Happy New Year - Bonne année carrée 2025 = (20+25)²=45²
2025 quelques propriétés de cet entier naturel
Le site Math93.com vous souhaite une heureuse, chaleureuse et studieuse année 2025. Profitons-en pour revenir sur quelques caractéristiques de cet entier naturel impair.
- Ecriture de 2025
- L'année 2025
- Diviseurs de 2025
- Années carrées et cube ? : 2025 première année carrée de ce siècle
- Quelques décompositions de 2025
- 2025 nombres abondant ou déficient
- Autres curiosités de 2025
- Avec 20 et 25
- Avec des coefficients binomiaux
- Ecrire 2025 avec tous les chiffres de 1 à 9 dans l'ordre
- Des formules magiques : \(\left\lfloor \pi^\text{e} +\text{e}^\pi \right\rfloor = 2025\) et \(\left\lfloor \left(\phi+1\right)^2 -i^0 - (\pi^3)^2 + \left(e^{\frac{8}{5}}\right)^5 \right\rfloor = 2025 \)
- Avec le PGCD
- 2025 est la somme des nombres dans la table de multiplication de 1 à 9
- Décomposition de 2025 en somme de cubes
Et carré d'une somme, le théorème de Nicomède - Décomposition de 2025 en sommes de carrés
- Décomposition en somme de 2 carrés
- En somme de 3 carrés (11 fois)
- En somme de 4 carrés (80 fois)
- En somme de 5 carrés (211 fois)
- En somme de 6 carrés (1 558 fois)
- Nombre de Kaprekar
1) Écriture du nombre 2025
Cet entier naturel s'écrit ainsi, en tenant compte de l'orthographe réformée par les recommandations de l'Académie Française publiées en 1990.
- Français : deux mille vingt-cinq
- Anglais : two thousand twenty-five or twenty twenty-five
- Espagnol : dos mil veinticinco
- Allemand : zweitausendfünfundzwanzig
- Italien : duemilaventicinque
- Portugais : dois mil e vinte e cinco
- Néerlandais : tweeduizendvijfentwintig
- Russe : две тысячи двадцать пять (dve tysyachi dvadtsat' pyat')
- Chinois (simplifié) : 二零二五年 (èr líng èr wǔ nián)
- Arabe : ألفان وخمسة وعشرون (alfān wa khamsa wa ʿishrūn)
Autres écriture de 2025
- Binaire : base 2 (utilisée en informatique)
11111101001 - Romain : numérotation historique utilisée dans l'Empire romain
MMXXV
- Origine : L'écriture des chiffres romains remonte à environ le VIIIᵉ siècle av. J.-C., pendant la période de la fondation de Rome.
- Apogée : L'utilisation des chiffres romains s'est généralisée pendant toute la période de la République romaine (509–27 av. J.-C.) et de l'Empire romain (27 av. J.-C.–476 apr. J.-C.).
- Les chiffres romains ont continué d'être utilisés en Europe pendant tout le Moyen Âge, même après la chute de l'Empire romain d'Occident (476 apr. J.-C.).
- À partir du Xᵉ siècle, les chiffres indiens (introduits pas les arabes), et la numération indienne de position (base 10), ont progressivement remplacé les chiffres romains pour les calculs mathématiques en raison de leur praticité.
- Hexadécimal : base 16 (utilisée en informatique pour simplifier les représentations binaires)
7E9
2) L'année 2025
Une année complète comprend généralement 52 semaines et 1 jour (ou 52 semaines et 2 jours pour une année bissextile).
- 2025 n'est pas bissextile (365 jours).
- Il y aura donc exactement 52 semaines et 1 jour en 2025.
Cela signifie que l'année se terminera également un mercredi (31 décembre 2025).
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).
- Il n'y aura qu’un seul vendredi 13 en 2025, qui sera le 13 juin 2025.
- 2025 est une année à 52 dimanches. On a eu 53 dimanches en : 2000, 2006, 2012, 2017 et 2023.
3) Diviseurs de 2 025 et nombres premiers
Décomposition en facteurs premiers
$$2 025 =3^4\times5^2$$
Diviseurs de 2025
Pour déterminer tous les diviseurs de 2025, il suffit d'écrire sa décopmposition en facteur premiers.
Cette dernière \(3^4\times5^2\) nous indique de suite qu'il y aura \(5\times3=15\) diviseurs, voyons pourquoi.
- Les diviseurs de 2025 sont obtenus en combinant les puissances des facteurs premiers :
Les diviseurs sont les produits possibles de : $$5^a \times 3^b \quad ;\quad a \in \{0, 1, 2\} \quad;\quad b \in \{0, 1, 2, 3, 4\}$$ - En faisant l'arbre suivant, on obtient tous les diviseurs de 2025
- L'entier impair 2025 admet donc 15 diviseurs (2023 n'en avait que 6 et 16 pour 2024 ) qui sont : $$1,3,5,9,15,25,27,45,75,81,135,225,405,675,2025$$
Nombre premier ?
- Nombre premiers : 2025 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.
- Le 2025e nombre premier est 17 609
4) Années carrées et cubes ?
2025 première année carrée de ce siècle
2025 est la première année carrée de ce siècle car \(2025 = (20+25)^2=45^2\) depuis 1936 (= 44²).
Tableau: les précédentes années carrées et les suivantes. Notez l'écart entre elles qui vaut \(2n – 1\).
\[
\text{Écart} = n^2 - (n-1)^2
\]
En développant :
\[
n^2 - (n-1)^2 = n^2 - (n^2 - 2n + 1)=2n-1
\]
2025 Année produit de carrés ?
Puisque $$2 025 =3^4\times5^2=\left(3^2\right)^2\times5^2$$
on peut écrire 2025 sous forme d'un produit de deux carrés :
$$2 025 =9^2\times5^2$$
Et les années cubes ?
Une année cube est une année qui peut être exprimée comme le cube d’un entier.
