American Citizen on Mon, 17 Nov 2025 02:32:58 +0100


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N = a^2 + b^2 + c^2 question

In trying to obtain a fast but exhaustive algorithm for finding 3 squares which sum to a given number n, I found that using n = 416666, I obtained 339 unique representations of [a,b,c] such that a^2+b^2+c^2 = n. 

Can anyone verify that this count is correct for n?

- Randall
k=[[0, 145, 629], [3, 76, 641], [3, 136, 631], [4, 25, 645], [4, 183, 619], [4, 225, 605], [4, 349, 543], [4, 367, 531], [4, 407, 501], [5, 196, 615], [5, 321, 560], [7, 176, 621], [7, 264, 589], [8, 91, 639], [8, 381, 521], [9, 43, 644], [9, 56, 643], [9, 104, 637], [9, 131, 632], [9, 149, 628], [9, 208, 611], [9, 299, 572], [9, 341, 548], [9, 352, 541], [9, 364, 533], [9, 413, 496], [9, 427, 484], [12, 281, 581], [13, 336, 551], [15, 179, 620], [16, 189, 617], [16, 219, 607], [17, 109, 636], [17, 144, 629], [20, 175, 621], [20, 329, 555], [21, 160, 625], [21, 247, 596], [21, 404, 503], [24, 143, 629], [24, 263, 589], [25, 300, 571], [27, 191, 616], [28, 351, 541], [29, 33, 644], [29, 212, 609], [29, 360, 535], [32, 69, 641], [35, 225, 604], [35, 440, 471], [36, 67, 641], [36, 199, 613], [36, 331, 553], [36, 371, 527], [37, 161, 624], [37, 289, 576], [41, 96, 637], [41, 452, 459], [43, 256, 591], [43, 311, 564], [43, 401, 504], [45, 71, 640], [47, 364, 531], [48, 59, 641], [48, 301, 569], [49, 141, 628], [49, 243, 596], [49, 264, 587], [49, 331, 552], [51, 121, 632], [51, 433, 476], [55, 171, 620], [55, 280, 579], [56, 167, 621], [56, 239, 597], [56, 393, 509], [57, 269, 584], [59, 228, 601], [59, 344, 543], [59, 423, 484], [60, 295, 571], [60, 365, 529], [61, 68, 639], [61, 183, 616], [61, 329, 552], [61, 383, 516], [64, 109, 633], [64, 441, 467], [65, 96, 635], [65, 229, 600], [67, 151, 624], [67, 321, 556], [67, 416, 489], [69, 217, 604], [69, 353, 536], [69, 359, 532], [71, 136, 627], [71, 165, 620], [71, 240, 595], [71, 332, 549], [71, 348, 539], [71, 397, 504], [71, 420, 485], [73, 451, 456], [75, 215, 604], [76, 101, 633], [76, 299, 567], [76, 387, 511], [77, 79, 636], [77, 281, 576], [77, 369, 524], [77, 376, 519], [79, 252, 589], [79, 320, 555], [80, 179, 615], [81, 256, 587], [81, 316, 557], [83, 441, 464], [84, 107, 631], [84, 293, 569], [85, 360, 529], [87, 116, 629], [87, 349, 536], [88, 249, 589], [89, 152, 621], [89, 251, 588], [91, 377, 516], [92, 399, 499], [93, 169, 616], [93, 439, 464], [95, 204, 605], [95, 421, 480], [96, 139, 623], [96, 329, 547], [96, 415, 485], [96, 433, 469], [97, 264, 581], [99, 196, 607], [99, 368, 521], [100, 105, 629], [100, 145, 621], [101, 319, 552], [104, 187, 609], [104, 375, 515], [107, 201, 604], [107, 271, 576], [107, 399, 496], [108, 241, 589], [108, 281, 571], [108, 419, 479], [108, 449, 451], [109, 368, 519], [109, 431, 468], [112, 299, 561], [113, 144, 619], [113, 371, 516], [116, 199, 603], [116, 363, 521], [116, 447, 451], [121, 128, 621], [121, 196, 603], [121, 205, 600], [121, 357, 