Bonne année 2026 - Propriétés du nombre 2026 - Happy New Year
Happy New Year - Bonne année presque carrée : 2026 = 452 + 1
2026 : quelques propriétés de cet entier naturel
Le site Math93.com vous souhaite une heureuse, chaleureuse et studieuse année 2026. Profitons-en pour revenir sur quelques caractéristiques de cet entier naturel (natural number) pair (even number).
Sommaire
3.1. Méthode pour déterminer tous les diviseurs d’un entier
3.2. Liste des diviseurs de 2026
3.3. Approche historique de la méthode
4. Années carrées et cubes : 2026 est-elle une année carrée ?
5. Quelques décompositions de 2026
5.1. Nombre de Harshad (ou de Niven)
5.3. Somme de nombres premiers
6. 2026 : nombre abondant ou déficient
7.2. Avec des coefficients binomiaux
7.3. Écrire 2026 avec tous les chiffres de 1 à 9 dans l’ordre
7.6. Somme des chiffres et racine numérique
8. 2026 et la table de multiplication de 1 à 9
9. Décomposition de 2026 en somme de cubes
10. Et carré d’une somme : le théorème de Nicomède
11. Décomposition de 2026 en sommes de carrés
1) Écriture du nombre 2026
Cet entier naturel s'écrit ainsi, en tenant compte de l'orthographe réformée par les recommandations de l'Académie française publiées en 1990 :
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- Français : deux mille vingt-six
- Anglais : two thousand twenty-six
- Espagnol : dos mil veintiséis
- Allemand : zweitausendsechsundzwanzig
- Italien : duemilaventisei
- Portugais : dois mil e vinte e seis
- Néerlandais : tweeduizendzesentwintig
- Russe : две тысячи двадцать шесть
- Chinois (simplifié) : 二零二六年
- Arabe : ألفان وستة وعشرون
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Autres écritures de 2026
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- Binaire (binary representation) : 11111101010
- Romain (Roman numerals) : MMXXVI
- Hexadécimal (hexadecimal representation) : 7EA
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Repère historique
Les chiffres romains se développent dans l'Antiquité romaine et restent très utilisés pendant le Moyen Âge en Europe. La numération de position en base 10, venue d'Inde et transmise par les savants arabes, finit par s'imposer dans les calculs (écriture plus efficace, algorithmes opératoires plus simples).
2) L'année 2026
Une année complète comprend généralement 52 semaines et 1 jour (ou 52 semaines et 2 jours pour une année bissextile).
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- 2026 n'est pas bissextile (leap year) : 2026 n'est pas divisible par 4.
- Il y aura donc exactement 52 semaines et 1 jour en 2026.
- Le 1er janvier 2026 est un jeudi, donc le 31 décembre 2026 est aussi un jeudi.
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Rappel — Années bissextiles
Une année est bissextile si elle est divisible par 4, sauf si c'est une année de siècle (divisible par 100), auquel cas elle doit être divisible par 400.
Règle visuelle des années bissextiles
Année └─ divisible par 400 ? ├─ oui → bissextile │ └─ non ↓ divisible par 100 ? ├─ oui → non bissextile │ └─ non ↓ divisible par 4 ? ├─ oui → bissextile │ └─ non → non bissextile
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- Il y aura trois vendredis 13 en 2026 : 13 février, 13 mars et 13 novembre.
- 2026 est une année à 52 dimanches.
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Origine du mot « bissextile »
Le mot « bissextile » vient du latin bis sextus, qui signifie « deux fois le sixième ». Dans le calendrier romain, le 24 février était appelé « sixième jour avant les calendes de mars ». Lors d’une année bissextile, ce jour était compté deux fois, d’où le nom.
Pourquoi dit-on « leap year » ? (approche historique)
En anglais, une année bissextile se dit leap year. Le mot leap signifie « saut » (et non « année sautée »). Cette expression fait référence au saut d’un jour supplémentaire dans le calendrier lors d’une année bissextile.
Une année « commune » (common year) compte 365 jours, soit \(52\) semaines \(+1\) jour : le 1er janvier de l’année suivante tombe alors un jour plus tard dans la semaine. Une année bissextile compte 366 jours, soit \(52\) semaines \(+2\) jours : le 1er janvier de l’année suivante tombe donc deux jours plus tard. Le calendrier effectue ainsi un véritable « saut » d’un jour.
Repère historique (1582) : la règle actuelle des années bissextiles est fixée par le calendrier grégorien, instauré en 1582 par le pape Grégoire XIII. Elle permet de corriger la dérive du calendrier julien et d’aligner l’année civile sur la durée réelle de l’année solaire (environ \(365{,}2422\) jours).
Traduction littérale : leap year = « année du saut », expression qui a donné en français le terme année bissextile, issu du latin bis sextus.
3) Diviseurs de 2 026 et nombres premiers
Définition — Diviseur (divisor)
Un entier d est un diviseur de n si et seulement s'il existe un entier k tel que n = d k.Les nombres premiers
Les nombres premiers (prime numbers) sont les entiers naturels supérieurs à 1 qui n’admettent que deux diviseurs positifs distincts : 1 et eux-mêmes. Ils constituent les « briques élémentaires » de l’arithmétique : tout entier naturel peut s’écrire de manière unique comme produit de nombres premiers.
Cette propriété fondamentale est connue depuis l’Antiquité et formalisée dans Les Éléments d’Euclide (IIIe siècle av. J.-C.). Le résultat associé, appelé aujourd’hui théorème fondamental de l’arithmétique (fundamental theorem of arithmetic), est l’un des piliers de la théorie des nombres.
Pour une présentation détaillée, historique et illustrée des nombres premiers, on pourra consulter la page dédiée : Les nombres premiers – Math93 .
Décomposition en facteurs premiers
Pour étudier les diviseurs d’un entier naturel, on commence par écrire sa décomposition en facteurs premiers (prime factorization).
Dans le cas de 2026, on obtient : \[ 2026 = 2 \times 1013, \] où 1013 est un nombre premier.
Diviseurs de 2026
Méthode générale
Une fois la décomposition en facteurs premiers connue, tous les diviseurs de l’entier s’obtiennent de manière systématique.
Chaque diviseur est obtenu en choisissant, pour chaque facteur premier, un exposant compris entre 0 et l’exposant maximal apparaissant dans la décomposition.
Dans le cas de 2026 :
- on choisit \(2^0 = 1\) ou \(2^1 = 2\) ;
- on choisit \(1013^0 = 1\) ou \(1013^1 = 1013\).
Chaque diviseur est alors le produit d’un choix possible pour chaque facteur premier.
Diviseurs de 2026
En combinant les choix précédents, on obtient exactement les quatre diviseurs positifs de 2026 : \[ 1,\quad 2,\quad 1013,\quad 2026. \]
Cas général
Plus généralement, si un entier naturel \(n\) admet la décomposition en facteurs premiers \[ n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}, \] alors tout diviseur de \(n\) s’écrit sous la forme \[ p_1^{\beta_1} p_2^{\beta_2} \cdots p_k^{\beta_k} \quad \text{avec} \quad 0 \le \beta_i \le \alpha_i. \]
Approche historique
La méthode consistant à déterminer tous les diviseurs d’un entier à partir de sa décomposition en facteurs premiers est connue depuis l’Antiquité. Elle apparaît implicitement dans les Éléments d’Euclide (IIIe siècle av. J.-C.).
Euclide y montre que tout entier peut s’écrire comme produit de nombres premiers et que cette écriture est essentiellement unique. Cette propriété est aujourd’hui appelée théorème fondamental de l’arithmétique (fundamental theorem of arithmetic).
À partir de cette idée, chaque diviseur est vu comme une combinaison possible des facteurs premiers, en faisant varier leurs exposants. Cette approche combinatoire se développe chez les mathématiciens arabes du Moyen Âge puis se formalise pleinement à l’époque moderne.
Elle marque le passage d’une arithmétique fondée sur le calcul direct à une arithmétique structurelle, où les nombres premiers jouent le rôle de briques élémentaires des entiers.
Nombre premier ?