- Année cube précédente : 1728
$$1728=12^3$$
- Année cube suivante : 2197
$$2197=13^3$$
5) 2025 et quelques décompositions
Nombre de Harshad ou de Niven
Comme 2024, 2025 est un nombre Harshad ou de Niven, c'est à dire un nombre divisible par la somme de ses chiffres soit ici : \(2+0+2+5=9\)
$$\dfrac{2025}{9}=3^2\times5^2=225$$
Pour en savoir plus sur ces nombres : nombres de Harshad ou de Niven
Par exemple :
2000, 2001, 2004, 2007, 2010, 2016, 2020, 2022, 2023, 2024, 2025, 2028, 2030, 2034, 2040, 2043, 2052, 2061, 2064, 2070, 2080, 2085, 2088, 2090, 2100 …
Prochaines plages de quatre années de Harshad consécutives:
[2022, 2023, 2024, 2025], [3030, 3031, 3032, 3033], [10307, 10308, 10309, 10310], …
Précédentes: [510, 511, 512, 513], [1014, 1015, 1016, 1017]
Nombre polis : 14 fois somme d'entiers consécutifs
Un nombre poli ou escalier est un nombre qui peut s'écrire sous la forme de une ou plusieurs sommes de deux ou plusieurs nombres consécutifs. Le degré de politesse indique combien de fois un nombre est sommes de nombres consécutifs.
2025 est un nombre 8-poli.
Décompositions possibles :
Le nombre \( 2025 \) est un nombre poli, car il peut être exprimé comme une somme de nombres consécutifs de plusieurs façons. Son degré de politesse est \( 14 \), ce qui signifie qu'il peut être décomposé en \( 14 \) manières distinctes.
Voici toutes les décompositions possibles de \( 2025 \) comme somme d'entiers consécutifs :
\[
\begin{aligned}
1. & \quad 2025 = 1012 + 1013 \quad (\text{2 nombres}) \\
2. & \quad 2025 = 674 + 675 + 676 \quad (\text{3 nombres}) \\
3. & \quad 2025 = 403 + 404 + 405 + 406 + 407 \quad (\text{5 nombres}) \\
4. & \quad 2025 = 335 + 336 + \dots + 340 \quad (\text{6 nombres}) \\
5. & \quad 2025 = 221 + 222 + \dots + 229 \quad (\text{9 nombres}) \\
6. & \quad 2025 = 198 + 199 + \dots + 207 \quad (\text{10 nombres}) \\
7. & \quad 2025 = 128 + 129 + \dots + 142 \quad (\text{15 nombres}) \\
8. & \quad 2025 = 104 + 105 + \dots + 121 \quad (\text{18 nombres}) \\
9. & \quad 2025 = 69 + 70 + \dots + 93 \quad (\text{25 nombres}) \\
10. & \quad 2025 = 62 + 63 + \dots + 88 \quad (\text{27 nombres}) \\
11. & \quad 2025 = 53 + 54 + \dots + 82 \quad (\text{30 nombres}) \\
12. & \quad 2025 = 23 + 24 + \dots + 67 \quad (\text{45 nombres}) \\
13. & \quad 2025 = 16 + 17 + \dots + 65 \quad (\text{50 nombres}) \\
14. & \quad 2025 = 11 + 12 + \dots + 64 \quad (\text{54 nombres})
\end{aligned}
\]
Pour en savoir plus sur ces nombres : nombres polis
Remarque : Tous les nombres peuvent se mettent sous la forme de somme de consécutifs sauf les puissances de 2.
Somme de deux premiers consécutifs
2025 (comme 2024) n'est pas somme de 2 premiers consécutifs mais 2022 l'était : $$2022=1009+1013$$
Somme de deux nombres premiers
2025 n'est jamais somme de nombre premier.
En effet, l'entier 2025 est impair.
Or la somme de 2 nombre premiers différents de 2 est paire, ce qui laisse comme unique possibilité une décomposition avec l'entier premier 2.
Or dans ce cas on aurait la décomposition \(2+2023\) mais le nombre \( 2023 \) n'est pas un nombre premier, car il peut être décomposé en facteurs premiers comme suit :
\[
2023 = 7 \times 289 = 7 \times 17^2
\]
2024 était somme de 2 nombres premiers 32 fois :
- \(2024= 7+2017\)
- \(2024= 13+2011\)
- ...
- \(2024= 991+1033\)
Nombre parfait : c'est un entier somme de ses diviseurs propres
Un nombre parfait est un entier égal à la somme de ses diviseurs propres (c'est-à-dire tous ses diviseurs sauf lui-même). Par exemple, \( 6 \) est un nombre parfait car \( 6 = 1 + 2 + 3 \).
Vérifions si \( 2025 \) est un nombre parfait.
Les diviseurs propres de \( 2025 \) sont :
\[
1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675
\]
La somme de ces diviseurs est :
\[
1 + 3 + 5 + 9 + 15 + 25 + 27 + 45 + 75 + 81 + 135 + 225 + 405 + 675 = 1726
\]
Puisque \( 1726 \neq 2025 \), nous concluons que \( 2025 \) \textbf{n'est pas un nombre parfait}.
Nombre semi-parfait : somme de certains de ses diviseurs.
2025 n'est pas semi parfait contrairement à 2022 : $$2 022 = 1011 + 674 + 337$$
6) 2025 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.
Les diviseurs propres de \( 2025 \) sont :
\[
1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675
\]
La somme de ces diviseurs est :
\[
1 + 3 + 5 + 9 + 15 + 25 + 27 + 45 + 75 + 81 + 135 + 225 + 405 + 675 = 1726
\]
Puisque \( 1726 < 2025 \), nous concluons que \( 2025 \) \textbf{n'est pas un nombre abondant}, mais un nombre déficient.
$$1+2+ 4+8+ 11+ 22+ 23+44+ 46+ 88+ 92+ 184+ 253+ 506+ 1012 =4320 > 2024$$
7) Autres Curiosités de 2025
Avec 20 et 25
$$2025=(20+25)(20+25)$$
On avait
$$2024 = (20+24)+(20+24)+(20+24)(20+24)$$
Coefficient binomiaux et 2025
Pour rappel, la formule d'un coefficient binomial est donnée par :
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
où : \( n! = 1\times2\times ... \times n \) est la factorielle de \( n \),
Cette formule est utilisée pour compter le nombre de façons de choisir \( k \) éléments parmi \( n \) éléments sans ordre.
