16, 17, 18, 33], [1, 8, 16, 19, 21, 30], [1, 8, 18, 23, 23, 24], [1, 8, 20, 21, 21, 26], [1, 9, 9, 20, 26, 28], [1, 9, 10, 10, 29, 30], [1, 9, 10, 16, 17, 36], [1, 9, 10, 18, 19, 34], [1, 9, 10, 18, 26, 29], [1, 9, 12, 17, 22, 32], [1, 9, 12, 22, 23, 28], [1, 9, 13, 14, 26, 30], [1, 9, 14, 14, 18, 35], [1, 9, 14, 16, 20, 33], [1, 9, 14, 19, 22, 30], [1, 9, 15, 16, 26, 28], [1, 9, 16, 16, 23, 30], [1, 9, 16, 18, 20, 31], [1, 9, 16, 22, 24, 25], [1, 9, 17, 20, 24, 26], [1, 10, 10, 15, 21, 34], [1, 10, 11, 12, 19, 36], [1, 10, 11, 15, 26, 30], [1, 10, 11, 21, 24, 28], [1, 10, 12, 15, 23, 32], [1, 10, 12, 17, 20, 33], [1, 10, 12, 19, 24, 29], [1, 10, 13, 18, 23, 30], [1, 10, 14, 14, 21, 33], [1, 10, 14, 18, 21, 31], [1, 10, 15, 22, 22, 27], [1, 10, 16, 19, 24, 27], [1, 10, 16, 21, 21, 28], [1, 11, 12, 12, 13, 38], [1, 11, 13, 16, 24, 30], [1, 11, 14, 15, 18, 34], [1, 11, 14, 20, 24, 27], [1, 11, 15, 18, 26, 26], [1, 11, 20, 21, 22, 24], [1, 12, 12, 13, 14, 37], [1, 12, 12, 17, 17, 34], [1, 12, 12, 17, 22, 31], [1, 12, 12, 22, 25, 25], [1, 12, 12, 23, 23, 26], [1, 12, 13, 14, 27, 28], [1, 12, 13, 18, 19, 32], [1, 12, 13, 21, 22, 28], [1, 12, 14, 20, 21, 29], [1, 12, 16, 19, 19, 30], [1, 12, 17, 17, 20, 30], [1, 12, 17, 22, 23, 24], [1, 12, 18, 20, 23, 25], [1, 12, 19, 20, 21, 26], [1, 13, 13, 18, 24, 28], [1, 13, 16, 16, 21, 30], [1, 13, 18, 18, 23, 26], [1, 13, 18, 20, 20, 27], [1, 14, 14, 15, 26, 27], [1, 14, 14, 17, 21, 30], [1, 14, 15, 16, 16, 33], [1, 14, 15, 20, 24, 25], [1, 14, 15, 21, 22, 26], [1, 14, 16, 20, 21, 27], [1, 14, 17, 18, 22, 27], [1, 15, 16, 16, 18, 31], [1, 15, 16, 17, 24, 26], [1, 15, 17, 18, 20, 28], [1, 16, 20, 21, 21, 22], [1, 17, 18, 21, 22, 22], [1, 18, 18, 19, 22, 23], [1, 18, 19, 19, 20, 24], [2, 2, 2, 7, 21, 39], [2, 2, 2, 9, 9, 43], [2, 2, 2, 9, 29, 33], [2, 2, 2, 21, 27, 29], [2, 2, 3, 3, 29, 34], [2, 2, 3, 6, 11, 43], [2, 2, 3, 6, 17, 41], [2, 2, 3, 10, 15, 41], [2, 2, 3, 11, 11, 42], [2, 2, 3, 11, 21, 38], [2, 2, 3, 11, 27, 34], [2, 2, 3, 14, 17, 39], [2, 2, 3, 14, 21, 37], [2, 2, 3, 15, 25, 34], [2, 2, 3, 18, 29, 29], [2, 2, 5, 6, 27, 35], [2, 2, 5, 15, 26, 33], [2, 2, 5, 18, 21, 35], [2, 2, 5, 19, 27, 30], [2, 2, 6, 7, 9, 43], [2, 2, 6, 7, 29, 33], [2, 2, 6, 9, 23, 37], [2, 2, 6, 13, 17, 39], [2, 2, 6, 13, 21, 37], [2, 2, 6, 15, 23, 35], [2, 2, 6, 17, 27, 31], [2, 2, 6, 19, 23, 33], [2, 2, 6, 25, 25, 27], [2, 2, 7, 