Définition — Nombre Premier (prime nunber)
Un entier est dit premier n si et seulement il admet exactement 2 diviseurs. Ex. : 2, 3, 5, 7, 11, 13 ...
- Pour en savoir plus sur les nombres premiers.
- 2026 n'est pas un nombre premier : il admet plus de deux diviseurs.
- La prochaine année première est 2027.
L'indicatrice d’Euler (Euler Totient Function)
Définition — L'indicatrice d’Euler (Euler Totient Function)
L' indicatrice d’Euler (Euler totient function) d’un entier naturel \(n\), noté \(\varphi(n)\), est le nombre d’entiers compris entre 1 et \(n\) qui sont premiers avec \(n\).
La décomposition en facteurs premiers de 2026 est : \[ 2026 = 2 \times 1013. \] On obtient alors : \[ \varphi(2026) = 2026\left(1-\frac12\right)\left(1-\frac{1}{1013}\right) = 1012. \]
Approche historique
La fonction \(\varphi\) a été introduite par Leonhard Euler au XVIIIe siècle dans ses travaux sur l’arithmétique et les congruences. Elle joue un rôle central en théorie des nombres, notamment dans le théorème d’Euler, généralisation du petit théorème de Fermat.
4) Années carrées et cube ?
2026 est-elle une année carrée ?
Définition — Carré parfait (perfect square)
Un entier est un carré parfait s'il existe un entier \(n\) tel que \(N=n^2\).On a \(45^2=2025\) et \(46^2=2116\). Donc 2026 n'est pas un carré parfait. En revanche, 2026 est "presque carrée" : \[ 2026 = 45^2 + 1. \]
Propriété — Écart entre deux carrés consécutifs
\[ n^2 - (n-1)^2 = 2n-1. \]Anecdote
Cette identité explique pourquoi les écarts entre deux carrés consécutifs grandissent linéairement : \(1,3,5,7,\dots\). On retrouve là une première apparition naturelle des nombres impairs.
5) Quelques décompositions de 2026
5.1 Nombre de Harshad ou de Niven (Harshad / Niven number)
Définition
Un entier est un nombre de Harshad s'il est divisible par la somme de ses chiffres.Contrairement à 2024 et 2025, l'entier 2026 n'est pas un nombre Harshad ou de Niven, c'est à dire un nombre divisible par la somme de ses chiffres.
Somme des chiffres de 2026 : \(2+0+2+6=10\). Or 2026 n'est pas divisible par 10. Donc 2026 n'est pas un nombre de Harshad.
Pour en savoir plus sur ces nombres : nombres de Harshad ou de Niven
Exemples de nombres de Harshad ou de Niven
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- 2000, 2001, 2004, 2007, 2010, 2016, 2020, 2022, 2023, 2024, 2025, 2028, 2030, 2034, 2040, 2043, 2052, 2061, 2064, 2070, 2080, 2085, 2088, 2090, 2100 …
- Prochaines plages de quatre années de Harshad consécutives:
- [2022, 2023, 2024, 2025], [3030, 3031, 3032, 3033], [10307, 10308, 10309, 10310], …
- [510, 511, 512, 513], [1014, 1015, 1016, 1017]
- 2000, 2001, 2004, 2007, 2010, 2016, 2020, 2022, 2023, 2024, 2025, 2028, 2030, 2034, 2040, 2043, 2052, 2061, 2064, 2070, 2080, 2085, 2088, 2090, 2100 …
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5.2 2026 nombre poli (polite number)
Définition — Nombre poli
Un nombre poli ou escalier est un nombre qui peut s'écrire sous la forme de une ou plusieurs sommes de deux ou plusieurs nombres consécutifs. Le degré de politesse indique combien de fois un nombre est sommes de nombres consécutifs.Théorème — Sommes consécutives et diviseurs impairs
Le nombre de décompositions d'un entier en somme d'entiers consécutifs est égal au nombre de ses diviseurs impairs strictement supérieurs à 1.
Approche historique
Cette correspondance est un classique des problèmes d'arithmétique : elle permet de voir immédiatement pourquoi les puissances de 2 sont exactement les "entiers impolis" (aucun diviseur impair > 1).
Pour 2026, les diviseurs impairs sont \(1\) et \(1013\). Il n'y a donc qu'un seul diviseur impair strictement supérieur à 1 : 1013. Ainsi, 2026 admet une seule décomposition en somme d'entiers consécutifs 5son degré de politesse est 1) :
\[ 2026 = 505 + 506 + 507 + 508. \]
| Année | Décomposition en facteurs premiers | Diviseurs impairs (> 1) | Degré de politesse | Exemple de décomposition en somme consécutive |
|---|---|---|---|---|
| 2024 | \(2^3 \times 11 \times 23\) | 11 ; 23 ; 253 | 3 | \(2024 = 504 + 505 + 506 + 507 + 508\) |
| 2025 | \(3^4 \times 5^2\) | 15 diviseurs impairs | 14 | \(2025 = 674 + 675 + 676\) |
| 2026 | \(2 \times 1013\) | 1013 | 1 | \(2026 = 505 + 506 + 507 + 508\) |
| 2027 | \(2027\) (nombre premier) | Aucun diviseur impair autre que 1 | 0 | Pas de décomposition en sommes consécutives |
| 2028 | \(2^2 \times 3 \times 13\) | 3 ; 13 ; 39 | 3 | \(2028 = 1014 + 1015\) |
5.3 Somme de nombres premiers (Goldbach decomposition)
La conjecture de Goldbach
La conjecture de Goldbach est l'un des problèmes les plus célèbres de l'arithmétique, formulée en 1742 par le mathématicien Christian Goldbach. Elle stipule que tous les nombres pairs supérieurs ou égaux à 4 peuvent être écrits comme la somme de deux nombres premiers.
Enoncé de la conjecture
Pour tout entier \(n \ge 4\), \(n\) peut être exprimé comme la somme de deux nombres premiers : \[ n = p + q \quad \text{avec} \quad p, q \in \mathbb{P} \] où \(\mathbb{P}\) désigne l'ensemble des nombres premiers.
Exemples vérifiés de la conjecture
Voici quelques exemples de nombres pairs et leurs décompositions :
- 4 = 2 + 2
- 6 = 3 + 3
- 8 = 3 + 5
- 10 = 3 + 7
- 12 = 5 + 7
- 14 = 3 + 11
- 16 = 3 + 13
- 18 = 5 + 13
Vérifications numériques de la conjecture de Goldbach
La conjecture forte de Goldbach, énoncée en 1742, n’a toujours pas été démontrée de manière générale. Toutefois, elle a fait l’objet de vérifications informatiques massives.
À ce jour, tous les nombres pairs jusqu’à \(4 \times 10^{18}\) ont été testés et vérifiés comme étant somme de deux nombres premiers, sans qu’aucun contre-exemple n’ait été trouvé.
Ces résultats reposent sur des calculs parallèles de grande ampleur et ne constituent pas une preuve mathématique au sens strict, mais renforcent considérablement la plausibilité de la conjecture.
Source : R. Bisseling, Calculs parallèles pour la vérification de la conjecture de Goldbach binaire, Université Claude Bernard Lyon 1, vérification publiée et confirmée en 2014 (état toujours valide en 2025).
Comme 2026 est pair et \(2026\ge 4\), on peut exhiber des écritures en somme de deux nombres premiers :
\[ 2026 = 23 + 2003 = 29 + 1997 = 47 + 1979. \]
5.4 Nombre parfait (perfect number)
Définition — Nombre parfait
Un nombre parfait est égal à la somme de ses diviseurs propres (proper divisors).Les diviseurs propres de 2026 sont \(1,2,1013\). Leur somme vaut \(1+2+1013 = 1016\), ce qui n'est pas égal à 2026. Donc 2026 n'est pas parfait.
6) 2026 nombres abondant ou déficient
Définition — Abondant / déficient
Un entier est abondant (abundant) si la somme de ses diviseurs propres est > à lui-même ; déficient (deficient) si elle est < ; parfait (perfect) si égalité.
Somme des diviseurs propres de 2026 : \[ 1+2+1013=1016<2026. \] Donc 2026 est un nombre déficient.