Écrivons \( 2025 \) comme le carré des coefficients binomiaux qui donnent \( 45 \) :
\[
2025 = \binom{10}{2}^2 = \binom{10}{8}^2 = \binom{45}{1}^2 = \binom{45}{44}^2
\]
Par exemple, en développant :
\[
\binom{10}{2}^2 = \left(\frac{10 \times 9}{2}\right)^2 = 45^2 = 2025.
\]
Ainsi, \( 2025 \) peut être exprimé comme le carré de ces coefficients binomiaux.
Ecrire 2025 avec tous les chiffres de 1 à 9 dans l'ordre
Je n'ai rien trouvé, mais on avais en 2024 :
$$2024=1234−5+6+789 $$ $$2024=123+4(5+6+78)+9$$ $$2024=9×8+7+6×54×3×2+1$$
Formules Magiques : Avec quelques symboles célèbres
Happy New Year
$$\huge \left\lfloor \left(\phi+1\right)^2 -i^0 - (\pi^3)^2 + \left(e^{\frac{8}{5}}\right)^5 \right\rfloor = 2025 $$
Expliquons un peut cette égalité :
- \(\phi\) est le nombre d'or et vaut :$$\phi=\dfrac{1+\sqrt{5}}{2}\approx 1,618$$
- Le nombre e est la base des logarithmes naturels, c'est-à-dire le nombre défini par \(\ln(e) = 1\). Cette constante mathématique, également appelée nombre d'Euler ou constante de Néper en référence aux mathématiciens Leonhard Euler et John Napier, vaut environ 2,71828.
- La fonction plancher (floor function), notée \( \lfloor x \rfloor \), est définie par encadrement : \[ \lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 \] Elle représente le plus grand entier inférieur ou égal à \(x\).
- La fonction plafond (ceil function), notée \( \lceil x \rceil \), est définie par encadrement : \[ \lfloor x \rfloor -1< x \leq \lfloor x \rfloor \] Elle représente le plus petit entier supérieur ou égal à \(x\).
- i nombre complexe dont le carré vaut \(-1\).
- On a donc : $$(\phi+1)^2-i^0-\left(\pi^3\right)^2+\left(e^\frac85\right)^5\approx 2025,42$$
Et la partie entière inférieure vaut bien 2025.
On a aussi :
$$\huge \left\lfloor \pi^\text{e} +\text{e}^\pi \right\rfloor = 2025 $$
$$\huge \left\lfloor \dfrac{\text{e}^{2+0+2+5}}{\left\lceil \pi\right\rceil}\right\rfloor = 2025 $$
Avec le PGCD
Pour rappel, le PGCD (GCD en anglais) de deux entiers est le Plus Grand Commun Diviseur (Greatest Common Divisor) de ces deux entiers.
On a l'égalité : $$\prod_{i=1}^{14} \text{PGCD}(i, 15) = 2025$$
En effet le produit des PGCD des entiers de 1 à 14 avec 15 donne 2025 :
\begin{aligned}
\gcd(1, 15) &= 1 \\
\gcd(2, 15) &= 1 \\
\gcd(3, 15) &= 3 \\
\gcd(4, 15) &= 1 \\
\gcd(5, 15) &= 5 \\
\gcd(6, 15) &= 3 \\
\gcd(7, 15) &= 1 \\
\gcd(8, 15) &= 1 \\
\gcd(9, 15) &= 3 \\
\gcd(10, 15) &= 5 \\
\gcd(11, 15) &= 1 \\
\gcd(12, 15) &= 3 \\
\gcd(13, 15) &= 1 \\
\gcd(14, 15) &= 1 \\
\end{aligned}
8) 2025 et la somme des nombres dans la table de multiplication de 1 à 9
Si on effectue la somme des nombres qui interviennent dans la tavles de multiplication de 1 à 9 on obtient 2025, ce qui peut s'écrire :
$$\sum_{n=1}^{9} \sum_{k=1}^{9} n \cdot k = 2025 $$
Voici pourquoi :
- Le nombre de points sur ce carré correspond à la somme de tous les nombres dans la table de 1 à 9.
- Ce carré est de côté 45, en effet en utilisant la formule de la somme des termes d'une suite arithmétique : $$1+2+\cdots+9=9\times\dfrac{1+9}{2}=9\times 5=45$$
- Donc le nombre total de points est : $$45^2=2025$$
9) 2025 en somme de cubes
$$2025 = \left(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9\right)^2=1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3$$
C'est un cas particulier du théorème de Nicomaque.
Le théorème de Nicomaque, attribué à Nicomachus de Gérase (environ 60 - 120 après J.-C.) énonce que la somme des cubes des \(n\) premiers entiers naturels est égale au carré de leur somme : $$ \sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2 $$ En d'autres termes : $$\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 $$
Nicomaque n'a pas démontré le théorème de manière rigoureuse. Il l'a probablement découvert par observation et vérifications empiriques.
On peut voir une annimation visuelle de la preuve de cette égalité :
10) 2025 Somme de carrés
2025 somme de deux carrés ?
2025 est 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.
Autrement dit : Un entier \(n\) est décomposable en une somme de deux carrés si, dans sa décomposition en facteurs premiers, tout facteur premier congru à \(3 \mod 4\) apparaît avec une puissance paire.
- Étape 1 : Décomposition en facteurs premiers
On décompose \(2025\) comme suit : $$ 2025 = 5^2 \cdot 3^4$$
Les facteurs premiers sont : \(3 \quad \text{et} \quad 5 \)- \(3 \mod 4 = 3\) apparaît avec une puissance \(4\), qui est paire.
- \(5 \mod 4 = 1\), qui ne pose aucune restriction.
- Par conséquent, \(2025\) est décomposable en une somme de deux carrés.
- Étape 2 : Recherche des deux carrés
Nous cherchons des entiers \(a\) et \(b\) tels que : $$ 2025 = a^2 + b^2 $$
En testant, on trouve deux décompositions : $$ 2025 = 0^2 + 45^2 $$ $$ 2025 = 27^2+36^2 $$
Remarque : on retrouve le fameux triplet pythagoricien \((3-4-5)\), convenablement dilaté par 9.
