Année 5778

Et 5778 est divisible par 27=5+7+7+8.

C'est si rare, les nombres entiers naturels qui sont sommes de quatre carrés entiers ?

Y'a une formule concernant le nombre de représentation ici

Ici, le nombre de représentation c'est $24$ fois la somme des diviseurs impairs de $5778$.

Ca fait un peu beaucoup quand même :-D

Bonne question :-)

$17^2+17^2+50^2+50^2$

Grâce à Apollonius et flipflop, on a $480$ représentations de $5778$.
Plus que $103200$ ! B-)-

Flipflop, tu as confondu $5778$ et $5578$.

On n'en est donc qu'à $384$ représentations !

Si on impose $1\leq a<b<c<d$ il n'y a que 233 solutions:

[01, 06, 29, 70], [01, 08, 23, 72], [01, 08, 33, 68], [01, 09, 40, 64], [01, 13, 42, 62], [01, 14, 35, 66], [01, 16, 36, 65], [01, 19, 50, 54], [01, 26, 50, 51], [01, 28, 32, 63], [01, 30, 34, 61], [02, 03, 17, 74], [02, 03, 49, 58], [02, 05, 50, 57], [02, 07, 10, 75], [02, 07, 37, 66], [02, 07, 53, 54], [02, 09, 43, 62], [02, 11, 18, 73], [02, 14, 33, 67], [02, 17, 42, 61], [02, 18, 31, 67], [02, 18, 35, 65], [02, 19, 38, 63], [02, 22, 23, 69], [02, 23, 42, 59], [02, 26, 43, 57], [02, 27, 41, 58], [02, 29, 33, 62], [02, 30, 43, 55], [02, 34, 37, 57], [02, 38, 39, 53], [03, 06, 42, 63], [03, 10, 38, 65], [03, 11, 32, 68], [03, 12, 21, 72], [03, 12, 45, 60], [03, 14, 47, 58], [03, 16, 32, 67], [03, 16, 52, 53], [03, 17, 46, 58], [03, 18, 33, 66], [03, 19, 28, 68], [03, 28, 43, 56], [03, 31, 38, 58], [03, 32, 44, 53], [03, 33, 42, 54], [03, 38, 46, 47], [04, 05, 51, 56], [04, 07, 23, 72], [04, 07, 33, 68], [04, 12, 17, 73], [04, 15, 49, 56], [04, 16, 45, 59], [04, 21, 35, 64], [04, 21, 40, 61], [04, 24, 31, 65], [04, 35, 44, 51], [05, 06, 26, 71], [05, 09, 14, 74], [05, 10, 18, 73], [05, 13, 20, 72], [05, 18, 23, 70], [05, 19, 36, 64], [05, 20, 27, 68], [05, 25, 42, 58], [05, 28, 37, 60], [05, 32, 45, 52], [05, 34, 41, 54], [05, 40, 43, 48], [06, 07, 43, 62], [06, 09, 30, 69], [06, 13, 47, 58], [06, 15, 51, 54], [06, 19, 34, 65], [06, 23, 37, 62], [06, 23, 43, 58], [06, 26, 29, 65], [06, 27, 42, 57], [06, 29, 49, 50], [06, 35, 46, 49], [07, 09, 32, 68], [07, 10, 27, 70], [07, 11, 42, 62], [07, 12, 16, 73], [07, 12, 31, 68], [07, 16, 17, 72], [07, 17, 48, 56], [07, 20, 48, 55], [07, 21, 38, 62], [07, 22, 42, 59], [07, 23, 24, 68], [07, 23, 40, 60], [07, 27, 34, 62], [07, 32, 48, 49], [07, 33, 44, 52], [07, 34, 42, 53], [07, 37, 38, 54], [07, 42, 43, 46], [08, 12, 23, 71], [08, 12, 43, 61], [08, 13, 19, 72], [08, 13, 28, 69], [08, 16, 47, 57], [08, 19, 27, 68], [08, 20, 33, 65], [08, 21, 28, 67], [08, 23, 33, 64], [08, 23, 44, 57], [08, 27, 43, 56], [08, 28, 31, 63], [08, 28, 41, 57], [08, 33, 40, 55], [09, 10, 11, 74], [09, 10, 46, 59], [09, 11, 26, 70], [09, 12, 48, 57],