524], [121, 420, 475], [123, 409, 484], [124, 257, 579], [124, 309, 553], [125, 129, 620], [125, 360, 521], [127, 189, 604], [127, 444, 451], [128, 231, 589], [128, 399, 491], [128, 411, 481], [129, 272, 571], [129, 421, 472], [131, 372, 511], [132, 161, 611], [133, 321, 544], [135, 404, 485], [136, 381, 503], [136, 433, 459], [137, 396, 491], [139, 151, 612], [139, 297, 556], [139, 399, 488], [140, 279, 565], [141, 416, 473], [144, 251, 577], [144, 281, 563], [144, 311, 547], [144, 391, 493], [145, 204, 595], [145, 296, 555], [145, 429, 460], [147, 184, 601], [147, 401, 484], [148, 339, 529], [149, 384, 497], [151, 256, 573], [151, 428, 459], [152, 321, 539], [153, 236, 581], [156, 229, 583], [156, 329, 533], [159, 163, 604], [159, 232, 581], [160, 221, 585], [160, 255, 571], [160, 429, 455], [161, 261, 568], [161, 404, 477], [164, 207, 589], [164, 347, 519], [165, 185, 596], [165, 304, 545], [165, 400, 479], [169, 437, 444], [171, 328, 529], [171, 416, 463], [172, 269, 561], [176, 367, 501], [177, 196, 589], [177, 299, 544], [179, 188, 591], [179, 249, 568], [179, 305, 540], [179, 360, 505], [179, 384, 487], [179, 433, 444], [180, 421, 455], [181, 207, 584], [181, 343, 516], [183, 404, 469], [183, 424, 451], [184, 243, 569], [185, 396, 475], [188, 361, 501], [189, 383, 484], [189, 407, 464], [191, 212, 579], [191, 336, 517], [191, 348, 509], [192, 409, 461], [193, 381, 484], [195, 304, 535], [195, 329, 520], [196, 265, 555], [196, 285, 545], [196, 365, 495], [196, 373, 489], [199, 269, 552], [199, 339, 512], [200, 225, 571], [201, 419, 448], [203, 209, 576], [204, 241, 563], [204, 287, 541], [204, 389, 473], [204, 427, 439], [209, 339, 508], [212, 219, 569], [212, 261, 551], [212, 399, 461], [212, 421, 441], [213, 311, 524], [213, 316, 521], [217, 296, 531], [219, 328, 511], [219, 377, 476], [220, 321, 515], [221, 223, 564], [221, 372, 479], [223, 256, 549], [225, 229, 560], [227, 279, 536], [229, 308, 519], [229, 313, 516], [229, 412, 441], [231, 251, 548], [233, 264, 541], [235, 300, 521], [236, 373, 471], [237, 284, 529], [239, 403, 444], [241, 267, 536], [241, 328, 501], [247, 299, 516], [249, 283, 524], [249, 347, 484], [251, 417, 424], [252, 349, 481], [255, 325, 496], [256, 263, 531], [256, 267, 529], [256, 389, 447], [256, 417, 421], [260, 271, 525], [264, 271, 523], [264, 293, 511], [264, 313, 499], [264, 359, 467], [268, 401, 429], [269, 396, 433], [275, 295, 504], [279, 368, 451], [280, 321, 485], [281, 284, 507], [281, 339, 472], [281, 364, 453], [281, 392, 429], [285, 404, 415], [288, 361, 451], [289, 316, 483], [292, 331, 471], [296, 357, 449], [299, 329, 468], [299, 336, 463], [299, 401, 408], [301, 303, 484], [303, 331, 464], [304, 311, 477], [304, 337, 459], [307, 399, 404], [308, 391, 411], [309, 368, 431], [316, 397, 399], [317, 329, 456], [319, 381, 412], [321, 332, 451], [321, 340, 445], [327, 364, 421], [329, 349, 432], [329, 380, 405], [336, 341, 433], [344, 373, 399], [345, 371, 400], [347, 351, 416], [369, 373, 376]];