Une autre expression du nombre de décompositions d'un entier premier impair a été donnée par le mathématicien français Pierre de Fermat (1601-1655).
On dispose d'un autre théorème dont une preuve exposée sur ce poly (niveau supérieur) :
Théorème des deux carrés de Fermat (cas des nombres premiers) —
Un nombre premier impair (c'est-à-dire tous les nombres premiers sauf 2) est une somme de deux carrés parfaits si et seulement si le reste de sa division euclidienne par 4 est 1 ; dans ce cas, les carrés sont déterminés de manière unique.
2024 somme de 3 carrés parfaits
Le nombre 2025 est somme de 3 carrés 11 fois (13 fois si on autorise un des termes nul).
Le théorème des trois carrés démontré par Carl Friedrich GAUSS (1777-1855) en 1801, et s'exprime par:
Théorème des trois carrés —
Un entier naturel est somme de trois carrés si, et seulement si, il n'est pas de la forme \(4^i \left(8 j -1\right)\) avec i et j entiers positifs ou nuls.
Voici les 11 décompositions de 2025 en somme de 3 carrés :
$$[4, 28, 35], [5, 8, 44], [5, 20, 40], [6, 15, 42], [6, 30, 33], [8, 19, 40], [13, 16, 40], [15, 30, 30], [16, 20, 37], [20, 20, 35], [20, 28, 29]$$
- \(2025 = 4^2 + 28^2 + 35^2\)
- \(2025 = 5^2 + 8^2 + 44^2\)
- \(2025 = 5^2 + 20^2 + 40^2\)
- \(2025 = 6^2 + 15^2 + 42^2\)
- \(2025 = 6^2 + 30^2 + 33^2\)
- \(2025 = 8^2 + 19^2 + 40^2\)
- \(2025 = 13^2 + 16^2 + 40^2\)
- \(2025 = 15^2 + 30^2 + 30^2\)
- \(2025 = 16^2 + 20^2 + 37^2\)
- \(2025 = 20^2 + 20^2 + 35^2\)
- \(2025 = 20^2 + 28^2 + 29^2\)
- \(2025 = 0^2 + 0^2 + 45^2\)
- \(2025 = 0^2 + 27^2 + 36^2\)
2025 somme de 4 carrés parfaits
Le nombre 2025 est un entier qui peut s'écrire comme la somme de quatre carrés 80 fois :
- \( 2025=1^2+2^2+16^2 +42^2\)
- \( 2025 = 1^2+2^2+24^2 +38^2=\cdots\)
- ...
- \( 2025=20^2+20^2+21^2 +28^2\)
Voici la liste des 80 quadruplets [a, b, c, d] tels que $$2024 = a^2 + b^2 + c^2 + d^2$$ [1, 2, 16, 42], [1, 2, 24, 38], [1, 8, 14, 42], [1, 10, 18, 40], [1, 10, 30, 32], [1, 16, 18, 38], [1, 18, 26, 32], [2, 2, 9, 44], [2, 4, 18, 41], [2, 4, 22, 39], [2, 6, 7, 44], [2, 6, 31, 32], [2, 7, 26, 36], [2, 9, 28, 34], [2, 10, 20, 39], [2, 10, 25, 36], [2, 12, 14, 41], [2, 14, 15, 40], [2, 14, 23, 36], [2, 16, 26, 33], [2, 17, 24, 34], [2, 22, 24, 31], [3, 4, 8, 44], [3, 4, 20, 40], [3, 12, 24, 36], [4, 4, 12, 43], [4, 6, 23, 38], [4, 7, 14, 42], [4, 8, 24, 37], [4, 9, 22, 38], [4, 12, 29, 32], [4, 16, 27, 32], [4, 18, 23, 34], [4, 21, 28, 28], [4, 22, 25, 30], [5, 12, 16, 40], [5, 20, 24, 32], [6, 7, 28, 34], [6, 9, 12, 42], [6, 10, 17, 40], [6, 12, 18, 39], [6, 16, 17, 38], [6, 17, 26, 32], [6, 18, 24, 33], [6, 23, 26, 28], [7, 12, 26, 34], [7, 14, 22, 36], [7, 20, 26, 30], [8, 10, 30, 31], [8, 14, 26, 33], [8, 18, 26, 31], [8, 19, 24, 32], [9, 10, 20, 38], [9, 12, 30, 30], [9, 18, 18, 36], [9, 22, 26, 28], [10, 10, 12, 41], [10, 10, 15, 40], [10, 10, 23, 36], [10, 12, 25, 34], [10, 15, 16, 38], [10, 15, 26, 32], [10, 20, 25, 30], [12, 12, 21, 36], [12, 14, 23, 34], [12, 16, 16, 37], [12, 16, 20, 35], [12, 16, 28, 29], [12, 23, 26, 26], [12, 24, 24, 27], [13, 16, 24, 32], [14, 16, 22, 33], [14, 20, 23, 30], [15, 18, 24, 30], [16, 16, 27, 28], [16, 17, 18, 34], [16, 18, 22, 31], [17, 22, 24, 26], [18, 20, 25, 26], [20, 20, 21, 28]
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.
2025 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.
2025 somme de 5 carrés parfaits
Le nombre 2025 est un entier qui peut s'écrire comme la somme de cinq carrés 211 fois :
- \( 2025=1^2+2^2+18^2 +20^2+36^2\)
- \( 2025 = 1^2+4^2+6^2 +6^2+44^2=\cdots\)
- ...