[09, 16, 20, 71], [09, 17, 28, 68], [09, 21, 30, 66], [09, 22, 37, 62], [09, 22, 43, 58], [09, 24, 39, 60], [09, 25, 44, 56], [09, 28, 47, 52], [09, 31, 40, 56], [10, 14, 21, 71], [10, 17, 30, 67], [10, 17, 45, 58], [10, 19, 31, 66], [10, 19, 49, 54], [10, 22, 35, 63], [10, 26, 39, 59], [10, 26, 49, 51], [10, 31, 46, 51],
[10, 33, 35, 58], [10, 38, 45, 47], [11, 12, 32, 67], [11, 12, 52, 53], [11, 23, 42, 58], [11, 24, 40, 59], [11, 25, 26, 66], [11, 25, 46, 54], [11, 35, 36, 56],
[12, 13, 29, 68], [12, 13, 37, 64], [12, 17, 47, 56], [12, 19, 28, 67], [12, 20, 47, 55], [12, 23, 49, 52], [12, 24, 33, 63], [12, 25, 28, 65], [12, 27, 48, 51],
[12, 28, 37, 59], [12, 32, 47, 49], [12, 33, 36, 57], [12, 35, 40, 53], [13, 14, 38, 63], [13, 15, 22, 70], [13, 16, 27, 68], [13, 22, 30, 65], [13, 22, 34, 63],
[13, 22, 47, 54], [13, 26, 33, 62], [13, 27, 28, 64], [13, 28, 35, 60], [13, 30, 47, 50], [13, 33, 34, 58], [13, 37, 44, 48], [13, 38, 42, 49], [14, 21, 46, 55], [14, 22, 43, 57], [14, 30, 31, 61], [14, 34, 45, 49], [15, 20, 23, 68], [15, 22, 35, 62], [15, 26, 34, 61], [16, 19, 20, 69], [16, 19, 45, 56], [16, 21, 40, 59], [16, 23, 32, 63], [16, 24, 35, 61], [16, 37, 43, 48], [16, 39, 40, 49], [17, 18, 26, 67], [17, 18, 38, 61], [17, 19, 42, 58], [17, 24, 47, 52], [17, 28, 48, 49], [17, 30, 35, 58], [17, 35, 42, 50], [18, 21, 42, 57], [18, 23, 38, 59], [18, 23, 46, 53], [18, 33, 42, 51], [19, 20, 36, 61], [19, 22, 33, 62], [19, 30, 46, 49], [19, 38, 42, 47], [20, 21, 29, 64], [20, 23, 40, 57], [20, 25, 28, 63], [20, 29, 44, 51], [20, 35, 43, 48], [20, 36, 41, 49], [21, 23, 38, 58], [21, 28, 43, 52], [21, 30, 39, 54], [21, 34, 41, 50], [22, 23, 43, 54], [22, 26, 37, 57], [22, 29, 33, 58], [22, 30, 37, 55], [22, 41, 42, 43], [23, 25, 32, 60], [23, 26, 27, 62], [23, 26, 42, 53], [23, 27, 34, 58], [23, 30, 43, 50], [23, 32, 33, 56], [23, 32, 39, 52], [23, 37, 42, 46], [24, 29, 35, 56], [24, 32, 37, 53], [25, 30, 38, 53], [26, 31, 35, 54], [26, 31, 45, 46], [28, 29, 43, 48], [28, 32, 37, 51], [28, 37, 40, 45], [30, 33, 42, 45], [30, 35, 38, 47], [31, 32, 33, 52]