9, 11, 42], [2, 2, 7, 9, 21, 38], [2, 2, 7, 9, 27, 34], [2, 2, 7, 11, 18, 39], [2, 2, 7, 15, 29, 30], [2, 2, 7, 21, 25, 30], [2, 2, 9, 9, 22, 37], [2, 2, 9, 13, 26, 33], [2, 2, 9, 15, 22, 35], [2, 2, 9, 19, 22, 33], [2, 2, 9, 23, 26, 27], [2, 2, 10, 13, 15, 39], [2, 2, 10, 15, 27, 31], [2, 2, 11, 15, 15, 38], [2, 2, 11, 18, 27, 29], [2, 2, 13, 21, 26, 27], [2, 2, 14, 15, 15, 37], [2, 2, 14, 17, 21, 33], [2, 2, 14, 19, 27, 27], [2, 2, 15, 18, 25, 29], [2, 2, 15, 19, 23, 30], [2, 2, 17, 18, 21, 31], [2, 2, 18, 21, 25, 25], [2, 2, 19, 21, 22, 27], [2, 3, 3, 4, 7, 44], [2, 3, 3, 4, 31, 32], [2, 3, 3, 8, 16, 41], [2, 3, 3, 10, 26, 35], [2, 3, 3, 13, 26, 34], [2, 3, 3, 14, 19, 38], [2, 3, 3, 16, 28, 31], [2, 3, 3, 19, 22, 34], [2, 3, 3, 22, 26, 29], [2, 3, 4, 7, 24, 37], [2, 3, 4, 8, 9, 43], [2, 3, 4, 8, 29, 33], [2, 3, 4, 11, 28, 33], [2, 3, 4, 12, 13, 41], [2, 3, 4, 12, 25, 35], [2, 3, 4, 13, 15, 40], [2, 3, 4, 13, 23, 36], [2, 3, 4, 15, 20, 37], [2, 3, 4, 21, 23, 32], [2, 3, 5, 5, 14, 42], [2, 3, 5, 6, 10, 43], [2, 3, 5, 8, 20, 39], [2, 3, 5, 8, 25, 36], [2, 3, 5, 10, 11, 42], [2, 3, 5, 10, 21, 38], [2, 3, 5, 10, 27, 34], [2, 3, 5, 17, 20, 36], [2, 3, 5, 24, 25, 28], [2, 3, 6, 11, 22, 37], [2, 3, 6, 13, 19, 38], [2, 3, 6, 17, 23, 34], [2, 3, 6, 22, 23, 31], [2, 3, 7, 10, 30, 31], [2, 3, 7, 14, 26, 33], [2, 3, 7, 18, 26, 31], [2, 3, 7, 19, 24, 32], [2, 3, 8, 8, 19, 39], [2, 3, 8, 9, 29, 32], [2, 3, 8, 11, 12, 41], [2, 3, 8, 11, 15, 40], [2, 3, 8, 11, 23, 36], [2, 3, 8, 12, 29, 31], [2, 3, 8, 13, 16, 39], [2, 3, 8, 16, 27, 31], [2, 3, 8, 17, 19, 36], [2, 3, 8, 23, 24, 29], [2, 3, 9, 11, 28, 32], [2, 3, 9, 14, 17, 38], [2, 3, 9, 17, 22, 34], [2, 3, 9, 19, 28, 28], [2, 3, 9, 22, 22, 31], [2, 3, 10, 10, 17, 39], [2, 3, 10, 10, 21, 37], [2, 3, 10, 13, 29, 30], [2, 3, 10, 14, 25, 33], [2, 3, 10, 15, 23, 34], [2, 3, 10, 18, 19, 35], [2, 3, 10, 18, 25, 31], [2, 3, 11, 11, 18, 38], [2, 3, 11, 12, 28, 31], [2, 3, 11, 14, 18, 37], [2, 3, 11, 17, 24, 32], [2, 3, 11, 20, 20, 33], [2, 3, 11, 22, 26, 27], [2, 3, 11, 23, 24, 28], [2, 3, 12, 20, 25, 29], [2, 3, 13, 16, 17, 36], [2, 3, 13, 18, 19, 34], [2, 3, 13, 18, 26, 29], [2, 3, 14, 14, 23, 33], [2, 3, 14, 17, 25, 30], [2, 3, 14, 18, 23, 31], [2, 3, 15, 19, 20, 32], [2, 3, 15, 22, 25, 26], [2, 3, 16, 17, 21, 32], [2, 3, 16, 20, 25, 27], [2, 3, 16, 21, 23, 28], [2, 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