- \( 2025=18^2+18^2+18^2+18^2 +27^2\)
Voici la liste des 211 quintuplets [a, b, c, d, e] tels que $$2025 = a^2 + b^2 + c^2 + d^2+ e^2$$ [1, 2, 18, 20, 36], [1, 4, 6, 6, 44], [1, 4, 6, 26, 36], [1, 4, 10, 12, 42], [1, 4, 18, 28, 30], [1, 6, 8, 18, 40], [1, 6, 8, 30, 32], [1, 6, 12, 20, 38], [1, 6, 16, 24, 34], [1, 8, 22, 24, 30], [1, 10, 12, 22, 36], [1, 10, 18, 24, 32], [1, 12, 14, 28, 30], [1, 12, 18, 20, 34], [1, 14, 24, 24, 26], [1, 18, 20, 20, 30], [2, 2, 12, 28, 33], [2, 2, 18, 18, 37], [2, 2, 21, 26, 30], [2, 4, 4, 15, 42], [2, 4, 4, 30, 33], [2, 4, 9, 18, 40], [2, 4, 9, 30, 32], [2, 4, 12, 30, 31], [2, 4, 15, 22, 36], [2, 4, 23, 24, 30], [2, 5, 6, 14, 42], [2, 5, 14, 30, 30], [2, 6, 6, 10, 43], [2, 6, 8, 20, 39], [2, 6, 8, 25, 36], [2, 6, 10, 11, 42], [2, 6, 10, 21, 38], [2, 6, 10, 27, 34], [2, 6, 17, 20, 36], [2, 6, 24, 25, 28], [2, 7, 8, 12, 42], [2, 7, 10, 24, 36], [2, 8, 15, 24, 34], [2, 9, 12, 14, 40], [2, 9, 16, 28, 30], [2, 10, 11, 30, 30], [2, 10, 12, 16, 39], [2, 10, 15, 20, 36], [2, 10, 16, 24, 33], [2, 10, 18, 21, 34], [2, 11, 18, 26, 30], [2, 12, 12, 17, 38], [2, 12, 18, 23, 32], [2, 12, 24, 25, 26], [2, 14, 14, 27, 30], [2, 14, 15, 24, 32], [2, 14, 21, 22, 30], [2, 16, 17, 24, 30], [2, 18, 22, 22, 27], [3, 4, 12, 16, 40], [3, 4, 20, 24, 32], [3, 8, 12, 28, 32], [3, 8, 16, 20, 36], [3, 16, 20, 24, 28], [4, 4, 11, 24, 36], [4, 4, 15, 18, 38], [4, 4, 16, 21, 36], [4, 4, 24, 24, 29], [4, 6, 6, 16, 41], [4, 6, 7, 18, 40], [4, 6, 7, 30, 32], [4, 6, 10, 28, 33], [4, 6, 14, 16, 39], [4, 6, 17, 28, 30], [4, 6, 18, 25, 32], [4, 6, 20, 22, 33], [4, 7, 22, 24, 30], [4, 8, 9, 10, 42], [4, 8, 10, 18, 39], [4, 8, 12, 24, 35], [4, 9, 14, 24, 34], [4, 9, 24, 26, 26], [4, 10, 12, 26, 33], [4, 10, 15, 28, 30], [4, 10, 17, 18, 36], [4, 11, 12, 12, 40], [4, 12, 13, 20, 36], [4, 12, 14, 15, 38], [4, 12, 15, 22, 34], [4, 12, 17, 26, 30], [4, 12, 20, 21, 32], [4, 14, 18, 20, 33], [4, 15, 18, 26, 28], [4, 15, 20, 22, 30], [4, 16, 18, 23, 30], [4, 18, 18, 20, 31], [4, 18, 22, 24, 25], [5, 6, 6, 22, 38], [5, 6, 10, 10, 42], [5, 6, 14, 18, 38], [5, 6, 18, 22, 34], [5, 8, 8, 24, 36], [5, 8, 24, 24, 28], [5, 10, 10, 30, 30], [5, 10, 18, 26, 30], [5, 12, 16, 24, 32], [5, 14, 18, 18, 34], [5, 18, 18, 26, 26], [6, 6, 8, 17, 40], [6, 6, 9, 24, 36], [6, 6, 10, 22, 37], [6, 6, 11, 26, 34], [6, 6, 18, 27, 30], [6, 6, 20, 23, 32], [6, 7, 12, 14, 40], [6, 7, 16, 28, 30], [6, 8, 8, 30, 31], [6, 8, 9, 20, 38], [6, 8, 10, 12, 41], [6, 8, 10, 15, 40], [6, 8, 10, 23, 36], [6, 8, 12, 25, 34], [6, 8, 15, 16, 38], [6, 8, 15, 26, 32], [6, 8, 20, 25, 30], [6, 9, 10, 28, 32], [6, 9, 20, 22, 32], [6, 10, 11, 18, 38], [6, 10, 12, 28, 31], [6, 10, 14, 18, 37], [6, 10, 17, 24, 32], [6, 10, 20, 20, 33], [6, 10, 22, 26, 27], [6, 10, 23, 24, 28], [6, 11, 22, 22, 30], [6, 12, 14, 25, 32], [6, 12, 15, 18, 36], [6, 12, 17, 20, 34], [6, 12, 20, 22, 31], [6, 14, 14, 21, 34], [6, 14, 15, 28, 28], [6, 14, 16, 24, 31], [6, 14, 21, 26, 26], [6, 15, 16, 22, 32], [6, 16, 18, 25, 28], [6, 17, 20, 20, 30], [6, 18, 18, 21, 30], [6, 20, 22, 23, 24], [7, 8, 12, 18, 38], [7, 10, 12, 24, 34], [7, 10, 16, 18, 36], [7, 10, 20, 24, 30], [7, 12, 16, 26, 30], [7, 12, 18, 22, 32], [7, 18, 20, 24, 26], [8, 8, 12, 27, 32], [8, 8, 18, 22, 33], [8, 9, 10, 22, 36], [8, 9, 14, 28, 30], [8, 9, 18, 20, 34], [8, 10, 12, 14, 39], [8, 10, 14, 24, 33], [8, 10, 18, 24, 31], [8, 11, 12, 20, 36], [8, 12, 20, 24, 29], [8, 14, 17, 24, 30], [8, 15, 16, 18, 34], [8, 15, 22, 24, 26], [8, 16, 20, 24, 27], [8, 17, 18, 18, 32], [8, 18, 18, 23, 28], [9, 10, 10, 12, 40], [9, 10, 12, 16, 38], [9, 10, 12, 26, 32], [9, 10, 22, 24, 28], [9, 12, 18, 24, 30], [9, 14, 14, 16, 36], [9, 14, 18, 20, 32], [9, 20, 22, 22, 24], [10, 10, 14, 27, 30], [10, 10, 15, 24, 32], [10, 10, 21, 22, 30], [10, 11, 18, 18, 34], [10, 12, 12, 26, 31], [10, 12, 14, 17, 36], [10, 12, 15, 20, 34], [10, 12, 16, 25, 30], [10, 12, 23, 24, 26], [10, 14, 18, 26, 27], [10, 15, 20, 20, 30], [10, 16, 16, 18, 33], [10, 17, 22, 24, 24], [10, 18, 20, 24, 25], [10, 18, 21, 22, 26], [11, 12, 20, 24, 28], [11, 14, 18, 22, 30], [12, 12, 13, 28, 28], [12, 12, 16, 16, 35], [12, 12, 18, 18, 33], [12, 14, 14, 20, 33], [12, 14, 15, 26, 28], [12, 14, 16, 23, 30], [12, 14, 18, 20, 31], [12, 14, 22, 24, 25], [12, 16, 18, 25, 26], [12, 16, 20, 21, 28], [12, 17, 18, 22, 28], [14, 15, 16, 18, 32], [14, 18, 20, 23, 24], [15, 16, 22, 22, 24], [15, 18, 18, 24, 24], [15, 18, 20, 20, 26], [16, 16, 17, 18, 30], [16, 16, 19, 24, 24], [18, 18, 18, 18, 27]
2025 somme de 6 carrés parfaits
Pour information, il y a 1 558 décompositions de 2025 en somme de 6 carrés :
- \( 2025=1^2+1^2+1^2 +2^2+13^2+43^2\)
- ...