7) Autres curiosités de 2026
7.1 Avec 20 et 26
\[ 2026 = 20\times 101 + 26. \]
7.2 Avec des coefficients binomiaux (binomial coefficients)
Rappel — Coefficient binomial
\[ \binom{n}{2} = \frac{n(n-1)}{2}. \]Deux décompositions élégantes : \[ 2026=\binom{5}{2}+\binom{64}{2}=\binom{35}{2}+\binom{54}{2}. \]
7.3 Ecrire 2026 avec tous les chiffres de 1 à 9 dans l'ordre
Une écriture "dans l'ordre" (uniquement avec \(+,-,\times\) et des parenthèses) :
\[ 2026 = (1+2+3) - 4\big(5-(6+7\times 8\times 9)\big). \]
7.4 Des formules magiques
Rappel — Partie entière (floor function)
La partie entière \(\lfloor x\rfloor\) est le plus grand entier inférieur ou égal à \(x\).Une formule simple : \[ \left\lfloor \pi\times 645 \right\rfloor = 2026. \]
7.5 Avec le PGCD (greatest common divisor)
Définition — PGCD
Le PGCD de deux entiers est leur plus grand diviseur commun.Comme \(2026=2\times 1013\), \[ \gcd(2026,1013)=1013 \quad \text{et}\quad \gcd(2026,2)=2. \]
8) 2026 et la table de multiplication de 1 à 9
Rappel — Somme 1 + 2 + ... + 9
\[ 1+2+\cdots+9 = 9\times\frac{1+9}{2}=45. \]On retrouve alors immédiatement : \[ 2026 = 45^2 + 1. \]
Interprétation : si l'on construit un carré \(45\times 45\) (2025 points), 2026 correspond à "un point de plus".
9) Décomposition de 2026 en somme de cubes
Et carré d'une somme, le théorème de Nicomède
Théorème de Nicomaque (Nicomachus' theorem)
\[ \sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2 = \left(\frac{n(n+1)}{2}\right)^2. \]Approche historique
L'identité est traditionnellement attribuée à Nicomaque de Gérase (Ier–IIe siècle). Elle illustre une idée récurrente de l'Antiquité : découvrir des régularités numériques par observation, puis les justifier par des raisonnements de plus en plus structurés.
On peut voir une annimation visuelle de la preuve de cette égalité :
Contrairement à 2025, l'année 2026 n'est pas un carré parfait, donc elle ne peut pas être exactement de la forme \(\left(\frac{n(n+1)}{2}\right)^2\).
10) Décomposition de 2026 en sommes de carrés
10.1 Décomposition en somme de 2 carrés
Théorème des deux carrés
Un entier naturel est somme de deux carrés si et seulement si, dans sa décomposition en facteurs premiers (prime factorization), tout facteur premier congru à \(3\) modulo \(4\) apparaît avec un exposant pair.
Anecdote historique
Le cas des nombres premiers \(p\equiv 1\pmod 4\) est classiquement attribué à Fermat. Les preuves modernes s'appuient sur les entiers de Gauss : l'idée profonde est que “somme de deux carrés” se comporte bien avec la multiplication.
Ici \(2026 = 2\times 1013\) et \(1013 \equiv 1 \pmod 4\), donc 2026 est somme de deux carrés :
\[ 2026 = 45^2 + 1^2. \]
Forme développée : \(2026 = 2025 + 1\).
10.2 En somme de 3 carrés
Théorème des trois carrés
Un entier naturel est somme de trois carrés si et seulement s'il n'est pas de la forme \(4^a(8b+7)\).
Approche historique
Ce critère est établi à la fin du XVIIIe siècle (Legendre), puis réapparaît naturellement dans les travaux de Gauss. L'obstruction \(8b+7\) est un motif classique : certains restes modulo 8 ne peuvent pas être des carrés.
Voici les 8 décompositions en 3 carrés non nuls (ordre non décroissant), avec la forme développée :
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- \(2026 = 1^2 + 27^2 + 36^2 = 1 + 729 + 1296\)
- \(2026 = 3^2 + 9^2 + 44^2 = 9 + 81 + 1936\)
- \(2026 = 8^2 + 21^2 + 39^2 = 64 + 441 + 1521\)
- \(2026 = 9^2 + 24^2 + 37^2 = 81 + 576 + 1369\)
- \(2026 = 12^2 + 19^2 + 39^2 = 144 + 361 + 1521\)
- \(2026 = 13^2 + 21^2 + 38^2 = 169 + 441 + 1444\)
- \(2026 = 14^2 + 29^2 + 33^2 = 196 + 841 + 1089\)
- \(2026 = 18^2 + 20^2 + 36^2 = 324 + 400 + 1296\)
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10.3 En somme de 4 carrés
Théorème des quatre carrés de Lagrange
Tout entier naturel peut s'écrire comme somme de quatre carrés (certains pouvant être nuls).
Anecdote historique
Lagrange publie une preuve en 1770 ; l'idée circule déjà dans les commentaires de Bachet sur Diophante. C'est un jalon majeur : il garantit une représentation universelle (4 carrés suffisent toujours).
Pour 2026, si l'on impose des carrés non nuls et que l'on classe les écritures par ordre non décroissant, on obtient exactement 69 décompositions.
Afficher la liste complète (forme non développée)
2026 = 1^2 + 4^2 + 28^2 + 35^2 2026 = 1^2 + 5^2 + 8^2 + 44^2 2026 = 1^2 + 5^2 + 20^2 + 40^2 2026 = 1^2 + 6^2 + 15^2 + 42^2 2026 = 1^2 + 6^2 + 30^2 + 33^2 2026 = 1^2 + 7^2 + 7^2 + 44^2 2026 = 1^2 + 7^2 + 23^2 + 39^2 2026 = 1^2 + 7^2 + 31^2 + 33^2 2026 = 1^2 + 8^2 + 12^2 + 43^2 2026 = 1^2 + 8^2 + 28^2 + 39^2 2026 = 1^2 + 9^2 + 12^2 + 42^2 2026 = 1^2 + 9^2 + 20^2 + 38^2 2026 = 1^2 + 10^2 + 13^2 + 42^2 2026 = 1^2 + 10^2 + 27^2 + 36^2 2026 = 1^2 + 11^2 + 22^2 + 40^2 2026 = 1^2 + 12^2 + 12^2 + 43^2 2026 = 1^2 + 12^2 + 18^2 + 39^2 2026 = 1^2 + 12^2 + 22^2 + 37^2 2026 = 1^2 + 13^2 + 21^2 + 39^2 2026 = 1^2 + 13^2 + 29^2 + 29^2 2026 = 1^2 + 14^2 + 17^2 + 40^2 2026 = 1^2 + 14^2 + 24^2 + 35^2 2026 = 1^2 + 15^2 + 17^2 + 39^2 2026 = 1^2 + 15^2 + 21^2 + 37^2 2026 = 1^2 + 15^2 + 29^2 + 31^2 2026 = 1^2 + 16^2 + 16^2 + 41^2 2026 = 1^2 + 16^2 + 23^2 + 36^2 2026 = 1^2 + 17^2 + 21^2 + 37^2 2026 = 1^2 + 18^2 + 18^2 + 39^2 2026 = 1^2 + 19^2 + 19^2 + 37^2 2026 = 1^2 + 19^2 + 23^2 + 35^2 2026 = 1^2 + 20^2 + 20^2 + 37^2 2026 = 1^2 + 21^2 + 22^2 + 36^2 2026 = 1^2 + 22^2 + 24^2 + 35^2 2026 = 1^2 + 23^2 + 23^2 + 35^2 2026 = 2^2 + 2^2 + 29^2 + 35^2 2026 = 2^2 + 3^2 + 18^2 + 41^2 2026 = 2^2 + 3^2 + 30^2 + 33^2 2026 = 2^2 + 4^2 + 10^2 + 44^2 2026 = 2^2 + 4^2 + 20^2 + 40^2 2026 = 2^2 + 4^2 + 30^2 + 32^2 2026 = 2^2 + 6^2 + 19^2 + 41^2 2026 = 2^2 + 6^2 + 21^2 + 39^2 2026 = 2^2 + 7^2 + 7^2 + 44^2 2026 = 2^2 + 7^2 + 25^2 + 38^2 2026 = 2^2 + 8^2 + 16^2 + 41^2 2026 = 2^2 + 8^2 + 18^2 + 39^2 2026 = 2^2 + 9^2 + 9^2 + 44^2 2026 = 2^2 + 9^2 + 21^2 + 38^2 2026 = 2^2 + 10^2 + 16^2 + 41^2 2026 = 2^2 + 10^2 + 24^2 + 36^2 2026 = 2^2 + 11^2 + 13^2 + 44^2 2026 = 2^2 + 12^2 + 17^2 + 41^2 2026 = 2^2 + 12^2 + 25^2 + 35^2 2026 = 2^2 + 13^2 + 19^2 + 40^2 2026 = 2^2 + 14^2 + 24^2 + 35^2 2026 = 2^2 + 15^2 + 21^2 + 38^2 2026 = 2^2 + 16^2 + 16^2 + 41^2 2026 = 2^2 + 16^2 + 20^2 + 38^2 2026 = 2^2 + 17^2 + 19^2 + 40^2 2026 = 2^2 + 18^2 + 20^2 + 38^2 2026 = 2^2 + 19^2 + 21^2 + 38^2 2026 = 3^2 + 4^2 + 7^2 + 44^2 2026 = 3^2 + 5^2 + 10^2 + 44^2 2026 = 3^2 + 7^2 + 21^2 + 39^2 2026 = 3^2 + 9^2 + 24^2 + 40^2 2026 = 3^2 + 10^2 + 16^2 + 41^2 2026 = 4^2 + 6^2 + 17^2 + 43^2 2026 = 5^2 + 10^2 + 16^2 + 41^2 2026 = 6^2 + 6^2 + 10^2 + 44^2 2026 = 7^2 + 12^2 + 13^2 + 44^2 2026 = 9^2 + 9^2 + 20^2 + 42^2 2026 = 12^2 + 12^2 + 18^2 + 40^2 2026 = 15^2 + 15^2 + 16^2 + 40^2 2026 = 19^2 + 19^2 + 20^2 + 36^2
10.4 En somme de 5 carrés
Remarque
En 5 carrés, le nombre de représentations explose : on obtient ici 495 décompositions (carrés non nuls, ordre non décroissant).
