质数20260319可以表示为四个非负整数w、x、y、z的平方和， 求该四个非负整数组合数.

popo987654

请教各位方法，如何解法？

风花飘飘

思路：
平方数的末位数字不可能是2,3,7,8，在一个有限大的数字内分类排列组合排查，
这个方法应该是可以的吧？

风花飘飘

思路：
平方数的末位数字不可能是2,3,7,8，在一个有限大的数字内分类排列组合排查，
这个方法确实是可以的！

分析了一下，其中的的一个只能固定不变是4501？
剩下的三个只能是：偶数+奇数+奇数，
最后“用夹逼使劲夹逼”把水挤干，结果只有3组解：


等着对答案……

风花飘飘

算一算：

风花飘飘

这个生日的人，在29岁达到人生巅峰，宇宙幸运密码是29。

王守恩

题目。质数 20260319 可以表示为四个非负整数 x, y, z, w 的平方和。求 0 ≤ x ≤ y ≤ z ≤ w 组合(无序)数。

答:  组合(无序)数 = 421793 + 594 = 422387。

4个数不同。x1 = a + b + c + d——有 421793 种可能。——这个——人工可以有。

2个数相同。x2 = a + a + c + d——有 594 种可能。——关键是这个——人工可以有吗？

2对数相同。x3 = a + a + c + c——有 0 种可能。——人工可以有。

3个数相同。x4 = a + a + a + d——有 0 种可能。——人工可以有。

4个数相同。x5 = a + a + a + a——有 0 种可能。——人工可以有。

基本公式——8*(20260319 + 1) = 384*x1 + 192*x2 + 96*x3 + 64*x4 + 16*x5。

其中:384, 192, 96, 64, 16————人工可以有。

2 个数相同——有 594 种可能。
1, 3, 3, 670, 4451,
2, 5, 927, 927, 4306,
3, 5, 1383, 1383, 4054,
4, 5, 1562, 2985, 2985,
5, 5, 1714, 2943, 2943,
6, 9, 9, 34, 4501,
7, 9, 9, 2506, 3739,
8, 13, 1310, 3045, 3045,
9, 13, 2475, 2475, 2830,
10, 19, 579, 579, 4426,
11, 21, 751, 751, 4374,
12, 26, 1797, 1797, 3715,
13, 26, 2189, 2781, 2781,
14, 29, 386, 3171, 3171,
15, 29, 459, 459, 4454,
16, 29, 910, 3117, 3117,
17, 30, 535, 535, 4437,
18, 33, 33, 1246, 4325,
19, 33, 33, 2165, 3946,
20, 35, 2234, 2763, 2763,
21, 37, 1870, 2895, 2895,
22, 49, 49, 294, 4491,
23, 49, 49, 699, 4446,
24, 49, 49, 1854, 4101,
25, 49, 49, 2709, 3594,
26, 51, 51, 1309, 4306,
27, 54, 141, 3181, 3181,
28, 54, 1867, 1867, 3645,
29, 59, 59, 594, 4461,
30, 59, 59, 1059, 4374,
31, 62, 2045, 2835, 2835,
32, 63, 63, 1525, 4234,
33, 69, 894, 3119, 3119,
34, 69, 1039, 1039, 4254,
35, 69, 1170, 3073, 3073,
36, 69, 2273, 2273, 3150,
37, 70, 1647, 1647, 3851,
38, 70, 2583, 2583, 2629,
39, 81, 81, 811, 4426,
40, 85, 85, 3150, 3213,
41, 86, 2291, 2739, 2739,
42, 91, 1689, 1689, 3814,
43, 93, 93, 1190, 4339,
44, 95, 95, 2130, 3963,
45, 99, 114, 3181, 3181,
46, 99, 2130, 2803, 2803,
47, 105, 105, 1238, 4325,
48, 105, 105, 1765, 4138,
49, 106, 539, 3159, 3159,
50, 110, 843, 843, 4339,
51, 110, 2067, 2067, 3421,
52, 111, 111, 539, 4466,
53, 114, 257, 257, 4485,
54, 115, 525, 525, 4438,
55, 121, 121, 1461, 4254,
56, 121, 121, 3066, 3291,
57, 123, 123, 2045, 4006,
58, 125, 387, 387, 4466,
59, 125, 2565, 2565, 2662,
60, 134, 1809, 1809, 3701,
61, 135, 135, 1090, 4363,
62, 135, 135, 1435, 4262,
63, 141, 149, 149, 4494,
64, 141, 2011, 2011, 3486,
65, 141, 2579, 2579, 2634,
66, 146, 811, 3129, 3129,
67, 147, 147, 2150, 3949,
68, 149, 921, 921, 4306,
69, 157, 157, 1875, 4086,