- \( 2025=16^2+16^2+17^2 +18^2+18^2+24^2\)
Voici la liste des 1850 sextuplets [a, b, c, d, e, f] tels que $$2025 = a^2 + b^2 + c^2 + d^2+ e^2+ f^2$$ ([[1, 1, 1, 2, 13, 43], [1, 1, 1, 5, 29, 34], [1, 1, 1, 7, 23, 38], [1, 1, 1, 10, 31, 31], [1, 1, 1, 11, 26, 35], [1, 1, 1, 13, 22, 37], [1, 1, 1, 17, 17, 38], [1, 1, 2, 5, 25, 37], [1, 1, 2, 7, 11, 43], [1, 1, 2, 7, 17, 41], [1, 1, 2, 11, 23, 37], [1, 1, 2, 13, 13, 41], [1, 1, 2, 13, 25, 35], [1, 1, 2, 17, 19, 37], [1, 1, 2, 23, 23, 31], [1, 1, 3, 3, 18, 41], [1, 1, 3, 3, 22, 39], [1, 1, 3, 5, 15, 42], [1, 1, 3, 5, 30, 33], [1, 1, 3, 9, 13, 42], [1, 1, 3, 13, 18, 39], [1, 1, 3, 14, 27, 33], [1, 1, 3, 18, 27, 31], [1, 1, 3, 21, 22, 33], [1, 1, 5, 5, 23, 38], [1, 1, 5, 6, 21, 39], [1, 1, 5, 7, 10, 43], [1, 1, 5, 10, 23, 37], [1, 1, 5, 11, 14, 41], [1, 1, 5, 14, 29, 31], [1, 1, 5, 17, 22, 35], [1, 1, 5, 19, 26, 31], [1, 1, 6, 9, 15, 41], [1, 1, 6, 13, 27, 33], [1, 1, 6, 23, 27, 27], [1, 1, 7, 11, 22, 37], [1, 1, 7, 13, 19, 38], [1, 1, 7, 17, 23, 34], [1, 1, 7, 22, 23, 31], [1, 1, 9, 9, 30, 31], [1, 1, 9, 14, 15, 39], [1, 1, 9, 18, 23, 33], [1, 1, 9, 22, 27, 27], [1, 1, 10, 11, 11, 41], [1, 1, 10, 11, 29, 31], [1, 1, 10, 13, 23, 35], [1, 1, 11, 11, 25, 34], [1, 1, 11, 13, 17, 38], [1, 1, 13, 13, 23, 34], [1, 1, 13, 14, 17, 37], [1, 1, 13, 15, 27, 30], [1, 1, 13, 18, 21, 33], [1, 1, 13, 22, 23, 29], [1, 1, 14, 19, 25, 29], [1, 1, 15, 15, 22, 33], [1, 1, 17, 17, 17, 34], [1, 1, 17, 17, 22, 31], [1, 1, 17, 22, 25, 25], [1, 1, 17, 23, 23, 26], [1, 1, 18, 21, 23, 27], [1, 1, 19, 19, 25, 26], [1, 2, 2, 4, 8, 44], [1, 2, 2, 4, 20, 40], [1, 2, 2, 12, 24, 36], [1, 2, 3, 7, 21, 39], [1, 2, 3, 9, 9, 43], [1, 2, 3, 9, 29, 33], [1, 2, 3, 21, 27, 29], [1, 2, 4, 8, 28, 34], [1, 2, 4, 14, 28, 32], [1, 2, 5, 5, 11, 43], [1, 2, 5, 5, 17, 41], [1, 2, 5, 23, 25, 29], [1, 2, 7, 7, 31, 31], [1, 2, 7, 11, 13, 41], [1, 2, 7, 11, 25, 35], [1, 2, 7, 13, 29, 31], [1, 2, 7, 15, 15, 39], [1, 2, 7, 17, 29, 29], [1, 2, 7, 21, 21, 33], [1, 2, 8, 10, 16, 40], [1, 2, 8, 16, 16, 38], [1, 2, 8, 16, 26, 32], [1, 2, 8, 20, 20, 34], [1, 2, 9, 11, 27, 33], [1, 2, 9, 15, 25, 33], [1, 2, 11, 13, 19, 37], [1, 2, 11, 21, 27, 27], [1, 2, 11, 23, 23, 29], [1, 2, 12, 12, 24, 34], [1, 2, 12, 16, 18, 36], [1, 2, 12, 20, 24, 30], [1, 2, 13, 13, 29, 29], [1, 2, 13, 19, 23, 31], [1, 2, 14, 16, 28, 28], [1, 2, 14, 20, 20, 32], [1, 2, 15, 15, 27, 29], [1, 2, 15, 21, 25, 27], [1, 2, 16, 16, 22, 32], [1, 2, 17, 19, 23, 29], [1, 3, 3, 3, 29, 34], [1, 3, 3, 6, 11, 43], [1, 3, 3, 6, 17, 41], [1, 3, 3, 10, 15, 41], [1, 3, 3, 11, 11, 42], [1, 3, 3, 11, 21, 38], [1, 3, 3, 11, 27, 34], [1, 3, 3, 14, 17, 39], [1, 3, 3, 14, 21, 37], [1, 3, 3, 15, 25, 34], [1, 3, 3, 18, 29, 29], [1, 3, 5, 6, 27, 35], [1, 3, 5, 15, 26, 33], [1, 3, 5, 18, 21, 35], [1, 3, 5, 19, 27, 30], [1, 3, 6, 7, 9, 43], [1, 3, 6, 7, 29, 33], [1, 3, 6, 9, 23, 37], [1, 3, 6, 13, 17, 39], [1, 3, 6, 13, 21, 37], [1, 3, 6, 15, 23, 35], [1, 3, 6, 17, 27, 31], [1, 3, 6, 19, 23, 33], [1, 3, 6, 25, 25, 27], [1, 3, 7, 9, 11, 42], [1, 3, 7, 9, 21, 38], [1, 