Afficher les 495 décompositions (forme non développée)
2026 ([[1, 1, 2, 16, 42], [1, 1, 2, 24, 38], [1, 1, 8, 14, 42], [1, 1, 10, 18, 40], [1, 1, 10, 30, 32], [1, 1, 16, 18, 38], [1, 1, 18, 26, 32], [1, 2, 2, 9, 44], [1, 2, 4, 18, 41], [1, 2, 4, 22, 39], [1, 2, 6, 7, 44], [1, 2, 6, 31, 32], [1, 2, 7, 26, 36], [1, 2, 9, 28, 34], [1, 2, 10, 20, 39], [1, 2, 10, 25, 36], [1, 2, 12, 14, 41], [1, 2, 14, 15, 40], [1, 2, 14, 23, 36], [1, 2, 16, 26, 33], [1, 2, 17, 24, 34], [1, 2, 22, 24, 31], [1, 3, 4, 8, 44], [1, 3, 4, 20, 40], [1, 3, 12, 24, 36], [1, 4, 4, 12, 43], [1, 4, 6, 23, 38], [1, 4, 7, 14, 42], [1, 4, 8, 24, 37], [1, 4, 9, 22, 38], [1, 4, 12, 29, 32], [1, 4, 16, 27, 32], [1, 4, 18, 23, 34], [1, 4, 21, 28, 28], [1, 4, 22, 25, 30], [1, 5, 12, 16, 40], [1, 5, 20, 24, 32], [1, 6, 7, 28, 34], [1, 6, 9, 12, 42], [1, 6, 10, 17, 40], [1, 6, 12, 18, 39], [1, 6, 16, 17, 38], [1, 6, 17, 26, 32], [1, 6, 18, 24, 33], [1, 6, 23, 26, 28], [1, 7, 12, 26, 34], [1, 7, 14, 22, 36], [1, 7, 20, 26, 30], [1, 8, 10, 30, 31], [1, 8, 14, 26, 33], [1, 8, 18, 26, 31], [1, 8, 19, 24, 32], [1, 9, 10, 20, 38], [1, 9, 12, 30, 30], [1, 9, 18, 18, 36], [1, 9, 22, 26, 28], [1, 10, 10, 12, 41], [1, 10, 10, 15, 40], [1, 10, 10, 23, 36], [1, 10, 12, 25, 34], [1, 10, 15, 16, 38], [1, 10, 15, 26, 32], [1, 10, 20, 25, 30], [1, 12, 12, 21, 36], [1, 12, 14, 23, 34], [1, 12, 16, 16, 37], [1, 12, 16, 20, 35], [1, 12, 16, 28, 29], [1, 12, 23, 26, 26], [1, 12, 24, 24, 27], [1, 13, 16, 24, 32], [1, 14, 16, 22, 33], [1, 14, 20, 23, 30], [1, 15, 18, 24, 30], [1, 16, 16, 27, 28], [1, 16, 17, 18, 34], [1, 16, 18, 22, 31], [1, 17, 22, 24, 26], [1, 18, 20, 25, 26], [1, 20, 20, 21, 28], [2, 2, 3, 28, 35], [2, 2, 5, 12, 43], [2, 2, 8, 27, 35], [2, 2, 9, 16, 41], [2, 2, 19, 19, 36], [2, 2, 20, 23, 33], [2, 3, 4, 29, 34], [2, 3, 8, 10, 43], [2, 3, 13, 20, 38], [2, 4, 6, 11, 43], [2, 4, 6, 17, 41], [2, 4, 10, 15, 41], [2, 4, 11, 11, 42], [2, 4, 11, 21, 38], [2, 4, 11, 27, 34], [2, 4, 14, 17, 39], [2, 4, 14, 21, 37], [2, 4, 15, 25, 34], [2, 4, 18, 29, 29], [2, 5, 5, 6, 44], [2, 5, 5, 26, 36], [2, 5, 6, 19, 40], [2, 5, 8, 13, 42], [2, 5, 12, 22, 37], [2, 5, 14, 24, 35], [2, 5, 16, 29, 30], [2, 5, 20, 21, 34], [2, 5, 22, 27, 28], [2, 6, 7, 16, 41], [2, 6, 8, 31, 31], [2, 6, 11, 29, 32], [2, 6, 16, 19, 37], [2, 6, 19, 20, 35], [2, 6, 19, 28, 29], [2, 6, 20, 25, 31], [2, 7, 7, 18, 40], [2, 7, 7, 30, 32], [2, 7, 10, 28, 33], [2, 7, 14, 16, 39], [2, 7, 17, 28, 30], [2, 7, 18, 25, 32], [2, 7, 20, 22, 33], [2, 8, 9, 14, 41], [2, 8, 15, 17, 38], [2, 8, 19, 21, 34], [2, 8, 21, 26, 29], [2, 8, 23, 23, 30], [2, 9, 14, 28, 31], [2, 9, 16, 23, 34], [2, 10, 11, 24, 35], [2, 10, 13, 27, 32], [2, 10, 16, 21, 35], [2, 11, 11, 22, 36], [2, 11, 13, 24, 34], [2, 11, 21, 26, 28], [2, 11, 22, 24, 29], [2, 12, 13, 22, 35], [2, 12, 14, 29, 29], [2, 12, 19, 19, 34], [2, 12, 19, 26, 29], [2, 13, 13, 28, 30], [2, 13, 14, 19, 36], [2, 13, 16, 21, 34], [2, 14, 16, 27, 29], [2, 14, 17, 24, 31], [2, 14, 19, 21, 32], [2, 14, 24, 25, 25], [2, 15, 17, 22, 32], [2, 15, 22, 23, 28], [2, 16, 19, 26, 27], [2, 16, 21, 22, 29], [2, 17, 18, 25, 28], [2, 19, 19, 20, 30], [3, 3, 6, 6, 44], [3, 3, 6, 26, 36], [3, 3, 10, 12, 42], [3, 3, 18, 28, 30], [3, 4, 4, 7, 44], [3, 4, 4, 31, 32], [3, 4, 8, 16, 41], [3, 4, 10, 26, 35], [3, 4, 13, 26, 34], [3, 4, 14, 19, 38], [3, 4, 16, 28, 31], [3, 4, 19, 22, 34], [3, 4, 22, 26, 29], [3, 5, 8, 22, 38], [3, 5, 14, 14, 40], [3, 5, 22, 22, 32], [3, 6, 6, 24, 37], [3, 6, 18, 19, 36], [3, 6, 24, 26, 27], [3, 7, 20, 28, 28], [3, 8, 8, 17, 40], [3, 8, 9, 24, 36], [3, 8, 10, 22, 37], [3, 8, 11, 26, 34], [3, 8, 18, 27, 30], [3, 8, 20, 23, 32], [3, 9, 24, 24, 28], [3, 10, 11, 14, 40], [3, 10, 19, 20, 34], [3, 10, 20, 26, 29], [3, 10, 21, 24, 30], [3, 11, 14, 16, 38], [3, 11, 14, 26, 32], [3, 11, 16, 22, 34], [3, 12, 18, 18, 35], [3, 14, 14, 16, 37], [3, 14, 14, 20, 35], [3, 14, 14, 28, 29], [3, 14, 19, 26, 28], [3, 16, 20, 20, 31], [3, 17, 24, 24, 24], [3, 18, 21, 24, 26], [4, 4, 7, 24, 37], [4, 4, 8, 9, 43], [4, 4, 8, 29, 33], [4, 4, 11, 28, 33], [4, 4, 12, 13, 41], [4, 4, 12, 25, 35], [4, 4, 13, 15, 40], [4, 4, 13, 23, 36], [4, 4, 15, 20, 37], [4, 4, 21, 23, 32], [4, 5, 5, 14, 42], [4, 5, 6, 10, 43], [4, 5, 8, 20, 39], [4, 5, 8, 25, 36], [4, 5, 10, 11, 42], [4, 