Avec $3$ entiers naturels (non nécessairement distincts) rangés dans l'ordre croissant, il y a $18$ solutions :

[1, 1, 76], [1, 41, 64], [3, 12, 75], [7, 20, 73], [7, 52, 55], [8, 35, 67], [11, 44, 61], [15, 48, 57], [19, 44, 59], [21, 24, 69],
[23, 25, 68], [23, 32, 65], [24, 51, 51], [29, 29, 64], [31, 41, 56], [33, 33, 60], [35, 43, 52], [37, 40, 53]

Avec $4$ entiers naturels (non nécessairement distincts) rangés dans l'ordre croissant, il y a $291$ solutions :

[1, 6, 29, 70], [1, 8, 23, 72], [1, 8, 33, 68], [1, 9, 40, 64], [1, 13, 42, 62], [1, 14, 35, 66], [1, 16, 36, 65], [1, 19, 50, 54], [1, 26, 50, 51], [1, 28, 32, 63], [1, 30, 34, 61], [2, 2, 21, 73], [2, 2, 27, 71], [2, 3, 17, 74], [2, 3, 49, 58], [2, 5, 50, 57], [2, 7, 10, 75], [2, 7, 37, 66], [2, 7, 53, 54], [2, 9, 43, 62], [2, 11, 18, 73], [2, 14, 33, 67], [2, 17, 42, 61], [2, 18, 31, 67], [2, 18, 35, 65], [2, 19, 38, 63], [2, 22, 23, 69], [2, 23, 42, 59], [2, 26, 43, 57], [2, 27, 41, 58], [2, 29, 33, 62], [2, 30, 43, 55], [2, 34, 37, 57], [2, 38, 39, 53], [3, 3, 24, 72], [3, 6, 42, 63], [3, 10, 38, 65], [3, 11, 32, 68], [3, 12, 21, 72], [3, 12, 45, 60], [3, 14, 47, 58], [3, 16, 32, 67], [3, 16, 52, 53], [3, 17, 46, 58], [3, 18, 33, 66], [3, 19, 28, 68], [3, 19, 52, 52], [3, 28, 43, 56], [3, 30, 30, 63], [3, 31, 38, 58], [3, 32, 32, 61], [3, 32, 44, 53], [3, 33, 42, 54], [3, 38, 46, 47], [4, 4, 11, 75], [4, 4, 39, 65], [4, 4, 45, 61], [4, 5, 51, 56], [4, 7, 23, 72], [4, 7, 33, 68], [4, 12, 17, 73], [4, 12, 53, 53], [4, 15, 49, 56], [4, 16, 45, 59], [4, 17, 17, 72], [4, 21, 35, 64], [4, 21, 40, 61], [4, 24, 31, 65], [4, 35, 44, 51], [5, 6, 26, 71], [5, 8, 8, 75], [5, 9, 14, 74], [5, 10, 18, 73], [5, 13, 20, 72], [5, 18, 23, 70], [5, 19, 36, 64], [5, 20, 27, 68], [5, 25, 42, 58], [5, 28, 37, 60], [5, 32, 45, 52], [5, 34, 41, 54], [5, 39, 46, 46], [5, 40, 43, 48], [6, 6, 9, 75], [6, 7, 43, 62], [6, 9, 30, 69], [6, 13, 47, 58], [6, 15, 51, 54], [6, 19, 34, 65], [6, 23, 37, 62], [6, 23, 43, 58], [6, 26, 29, 65], [6, 27, 42, 57], [6, 29, 49, 50], [6, 35, 46, 49], [7, 9, 32, 68], [7, 10, 27, 70], [7, 11, 42, 62], [7, 12, 16, 73], [7, 12, 31, 68], [7, 16, 17, 72], [7, 17, 48, 56], [7, 20, 48, 55], [7, 21, 38, 62], [7, 22, 22, 69], [7, 22, 42, 59], [7, 23, 24, 68], [7, 23, 40, 60], [7, 27, 34, 62], [7, 27, 50, 50], [7, 32, 48, 49], [7, 33, 44, 52], [7, 34, 42, 53], [7, 37, 38, 54], [7, 42, 43, 46], [8, 8, 41, 63], [8, 8, 49, 57], [8, 12, 23, 71], [8, 12, 43, 61], [8, 13, 19, 72], [8, 13, 28, 69], [8, 16, 47, 57], [8, 19, 27, 68], [8, 20, 33, 65], [8, 21, 28, 67], [8, 23, 33, 64], [8, 23, 44, 57], [8, 27, 43, 56], [8, 28, 31, 63], [8, 28, 41, 57], [8, 33, 40, 55], [8, 36, 47, 47], [9, 10, 11, 74], [9, 10, 46, 59], [9, 11, 26, 70], [9, 12, 48, 57], [9, 16, 20, 71], [9, 17, 28, 68], [9, 17, 52, 52], [9, 21, 30, 66], [9, 22, 