70, 157, 157, 2925, 3414,
71, 159, 159, 3086, 3269,
72, 163, 1150, 3075, 3075,
73, 165, 165, 2738, 3565,
74, 165, 165, 3170, 3187,
75, 166, 459, 459, 4451,
76, 166, 2515, 2637, 2637,
77, 174, 1039, 1039, 4251,
78, 174, 1059, 3091, 3091,
79, 179, 2450, 2667, 2667,
80, 183, 183, 1771, 4130,
81, 187, 1430, 3015, 3015,
82, 189, 869, 869, 4326,
83, 195, 294, 3173, 3173,
84, 201, 201, 2546, 3701,
85, 201, 201, 2606, 3659,
86, 203, 203, 1974, 4035,
87, 205, 1594, 2973, 2973,
88, 205, 1845, 1845, 3662,
89, 207, 207, 386, 4475,
90, 211, 1174, 3069, 3069,
91, 219, 810, 3127, 3127,
92, 226, 303, 303, 4475,
93, 226, 691, 3141, 3141,
94, 226, 1647, 1647, 3845,
95, 226, 2099, 2811, 2811,
96, 229, 229, 2154, 3939,
97, 229, 229, 2829, 3486,
98, 230, 1773, 1773, 3731,
99, 237, 1075, 1075, 4230,
100, 237, 2525, 2525, 2730,
101, 238, 1065, 1065, 4235,
102, 245, 303, 303, 4474,
103, 247, 247, 1710, 4149,
104, 257, 257, 930, 4389,
105, 257, 257, 2010, 4011,
106, 257, 257, 2886, 3435,
107, 273, 273, 3070, 3269,
108, 275, 275, 813, 4410,
109, 275, 837, 837, 4334,
110, 275, 1005, 1005, 4262,
111, 286, 469, 3159, 3159,
112, 286, 1071, 1071, 4229,
113, 297, 297, 901, 4390,
114, 297, 297, 1174, 4325,
115, 299, 801, 801, 4346,
116, 299, 1990, 2847, 2847,
117, 301, 1059, 1059, 4234,
118, 301, 1246, 3051, 3051,
119, 306, 1831, 1831, 3669,
120, 309, 1734, 2929, 2929,
121, 309, 2263, 2263, 3150,
122, 310, 2475, 2475, 2813,
123, 311, 311, 2694, 3579,
124, 314, 1685, 2943, 2943,
125, 314, 2307, 2307, 3085,
126, 315, 315, 605, 4438,
127, 315, 315, 2387, 3790,
128, 319, 319, 1686, 4149,
129, 327, 327, 2294, 3845,
130, 331, 2511, 2511, 2746,
131, 333, 333, 650, 4429,
132, 334, 1509, 1509, 3949,
133, 338, 1525, 2985, 2985,
134, 339, 339, 1069, 4346,
135, 345, 345, 1930, 4037,
136, 349, 1450, 3003, 3003,
137, 349, 2229, 2229, 3194,
138, 362, 1125, 1125, 4195,
139, 362, 1835, 2895, 2895,
140, 366, 1579, 1579, 3891,
141, 366, 2325, 2713, 2713,
142, 383, 383, 2154, 3915,
143, 386, 1449, 1449, 3989,
144, 386, 1811, 2901, 2901,
145, 387, 387, 1085, 4334,
146, 387, 450, 3155, 3155,
147, 387, 1590, 2965, 2965,
148, 390, 1823, 1823, 3669,
149, 390, 2263, 2263, 3141,
150, 394, 611, 3141, 3141,
151, 394, 1923, 1923, 3565,
152, 399, 399, 2954, 3349,
153, 405, 1993, 1993, 3486,
154, 406, 1555, 2973, 2973,
155, 406, 2169, 2169, 3269,
156, 409, 409, 891, 4374,
157, 411, 411, 854, 4381,
158, 411, 411, 1874, 4051,
159, 414, 1543, 1543, 3915,
160, 419, 2254, 2739, 2739,
161, 423, 423, 706, 4405,
162, 423, 423, 1070, 4331,
163, 423, 423, 1094, 4325,