5, 10, 21, 38], [4, 5, 10, 27, 34], [4, 5, 17, 20, 36], [4, 5, 24, 25, 28], [4, 6, 11, 22, 37], [4, 6, 13, 19, 38], [4, 6, 17, 23, 34], [4, 6, 22, 23, 31], [4, 7, 10, 30, 31], [4, 7, 14, 26, 33], [4, 7, 18, 26, 31], [4, 7, 19, 24, 32], [4, 8, 8, 19, 39], [4, 8, 9, 29, 32], [4, 8, 11, 12, 41], [4, 8, 11, 15, 40], [4, 8, 11, 23, 36], [4, 8, 12, 29, 31], [4, 8, 13, 16, 39], [4, 8, 16, 27, 31], [4, 8, 17, 19, 36], [4, 8, 23, 24, 29], [4, 9, 11, 28, 32], [4, 9, 14, 17, 38], [4, 9, 17, 22, 34], [4, 9, 19, 28, 28], [4, 9, 22, 22, 31], [4, 10, 10, 17, 39], [4, 10, 10, 21, 37], [4, 10, 13, 29, 30], [4, 10, 14, 25, 33], [4, 10, 15, 23, 34], [4, 10, 18, 19, 35], [4, 10, 18, 25, 31], [4, 11, 11, 18, 38], [4, 11, 12, 28, 31], [4, 11, 14, 18, 37], [4, 11, 17, 24, 32], [4, 11, 20, 20, 33], [4, 11, 22, 26, 27], [4, 11, 23, 24, 28], [4, 12, 20, 25, 29], [4, 13, 16, 17, 36], [4, 13, 18, 19, 34], [4, 13, 18, 26, 29], [4, 14, 14, 23, 33], [4, 14, 17, 25, 30], [4, 14, 18, 23, 31], [4, 15, 19, 20, 32], [4, 15, 22, 25, 26], [4, 16, 17, 21, 32], [4, 16, 20, 25, 27], [4, 16, 21, 23, 28], [4, 17, 19, 24, 28], [4, 18, 19, 22, 29], [5, 5, 6, 28, 34], [5, 5, 12, 26, 34], [5, 5, 14, 22, 36], [5, 5, 20, 26, 30], [5, 6, 8, 26, 35], [5, 6, 10, 29, 32], [5, 6, 11, 20, 38], [5, 6, 13, 14, 40], [5, 6, 14, 20, 37], [5, 6, 16, 22, 35], [5, 7, 12, 28, 32], [5, 7, 16, 20, 36], [5, 8, 8, 28, 33], [5, 8, 9, 16, 40], [5, 8, 13, 18, 38], [5, 8, 14, 29, 30], [5, 8, 19, 26, 30], [5, 8, 20, 24, 31], [5, 10, 10, 24, 35], [5, 10, 11, 22, 36], [5, 10, 13, 24, 34], [5, 10, 21, 26, 28], [5, 10, 22, 24, 29], [5, 11, 14, 28, 30], [5, 11, 18, 20, 34], [5, 12, 17, 28, 28], [5, 13, 16, 26, 30], [5, 13, 18, 22, 32], [5, 14, 16, 18, 35], [5, 14, 20, 26, 27], [5, 16, 16, 20, 33], [5, 16, 19, 22, 30], [5, 20, 20, 24, 25], [5, 20, 21, 22, 26], [6, 6, 9, 28, 33], [6, 6, 12, 17, 39], [6, 6, 12, 21, 37], [6, 6, 17, 24, 33], [6, 6, 21, 27, 28], [6, 7, 8, 14, 41], [6, 7, 14, 28, 31], [6, 7, 16, 23, 34], [6, 8, 9, 9, 42], [6, 8, 9, 18, 39], [6, 8, 11, 19, 38], [6, 8, 14, 19, 37], [6, 8, 17, 26, 31], [6, 8, 25, 25, 26], [6, 9, 12, 26, 33], [6, 9, 15, 28, 30], [6, 9, 17, 18, 36], [6, 10, 11, 13, 40], [6, 10, 11, 20, 37], [6, 10, 12, 15, 39], [6, 10, 15, 24, 33], [6, 10, 20, 23, 31], [6, 11, 13, 16, 38], [6, 11, 13, 26, 32], [6, 11, 19, 22, 32], [6, 12, 21, 26, 27], [6, 13, 14, 16, 37], [6, 13, 14, 20, 35], [6, 13, 14, 28, 29], [6, 13, 19, 26, 28], [6, 15, 17, 24, 30], [6, 16, 17, 17, 34], [6, 16, 17, 22, 31], [6, 16, 22, 25, 25], [6, 16, 23, 23, 26], [6, 17, 20, 25, 26], [6, 18, 19, 24, 27], [6, 18, 21, 21, 28], [6, 19, 19, 22, 28], [7, 7, 8, 10, 42], [7, 7, 14, 24, 34], [7, 7, 24, 26, 26], [7, 8, 9, 26, 34], [7, 8, 12, 13, 40], [7, 8, 12, 20, 37], [7, 8, 14, 14, 39], [7, 8, 16, 19, 36], [7, 8, 20, 27, 28], [7, 8, 22, 23, 30], [7, 9, 10, 14, 40], [7, 9, 14, 16, 38], [7, 9, 14, 26, 32], [7, 9, 16, 22, 34], [7, 10, 10, 16, 39], [7, 10, 12, 17, 38], [7, 10, 18, 23, 32], [7, 10, 24, 25, 26], [7, 11, 16, 24, 32], [7, 12, 14, 26, 31], [7, 13, 16, 16, 36], [7, 14, 14, 17, 36], [7, 14, 15, 20, 34], [7, 14, 16, 25, 30], [7, 14, 23, 24, 26], [7, 15, 20, 26, 26], [7, 15, 22, 22, 28], [7, 16, 16, 21, 32], [7, 16, 19, 24, 28], [8, 8, 11, 16, 39], [8, 8, 12, 23, 35], [8, 8, 19, 24, 31], [8, 9, 9, 30, 30], [8, 9, 10, 10, 41], [8, 9, 10, 25, 34], [8, 9, 12, 21, 36], [8, 9, 14, 23, 34], [8, 9, 16, 16, 37], [8, 9, 16, 20, 35], [8, 9, 16, 28, 29], [8, 9, 23, 26, 26], [8, 9, 24, 24, 27], [8, 10, 11, 29, 30], [8, 10, 13, 18, 37], [8, 10, 14, 21, 35], [8, 10, 15, 26, 31], [8, 10, 17, 22, 33], [8, 11, 16, 17, 36], [8, 11, 18, 19, 34], [8, 11, 18, 26, 29], [8, 12, 13, 25, 32], [8, 13, 14, 21, 34], [8, 13, 15, 28, 28], [8, 13, 16, 24, 31], [8, 13, 21, 26, 26], [8, 14, 14, 27, 29], [8, 14, 19, 26, 27], [8, 14, 21, 22, 29], [8, 15, 16, 16, 35], [8, 15, 18, 18, 33], [8, 16, 16, 19, 33], [8, 16, 17, 24, 29], [8, 17, 17, 22, 30], [8, 18, 22, 23, 25], [8, 19, 20, 24, 25], [8, 19, 21, 22, 26], [9, 10, 10, 28, 31], [9, 10, 14, 25, 32], [9, 10, 15, 18, 36], [9, 10, 17, 20, 34], [9, 10, 20, 22, 31], [9, 11, 16, 28, 28], [9, 11, 20, 20, 32], [9, 12, 12, 19, 36], [9, 12, 15, 26, 30], [9, 12, 21, 24, 28], [9, 14, 14, 23, 32], [9, 14, 17, 26, 28], [9, 16, 22, 23, 26], [9, 19, 20, 20, 28], [10, 10, 12, 