37, 62], [9, 22, 43, 58], [9, 24, 39, 60], [9, 25, 44, 56], [9, 28, 47, 52], [9, 31, 40, 56], [9, 33, 48, 48], [9, 38, 38, 53], [10, 10, 33, 67], [10, 14, 21, 71], [10, 17, 30, 67], [10, 17, 45, 58], [10, 19, 31, 66], [10, 19, 49, 54], [10, 22, 35, 63], [10, 26, 39, 59], [10, 26, 49, 51], [10, 31, 46, 51], [10, 33, 35, 58], [10, 38, 45, 47], [11, 11, 44, 60], [11, 12, 32, 67], [11, 12, 52, 53], [11, 23, 42, 58], [11, 24, 40, 59], [11, 25, 26, 66], [11, 25, 46, 54], [11, 35, 36, 56], [12, 12, 27, 69], [12, 12, 39, 63], [12, 13, 29, 68], [12, 13, 37, 64], [12, 15, 15, 72], [12, 17, 47, 56], [12, 19, 28, 67], [12, 20, 47, 55], [12, 23, 49, 52], [12, 24, 33, 63], [12, 25, 28, 65], [12, 27, 48, 51], [12, 28, 37, 59], [12, 32, 47, 49], [12, 33, 36, 57], [12, 35, 40, 53], [12, 43, 43, 44], [13, 13, 16, 72], [13, 13, 48, 56], [13, 14, 38, 63], [13, 15, 22, 70], [13, 16, 27, 68], [13, 22, 30, 65], [13, 22, 34, 63], [13, 22, 47, 54], [13, 26, 33, 62], [13, 27, 28, 64], [13, 28, 35, 60], [13, 30, 47, 50], [13, 33, 34, 58], [13, 37, 44, 48], [13, 38, 42, 49], [14, 14, 25, 69], [14, 21, 46, 55], [14, 22, 43, 57], [14, 30, 31, 61], [14, 34, 45, 49], [15, 16, 16, 71], [15, 20, 23, 68], [15, 22, 35, 62], [15, 26, 34, 61], [15, 41, 44, 44], [15, 42, 42, 45], [16, 19, 20, 69], [16, 19, 45, 56], [16, 21, 40, 59], [16, 23, 32, 63], [16, 24, 35, 61], [16, 31, 31, 60], [16, 37, 43, 48], [16, 39, 40, 49], [17, 17, 24, 68], [17, 17, 40, 60], [17, 18, 26, 67], [17, 18, 38, 61], [17, 19, 42, 58], [17, 24, 47, 52], [17, 28, 48, 49], [17, 30, 35, 58], [17, 35, 42, 50], [17, 38, 38, 51], [18, 21, 42, 57], [18, 23, 38, 59], [18, 23, 46, 53], [18, 33, 42, 51], [19, 20, 36, 61], [19, 22, 33, 62], [19, 30, 46, 49], [19, 38, 42, 47], [20, 21, 29, 64], [20, 23, 40, 57], [20, 24, 49, 49], [20, 25, 28, 63], [20, 29, 44, 51], [20, 35, 43, 48], [20, 36, 41, 49], [21, 21, 36, 60], [21, 23, 38, 58], [21, 27, 48, 48], [21, 28, 43, 52], [21, 30, 39, 54], [21, 34, 34, 55], [21, 34, 41, 50], [22, 22, 29, 63], [22, 22, 33, 61], [22, 22, 47, 51], [22, 23, 43, 54], [22, 26, 37, 57], [22, 29, 33, 58], [22, 30, 37, 55], [22, 41, 42, 43], [23, 25, 32, 60], [23, 26, 27, 62], [23, 26, 42, 53], [23, 27, 34, 58], [23, 30, 43, 50], [23, 32, 33, 56], [23, 32, 39, 52], [23, 37, 42, 46], [24, 24, 45, 51], [24, 28, 47, 47], [24, 29, 35, 56], [24, 32, 37, 53], [25, 30, 38, 53], [26, 26, 45, 49], [26, 31, 35, 54], [26, 31, 45, 46], [27, 28, 28, 59], [27, 30, 30, 57], [27, 39, 42, 42], [27, 40, 40, 43], [28, 28, 31, 57], [28, 29, 43, 48], [28, 32, 37, 51], [28, 36, 43, 43], [28, 37, 40, 45], [30, 33, 42, 45], [30, 35, 38, 47], [31, 32, 33, 52], [32, 32, 39, 47], [32, 35, 35, 48], [33, 33, 36, 48], [37, 38, 38, 39]

Moi c'est la remarque de Sylvain que je trouve intéressante
https://oeis.org/A005349
https://en.wikipedia.org/wiki/Harshad_number