3, 7, 9, 27, 34], [1, 3, 7, 11, 18, 39], [1, 3, 7, 15, 29, 30], [1, 3, 7, 21, 25, 30], [1, 3, 9, 9, 22, 37], [1, 3, 9, 13, 26, 33], [1, 3, 9, 15, 22, 35], [1, 3, 9, 19, 22, 33], [1, 3, 9, 23, 26, 27], [1, 3, 10, 13, 15, 39], [1, 3, 10, 15, 27, 31], [1, 3, 11, 15, 15, 38], [1, 3, 11, 18, 27, 29], [1, 3, 13, 21, 26, 27], [1, 3, 14, 15, 15, 37], [1, 3, 14, 17, 21, 33], [1, 3, 14, 19, 27, 27], [1, 3, 15, 18, 25, 29], [1, 3, 15, 19, 23, 30], [1, 3, 17, 18, 21, 31], [1, 3, 18, 21, 25, 25], [1, 3, 19, 21, 22, 27], [1, 4, 4, 8, 22, 38], [1, 4, 4, 14, 14, 40], [1, 4, 4, 22, 22, 32], [1, 4, 6, 8, 12, 42], [1, 4, 6, 10, 24, 36], [1, 4, 8, 10, 20, 38], [1, 4, 8, 12, 30, 30], [1, 4, 8, 18, 18, 36], [1, 4, 8, 22, 26, 28], [1, 4, 10, 10, 28, 32], [1, 4, 10, 20, 22, 32], [1, 4, 14, 16, 20, 34], [1, 4, 16, 20, 26, 26], [1, 4, 16, 22, 22, 28], [1, 4, 18, 18, 24, 28], [1, 5, 5, 5, 10, 43], [1, 5, 5, 11, 22, 37], [1, 5, 5, 13, 19, 38], [1, 5, 5, 17, 23, 34], [1, 5, 5, 22, 23, 31], [1, 5, 6, 9, 19, 39], [1, 5, 7, 7, 26, 35], [1, 5, 7, 10, 13, 41], [1, 5, 7, 10, 25, 35], [1, 5, 7, 13, 25, 34], [1, 5, 7, 14, 23, 35], [1, 5, 7, 22, 25, 29], [1, 5, 9, 10, 27, 33], [1, 5, 9, 15, 18, 37], [1, 5, 9, 17, 27, 30], [1, 5, 10, 13, 19, 37], [1, 5, 10, 21, 27, 27], [1, 5, 10, 23, 23, 29], [1, 5, 11, 13, 22, 35], [1, 5, 11, 14, 29, 29], [1, 5, 11, 19, 19, 34], [1, 5, 11, 19, 26, 29], [1, 5, 13, 23, 25, 26], [1, 5, 14, 17, 17, 35], [1, 5, 15, 15, 18, 35], [1, 5, 15, 18, 19, 33], [1, 5, 19, 22, 23, 25], [1, 5, 21, 21, 21, 26], [1, 6, 6, 12, 28, 32], [1, 6, 6, 16, 20, 36], [1, 6, 7, 11, 27, 33], [1, 6, 7, 15, 25, 33], [1, 6, 8, 12, 22, 36], [1, 6, 8, 18, 24, 32], [1, 6, 9, 15, 29, 29], [1, 6, 9, 17, 23, 33], [1, 6, 9, 21, 25, 29], [1, 6, 10, 12, 12, 40], [1, 6, 11, 11, 15, 39], [1, 6, 12, 12, 16, 38], [1, 6, 12, 12, 26, 32], [1, 6, 12, 22, 24, 28], [1, 6, 13, 15, 15, 37], [1, 6, 13, 17, 21, 33], [1, 6, 13, 19, 27, 27], [1, 6, 15, 19, 21, 31], [1, 6, 16, 16, 24, 30], [1, 6, 17, 21, 23, 27], [1, 7, 7, 7, 14, 41], [1, 7, 7, 9, 9, 42], [1, 7, 7, 9, 18, 39], [1, 7, 7, 11, 19, 38], [1, 7, 7, 14, 19, 37], [1, 7, 7, 17, 26, 31], [1, 7, 7, 25, 25, 26], [1, 7, 9, 15, 15, 38], [1, 7, 9, 18, 27, 29], [1, 7, 10, 11, 23, 35], [1, 7, 10, 17, 19, 35], [1, 7, 10, 17, 25, 31], [1, 7, 10, 25, 25, 25], [1, 7, 11, 11, 17, 38], [1, 7, 11, 13, 23, 34], [1, 7, 11, 14, 17, 37], [1, 7, 11, 15, 27, 30], [1, 7, 11, 18, 21, 33], [1, 7, 11, 22, 23, 29], [1, 7, 13, 13, 26, 31], [1, 7, 13, 17, 19, 34], [1, 7, 13, 17, 26, 29], [1, 7, 13, 19, 22, 31], [1, 7, 14, 17, 23, 31], [1, 7, 14, 23, 25, 25], [1, 7, 15, 15, 25, 30], [1, 7, 17, 19, 22, 29], [1, 8, 8, 8, 26, 34], [1, 8, 8, 10, 14, 40], [1, 8, 8, 14, 16, 38], [1, 8, 8, 14, 26, 32], [1, 8, 8, 16, 22, 34], [1, 8, 10, 20, 26, 28], [1, 8, 12, 14, 18, 36], [1, 8, 14, 14, 28, 28], [1, 8, 14, 16, 22, 32], [1, 8, 18, 22, 24, 24], [1, 8, 20, 20, 22, 26], [1, 9, 9, 9, 10, 41], [1, 9, 9, 9, 25, 34], [1, 9, 9, 11, 29, 30], [1, 9, 9, 13, 18, 37], [1, 9, 9, 14, 21, 35], [1, 9, 9, 15, 26, 31], [1, 9, 9, 17, 22, 33], [1, 