29, 29], [10, 10, 13, 19, 36], [10, 10, 16, 27, 29], [10, 10, 17, 24, 31], [10, 10, 19, 21, 32], [10, 10, 24, 25, 25], [10, 11, 11, 28, 30], [10, 11, 16, 18, 35], [10, 11, 20, 26, 27], [10, 12, 13, 13, 38], [10, 12, 14, 19, 35], [10, 12, 14, 25, 31], [10, 12, 18, 27, 27], [10, 12, 21, 21, 30], [10, 14, 23, 24, 25], [10, 15, 15, 24, 30], [10, 15, 16, 17, 34], [10, 15, 16, 22, 31], [10, 15, 20, 25, 26], [10, 17, 17, 18, 32], [10, 17, 18, 23, 28], [10, 18, 19, 20, 29], [11, 11, 12, 14, 38], [11, 11, 12, 22, 34], [11, 11, 18, 26, 28], [11, 11, 20, 22, 30], [11, 12, 14, 14, 37], [11, 12, 20, 20, 31], [11, 13, 16, 18, 34], [11, 13, 22, 24, 26], [11, 14, 14, 27, 28], [11, 14, 18, 19, 32], [11, 14, 21, 22, 28], [11, 15, 16, 20, 32], [11, 16, 17, 24, 28], [11, 16, 18, 22, 29], [11, 19, 22, 22, 24], [11, 20, 20, 23, 24], [12, 12, 12, 15, 37], [12, 12, 15, 27, 28], [12, 13, 14, 19, 34], [12, 13, 14, 26, 29], [12, 13, 17, 20, 32], [12, 13, 19, 26, 26], [12, 13, 20, 23, 28], [12, 14, 14, 23, 31], [12, 14, 19, 22, 29], [12, 16, 16, 23, 29], [12, 17, 21, 24, 24], [12, 17, 22, 22, 25], [12, 18, 21, 21, 26], [12, 19, 19, 22, 26], [13, 13, 14, 14, 36], [13, 14, 14, 21, 32], [13, 14, 16, 26, 27], [13, 14, 19, 20, 30], [13, 16, 16, 16, 33], [13, 16, 20, 24, 25], [13, 16, 21, 22, 26], [14, 14, 15, 25, 28], [14, 14, 16, 17, 33], [14, 14, 23, 23, 24], [14, 15, 20, 23, 26], [14, 16, 17, 18, 31], [14, 16, 18, 25, 25], [14, 16, 19, 22, 27], [14, 17, 17, 24, 26], [14, 18, 19, 19, 28], [15, 15, 18, 24, 26], [15, 16, 19, 20, 28], [16, 16, 16, 23, 27], [16, 16, 17, 21, 28], [16, 19, 21, 22, 22], [16, 20, 20, 21, 23], [17, 18, 20, 22, 23], [17, 19, 20, 20, 24], [18, 18, 19, 21, 24]], 495)
10.5 En somme de 6 carrés
Remarque
En 6 carrés, on obtient 1 554 décompositions (carrés non nuls, ordre non décroissant).
Afficher les 1 554 décompositions (forme non développée)
2026 ([[1, 1, 2, 18, 20, 36], [1, 1, 4, 6, 6, 44], [1, 1, 4, 6, 26, 36], [1, 1, 4, 10, 12, 42], [1, 1, 4, 18, 28, 30], [1, 1, 6, 8, 18, 40], [1, 1, 6, 8, 30, 32], [1, 1, 6, 12, 20, 38], [1, 1, 6, 16, 24, 34], [1, 1, 8, 22, 24, 30], [1, 1, 10, 12, 22, 36], [1, 1, 10, 18, 24, 32], [1, 1, 12, 14, 28, 30], [1, 1, 12, 18, 20, 34], [1, 1, 14, 24, 24, 26], [1, 1, 18, 20, 20, 30], [1, 2, 2, 12, 28, 33], [1, 2, 2, 18, 18, 37], [1, 2, 2, 21, 26, 30], [1, 2, 4, 4, 15, 42], [1, 2, 4, 4, 30, 33], [1, 2, 4, 9, 18, 40], [1, 2, 4, 9, 30, 32], [1, 2, 4, 12, 30, 31], [1, 2, 4, 15, 22, 36], [1, 2, 4, 23, 24, 30], [1, 2, 5, 6, 14, 42], [1, 2, 5, 14, 30, 30], [1, 2, 6, 6, 10, 43], [1, 2, 6, 8, 20, 39], [1, 2, 6, 8, 25, 36], [1, 2, 6, 10, 11, 42], [1, 2, 6, 10, 21, 38], [1, 2, 6, 10, 27, 34], [1, 2, 6, 17, 20, 36], [1, 2, 6, 24, 25, 28], [1, 2, 7, 8, 12, 42], [1, 2, 7, 10, 24, 36], [1, 2, 8, 15, 24, 34], [1, 2, 9, 12, 14, 40], [1, 2, 9, 16, 28, 30], [1, 2, 10, 11, 30, 30], [1, 2, 10, 12, 16, 39], [1, 2, 10, 15, 20, 36], [1, 2, 10, 16, 24, 33], [1, 2, 10, 18, 21, 34], [1, 2, 11, 18, 26, 30], [1, 2, 12, 12, 17, 38], [1, 2, 12, 18, 23, 32], [1, 2, 12, 24, 25, 26], [1, 2, 14, 14, 27, 30], [1, 2, 14, 15, 24, 32], [1, 2, 14, 21, 22, 30], [1, 2, 16, 17, 24, 30], [1, 2, 18, 22, 22, 27], [1, 3, 4, 12, 16, 40], [1, 3, 4, 20, 24, 32], [1, 3, 8, 12, 28, 32], [1, 3, 8, 16, 20, 36], [1, 3, 16, 20, 24, 28], [1, 4, 4, 11, 24, 36], [1, 4, 4, 15, 18, 38], [1, 4, 4, 16, 21, 36], [1, 4, 4, 24, 24, 29], [1, 4, 6, 6, 16, 41], [1, 4, 6, 7, 18, 40], [1, 4, 6, 7, 30, 32], [1, 4, 6, 10, 28, 33], [1, 4, 6, 14, 16, 39], [1, 4, 6, 17, 28, 30], [1, 4, 6, 18, 25, 32], [1, 4, 6, 20, 22, 33], [1, 4, 7, 22, 24, 30], [1, 4, 8, 9, 10, 42], [1, 4, 8, 10, 18, 39], [1, 4, 8, 12, 24, 35], [1, 4, 9, 14, 24, 34], [1, 4, 9, 24, 26, 26], [1, 4, 10, 12, 26, 33], [1, 4, 10, 15, 28, 30], [1, 4, 10, 17, 18, 36], [1, 4, 11, 12, 12, 40], [1, 4, 12, 13, 20, 36], [1, 4, 12, 14, 15, 38], [1, 4, 12, 15, 22, 34], [1, 4, 12, 17, 26, 30], [1, 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[6, 12, 13, 14, 16, 35], [6, 12, 14, 17, 20, 31], [6, 12, 14, 20, 25, 25], [6, 12, 16, 17, 25, 26], [6, 12, 17, 17, 22, 28], [6, 13, 13, 20, 24, 26], [6, 13, 14, 14, 23, 30], [6, 13, 14, 18, 25, 26], [6, 13, 14, 20, 21, 28], [6, 13, 16, 18, 20, 29], [6, 13, 18, 22, 22, 23], [6, 13, 19, 20, 22, 24], [6, 14, 14, 14, 21, 31], [6, 14, 15, 16, 17, 32], [6, 14, 15, 16, 23, 28], [6, 14, 16, 16, 21, 29], [6, 14, 17, 20, 23, 24], [6, 14, 18, 19, 22, 25], [6, 15, 16, 20, 22, 25], [6, 15, 17, 18, 24, 24], [6, 15, 17, 20, 20, 26], [6, 15, 18, 18, 21, 26], [6, 16, 16, 17, 17, 30], [6, 16, 16, 18, 23, 25], [6, 16, 16, 19, 21, 26], [6, 17, 18, 18, 18, 27], [7, 7, 8, 8, 30, 30], [7, 7, 10, 12, 28, 30], [7, 7, 10, 24, 24, 26], [7, 7, 12, 12, 14, 38], [7, 7, 12, 12, 22, 34], [7, 7, 12, 18, 26, 28], [7, 7, 12, 20, 22, 30], [7, 7, 14, 16, 24, 30], [7, 7, 16, 18, 18, 32], [7, 8, 8, 9, 18, 38], [7, 8, 8, 18, 25, 30], [7, 8, 9, 10, 24, 34], [7, 8, 9, 16, 26, 30], [7, 8, 9, 18, 22, 32], [7, 8, 10, 12, 15, 38], [7, 8, 10, 18, 20, 33], [7, 8, 12, 12, 16, 37], [7, 8, 12, 12, 20, 35], [7, 8, 12, 12, 28, 29], [7, 8, 12, 13, 24, 32], [7, 8, 12, 14, 22, 33], [7, 8, 12, 16, 27, 28], [7, 8, 12, 17, 18, 34], [7, 8, 12, 18, 22, 31], [7, 8, 14, 14, 15, 36], [7, 8, 17, 18, 20, 30], [7, 8, 18, 22, 23, 24], [7, 8, 19, 20, 24, 24], [7, 9, 10, 10, 20, 36], [7, 9, 10, 14, 24, 32], [7, 9, 12, 14, 20, 34], [7, 9, 12, 20, 26, 26], [7, 9, 12, 22, 22, 28], [7, 9, 14, 20, 20, 30], [7, 9, 16, 16, 22, 30], [7, 10, 10, 15, 16, 36], [7, 10, 10, 24, 24, 25], [7, 10, 12, 14, 24, 31], [7, 10, 12, 15, 22, 32], [7, 10, 12, 18, 25, 28], [7, 10, 13, 18, 22, 30], [7, 10, 14, 23, 24, 24], [7, 10, 15, 20, 24, 26], [7, 11, 12, 12, 28, 28], [7, 12, 12, 22, 23, 26], [7, 12, 14, 15, 16, 34], [7, 12, 14, 17, 18, 32], [7, 12, 14, 18, 23, 28], [7, 12, 15, 16, 26, 26], [7, 12, 17, 22, 22, 24], [7, 12, 18, 20, 22, 25], [7, 13, 16, 20, 24, 24], [7, 13, 18, 18, 22, 26], [7, 14, 14, 15, 24, 28], [7, 14, 14, 18, 19, 30], [7, 14, 15, 16, 20, 30], [7, 14, 16, 18, 24, 25], [7, 16, 17, 18, 18, 28], [7, 18, 18, 19, 22, 22], [7, 18, 18, 20, 20, 23], [8, 8, 8, 9, 27, 32], [8, 8, 8, 12, 13, 39], [8, 8, 8, 12, 27, 31], [8, 8, 8, 13, 24, 33], [8, 8, 8, 23, 24, 27], [8, 8, 9, 10, 14, 39], [8, 8, 9, 11, 20, 36], [8, 8, 9, 20, 24, 29], [8, 8, 10, 13, 27, 30], [8, 8, 10, 15, 22, 33], [8, 8, 11, 15, 16, 36], [8, 8, 11, 24, 24, 25], [8, 8, 12, 13, 17, 36], [8, 8, 12, 17, 21, 32], [8, 8, 12, 20, 25, 27], [8, 8, 12, 21, 23, 28], [8, 8, 13, 18, 26, 27], [8, 8, 14, 17, 18, 33], [8, 8, 14, 19, 21, 30], [8, 8, 15, 16, 24, 29], [8, 8, 15, 17, 22, 30], [8, 8, 16, 21, 24, 25], [8, 8, 17, 18, 18, 31], [8, 8, 18, 18, 25, 25], [8, 8, 18, 19, 22, 27], [8, 9, 9, 10, 10, 40], [8, 9, 9, 10, 16, 38], [8, 9, 9, 10, 26, 32], [8, 9, 9, 18, 24, 30], [8, 9, 10, 12, 26, 31], [8, 9, 10, 14, 17, 36], [8, 9, 10, 15, 20, 34], [8, 9, 10, 16, 25, 30], [8, 9, 10, 23, 24, 26], [8, 9, 11, 20, 24, 28], [8, 9, 12, 13, 28, 28], [8, 9, 12, 16, 16, 35], [8, 9, 12, 18, 18, 33], [8, 9, 14, 14, 20, 33], [8, 9, 14, 15, 26, 28], [8, 9, 14, 16, 23, 30], [8, 9, 14, 18, 20, 31], [8, 9, 14, 22, 24, 25], [8, 9, 16, 18, 25, 26], [8, 9, 16, 20, 21, 28], [8, 9, 17, 18, 22, 28], [8, 10, 10, 12, 23, 33], [8, 10, 10, 15, 24, 31], [8, 10, 11, 11, 18, 36], [8, 10, 11, 12, 21, 34], [8, 10, 11, 18, 24, 29], [8, 10, 11, 20, 21, 30], [8, 10, 12, 13, 18, 35], [8, 10, 12, 17, 23, 30], [8, 10, 13, 21, 24, 26], [8, 10, 14, 19, 24, 27], [8, 10, 14, 21, 21, 28], [8, 10, 15, 15, 16, 34], [8, 10, 15, 17, 18, 32], [8, 10, 15, 18, 23, 28], [8, 10, 16, 18, 21, 29], [8, 10, 19, 21, 22, 24], [8, 11, 11, 12, 26, 30], [8, 11, 12, 12, 23, 32], [8, 11, 12, 22, 22, 27], [8, 11, 13, 14, 24, 30], [8, 11, 13, 18, 18, 32], [8, 11, 15, 16, 24, 28], [8, 11, 16, 18, 19, 30], [8, 11, 17, 20, 24, 24], [8, 11, 18, 20, 21, 26], [8, 12, 12, 12, 21, 33], [8, 12, 12, 17, 19, 32], [8, 12, 12, 19, 23, 28], [8, 12, 13, 13, 18, 34], [8, 12, 13, 15, 20, 32], [8, 12, 13, 17, 24, 28], [8, 12, 13, 18, 22, 29], [8, 12, 14, 19, 19, 30], [8, 12, 15, 21, 24, 24], [8, 12, 15, 22, 22, 25], [8, 12, 16, 19, 24, 25], [8, 12, 17, 18, 23, 26], [8, 12, 17, 20, 20, 27], [8, 12, 18, 18, 21, 27], [8, 13, 13, 18, 20, 30], [8, 13, 14, 16, 21, 30], [8, 13, 16, 18, 22, 27], [8, 13, 18, 18, 19, 28], [8, 14, 14, 15, 16, 33], [8, 14, 14, 20, 21, 27], [8, 14, 15, 16, 18, 31], [8, 14, 15, 17, 24, 26], [8, 14, 20, 21, 21, 22], [8, 15, 16, 16, 21, 28], [8, 15, 18, 20, 22, 23], [8, 15, 19, 20, 20, 24], [8, 16, 17, 20, 21, 24], [8, 16, 18, 18, 23, 23], [8, 17, 17, 18, 22, 24], [8, 17, 18, 18, 20, 25], [9, 9, 10, 12, 18, 36], [9, 9, 10, 14, 28, 28], [9, 