9, 10, 15, 23, 33], [1, 9, 10, 21, 21, 31], [1, 9, 11, 15, 21, 34], [1, 9, 13, 15, 18, 35], [1, 9, 13, 18, 19, 33], [1, 9, 14, 17, 27, 27], [1, 9, 15, 17, 23, 30], [1, 9, 17, 21, 22, 27], [1, 9, 18, 19, 23, 27], [1, 10, 10, 16, 28, 28], [1, 10, 10, 20, 20, 32], [1, 10, 11, 11, 29, 29], [1, 10, 11, 17, 17, 35], [1, 10, 13, 13, 19, 35], [1, 10, 13, 13, 25, 31], [1, 10, 13, 15, 21, 33], [1, 10, 13, 17, 25, 29], [1, 10, 14, 24, 24, 24], [1, 10, 15, 21, 23, 27], [1, 10, 16, 16, 16, 34], [1, 10, 16, 20, 22, 28], [1, 10, 19, 19, 19, 29], [1, 11, 11, 13, 13, 38], [1, 11, 11, 14, 19, 35], [1, 11, 11, 14, 25, 31], [1, 11, 11, 18, 27, 27], [1, 11, 11, 21, 21, 30], [1, 11, 13, 13, 14, 37], [1, 11, 13, 17, 17, 34], [1, 11, 13, 17, 22, 31], [1, 11, 13, 22, 25, 25], [1, 11, 13, 23, 23, 26], [1, 11, 15, 18, 25, 27], [1, 11, 17, 17, 22, 29], [1, 11, 19, 22, 23, 23], [1, 12, 12, 16, 18, 34], [1, 12, 12, 22, 24, 26], [1, 12, 14, 18, 24, 28], [1, 12, 16, 18, 20, 30], [1, 13, 13, 13, 19, 34], [1, 13, 13, 13, 26, 29], [1, 13, 13, 14, 23, 31], [1, 13, 13, 19, 22, 29], [1, 13, 14, 17, 23, 29], [1, 13, 15, 15, 26, 27], [1, 13, 15, 17, 21, 30], [1, 13, 17, 19, 23, 26], [1, 13, 18, 19, 21, 27], [1, 14, 15, 15, 17, 33], [1, 14, 17, 17, 17, 31], [1, 14, 17, 17, 25, 25], [1, 15, 15, 17, 18, 31], [1, 15, 15, 18, 25, 25], [1, 15, 15, 19, 22, 27], [1, 16, 20, 20, 22, 22], [1, 17, 18, 21, 21, 23], [1, 17, 19, 19, 22, 23], [1, 18, 18, 20, 20, 24], [2, 2, 2, 2, 28, 35], [2, 2, 2, 4, 29, 34], [2, 2, 2, 8, 10, 43], [2, 2, 2, 13, 20, 38], [2, 2, 3, 6, 6, 44], [2, 2, 3, 6, 26, 36], [2, 2, 3, 10, 12, 42], [2, 2, 3, 18, 28, 30], [2, 2, 4, 4, 7, 44], [2, 2, 4, 4, 31, 32], [2, 2, 4, 8, 16, 41], [2, 2, 4, 10, 26, 35], [2, 2, 4, 13, 26, 34], [2, 2, 4, 14, 19, 38], [2, 2, 4, 16, 28, 31], [2, 2, 4, 19, 22, 34], [2, 2, 4, 22, 26, 29], [2, 2, 5, 8, 22, 38], [2, 2, 5, 14, 14, 40], [2, 2, 5, 22, 22, 32], [2, 2, 6, 6, 24, 37], [2, 2, 6, 18, 19, 36], [2, 2, 6, 24, 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11) Nombre de Kaprekar
Nombre de Kaprekar —
Un nombre \(N\) à \(D\) chiffres est un nombre de Kaprekar si en partageant son carré en deux nombres de \(D\) chiffres, la somme de ces deux nombres est égale à \(N\) .
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Un nombre de Kaprekar est un entier naturel non négatif \(N\) tel que, si on élève \(N\) au carré, on peut diviser le résultat en deux parties (gauche et droite), et la somme de ces deux parties est égale à \(N\).
Exemple avec 45
- \(45^2 = 2025\),
- Séparons \(2025\) en \(A = 20\) et \(B = 25\),
- La somme \(A + B = 20 + 25 = 45\),
- Donc \(45\) est un nombre de Kaprekar.
Racines carrés magiques
Pour ces nombres, la racine se calcule facilement:
- Prendre la partie gauche du nombre, et
- Ajouter la partie droite.
Exemples avec \(45^2 = 2025\)
- Séparons \(2025\) en deux parties : \(A = 20\) et \(B = 25\).
- La racine carrée est donnée par : \[ A + B = 20 + 25 = 45 \]
- Ainsi, \(45\) est la racine carrée de \(2025\).
Pour en savoir plus => Les nombres de Kaprekar.