9, 10, 16, 22, 32], [9, 9, 12, 12, 26, 30], [9, 9, 14, 16, 16, 34], [9, 9, 14, 20, 22, 28], [9, 9, 16, 16, 26, 26], [9, 10, 10, 13, 26, 30], [9, 10, 10, 14, 18, 35], [9, 10, 10, 16, 20, 33], [9, 10, 10, 19, 22, 30], [9, 10, 12, 15, 24, 30], [9, 10, 12, 16, 17, 34], [9, 10, 12, 16, 22, 31], [9, 10, 12, 20, 25, 26], [9, 10, 13, 14, 18, 34], [9, 10, 13, 18, 26, 26], [9, 10, 14, 15, 20, 32], [9, 10, 14, 17, 24, 28], [9, 10, 14, 18, 22, 29], [9, 10, 16, 17, 20, 30], [9, 10, 16, 22, 23, 24], [9, 10, 18, 19, 22, 26], [9, 11, 12, 16, 20, 32], [9, 12, 12, 12, 12, 37], [9, 12, 12, 12, 27, 28], [9, 12, 14, 14, 25, 28], [9, 12, 14, 20, 23, 26], [9, 12, 15, 18, 24, 26], [9, 12, 16, 19, 20, 28], [9, 13, 14, 14, 22, 30], [9, 13, 18, 22, 22, 22], [9, 13, 20, 20, 20, 24], [9, 14, 15, 16, 22, 28], [9, 14, 17, 20, 22, 24], [9, 14, 18, 20, 20, 25], [9, 16, 16, 18, 22, 25], [9, 16, 17, 18, 20, 26], [10, 10, 10, 14, 21, 33], [10, 10, 10, 18, 21, 31], [10, 10, 11, 15, 18, 34], [10, 10, 11, 20, 24, 27], [10, 10, 12, 12, 13, 37], [10, 10, 12, 13, 27, 28], [10, 10, 12, 20, 21, 29], [10, 10, 14, 15, 26, 27], [10, 10, 14, 17, 21, 30], [10, 10, 15, 16, 16, 33], [10, 10, 15, 20, 24, 25], [10, 10, 15, 21, 22, 26], [10, 10, 16, 20, 21, 27], [10, 10, 17, 18, 22, 27], [10, 11, 11, 18, 24, 28], [10, 11, 12, 12, 19, 34], [10, 11, 12, 12, 26, 29], [10, 11, 12, 13, 14, 36], [10, 11, 12, 14, 21, 32], [10, 11, 12, 16, 26, 27], [10, 11, 12, 19, 20, 30], [10, 11, 13, 22, 24, 24], [10, 11, 14, 14, 18, 33], [10, 11, 14, 15, 22, 30], [10, 11, 14, 18, 18, 31], [10, 11, 16, 18, 21, 28], [10, 12, 12, 15, 18, 33], [10, 12, 12, 22, 23, 25], [10, 12, 13, 14, 24, 29], [10, 12, 13, 19, 24, 26], [10, 12, 13, 20, 22, 27], [10, 12, 14, 15, 20, 31], [10, 12, 14, 19, 21, 28], [10, 12, 15, 16, 25, 26], [10, 12, 15, 17, 22, 28], [10, 12, 16, 18, 19, 29], [10, 12, 18, 20, 23, 23], [10, 12, 18, 21, 21, 24], [10, 12, 19, 19, 22, 24], [10, 13, 14, 16, 24, 27], [10, 13, 16, 21, 22, 24], [10, 13, 18, 18, 22, 25], [10, 14, 15, 15, 16, 32], [10, 14, 15, 20, 23, 24], [10, 14, 17, 17, 24, 24], [10, 14, 17, 18, 21, 26], [10, 15, 15, 18, 24, 24], [10, 15, 15, 20, 20, 26], [10, 15, 16, 16, 17, 30], [10, 15, 18, 18, 18, 27], [10, 16, 16, 18, 19, 27], [10, 18, 19, 20, 20, 21], [11, 11, 12, 16, 22, 30], [11, 11, 18, 20, 22, 24], [11, 12, 12, 14, 14, 35], [11, 12, 12, 16, 20, 31], [11, 12, 13, 18, 22, 28], [11, 12, 14, 18, 20, 29], [11, 12, 15, 16, 16, 32], [11, 12, 16, 20, 23, 24], [11, 12, 18, 19, 20, 26], [11, 13, 16, 16, 18, 30], [11, 14, 14, 17, 18, 30], [11, 14, 14, 19, 24, 24], [11, 14, 15, 18, 22, 26], [11, 14, 16, 18, 20, 27], [11, 16, 18, 20, 21, 22], [11, 17, 18, 18, 22, 22], [12, 12, 12, 12, 15, 35], [12, 12, 12, 12, 19, 33], [12, 12, 12, 17, 24, 27], [12, 12, 13, 16, 17, 32], [12, 12, 13, 16, 23, 28], [12, 12, 14, 22, 23, 23], [12, 12, 15, 17, 18, 30], [12, 12, 15, 19, 24, 24], [12, 12, 17, 17, 22, 26], [12, 12, 18, 18, 19, 27], [12, 13, 13, 14, 18, 32], [12, 13, 13, 22, 22, 24], [12, 13, 14, 16, 19, 30], [12, 13, 14, 20, 21, 26], [12, 13, 16, 16, 24, 25], [12, 14, 14, 17, 24, 25], [12, 14, 14, 19, 20, 27], [12, 14, 15, 16, 23, 26], [12, 14, 17, 17, 18, 28], [12, 14, 19, 20, 21, 22], [12, 15, 15, 18, 18, 28], [12, 15, 16, 16, 19, 28], [12, 15, 17, 20, 22, 22], [12, 16, 16, 20, 21, 23], [12, 16, 17, 18, 22, 23], [12, 16, 17, 19, 20, 24], [13, 13, 16, 18, 18, 28], [13, 14, 18, 18, 22, 23], [13, 14, 18, 19, 20, 24], [13, 15, 16, 20, 20, 24], [14, 14, 14, 15, 22, 27], [14, 14, 15, 15, 20, 28], [14, 14, 15, 21, 22, 22], [14, 14, 16, 19, 21, 24], [14, 14, 18, 18, 19, 25], [14, 15, 16, 17, 22, 24], [14, 15, 16, 18, 20, 25], [15, 16, 16, 17, 18, 26], [16, 18, 18, 19, 19, 20], [17, 17, 18, 18, 20, 20], [17, 18, 18, 18, 18, 21]], 1554)
11) Nombre de Kaprekar (Kaprekar number)
Définition
Un entier \(n\) est un nombre de Kaprekar si l'on peut découper l'écriture décimale de \(n^2\) en deux parties dont la somme vaut \(n\) (suivant une convention de découpage non triviale).
Pour en savoir plus => Les nombres de Kaprekar.
Calculons : \[ 2026^2 = 4\,104\,676. \]
On teste par exemple le découpage en séparant les 3 derniers chiffres : \[ 4104 + 676 = 4780 \neq 2026. \] Les autres découpages usuels ne conviennent pas non plus : 2026 n'est donc pas un nombre de Kaprekar.