164, 423, 423, 1450, 4219,
165, 426, 789, 3119, 3119,
166, 427, 427, 669, 4410,
167, 427, 427, 1485, 4206,
168, 427, 427, 1869, 4050,
169, 427, 427, 2694, 3555,
170, 429, 1314, 3029, 3029,
171, 434, 789, 789, 4339,
172, 437, 437, 1266, 4275,
173, 445, 445, 915, 4362,
174, 445, 445, 3138, 3165,
175, 446, 2389, 2679, 2679,
176, 450, 1507, 1507, 3939,
177, 459, 459, 1006, 4339,
178, 459, 459, 1139, 4306,
179, 462, 1875, 2875, 2875,
180, 465, 465, 890, 4363,
181, 469, 1521, 1521, 3926,
182, 470, 1347, 1347, 4051,
183, 470, 2539, 2607, 2607,
184, 477, 477, 2590, 3619,
185, 481, 481, 1314, 4251,
186, 481, 481, 1926, 4011,
187, 485, 862, 3105, 3105,
188, 485, 1977, 1977, 3494,
189, 491, 814, 3111, 3111,
190, 495, 495, 2245, 3838,
191, 495, 495, 2330, 3787,
192, 499, 499, 1779, 4074,
193, 499, 499, 3066, 3219,
194, 499, 2109, 2109, 3334,
195, 499, 2571, 2571, 2606,
196, 501, 501, 2206, 3859,
197, 510, 607, 607, 4389,
198, 510, 2065, 2065, 3387,
199, 513, 513, 2050, 3941,
200, 513, 513, 2555, 3634,
201, 525, 525, 862, 4355,
202, 525, 525, 1987, 3970,
203, 525, 525, 2810, 3437,
204, 525, 653, 653, 4374,
205, 525, 797, 797, 4326,
206, 526, 1953, 1953, 3515,
207, 526, 2541, 2541, 2659,
208, 530, 627, 627, 4381,
209, 531, 2394, 2669, 2669,
210, 535, 535, 2805, 3438,
211, 539, 1311, 1311, 4066,
212, 539, 1839, 1839, 3634,
213, 539, 2090, 2793, 2793,
214, 539, 2457, 2457, 2810,
215, 541, 1509, 1509, 3926,
216, 541, 1941, 1941, 3526,
217, 553, 553, 2325, 3774,
218, 555, 2154, 2767, 2767,
219, 555, 2425, 2425, 2862,
220, 562, 2235, 2235, 3155,
221, 562, 2565, 2565, 2605,
222, 565, 663, 663, 4366,
223, 565, 1137, 1137, 4166,
224, 566, 2151, 2151, 3269,
225, 566, 2181, 2181, 3229,
226, 567, 567, 1490, 4171,
227, 574, 1981, 2829, 2829,
228, 581, 581, 3021, 3234,
229, 581, 831, 831, 4306,
230, 581, 999, 999, 4234,
231, 585, 585, 1430, 4187,
232, 589, 1851, 1851, 3614,
233, 591, 591, 821, 4346,
234, 599, 599, 2379, 3726,
235, 599, 599, 2466, 3669,
236, 605, 1162, 3045, 3045,
237, 607, 607, 1461, 4170,
238, 607, 607, 2715, 3486,
239, 607, 607, 2805, 3414,
240, 611, 2469, 2469, 2774,
241, 621, 621, 2126, 3869,
242, 621, 621, 3094, 3149,
243, 621, 930, 3083, 3083,
244, 627, 627, 1594, 4115,
245, 630, 2317, 2317, 3021,
246, 635, 635, 1437, 4170,
247, 635, 635, 3075, 3162,
248, 635, 2097, 2097, 3326,
249, 646, 1261, 3021, 3021,
250, 651, 651, 1219, 4234,
251, 651, 651, 1954, 3949,
252, 653, 653, 2610, 3549,
253, 663, 663, 1666, 4075,
254, 663, 663, 2485, 3634,
255, 663, 663, 2875, 3334,
256, 666, 1659, 2921, 2921,
257, 666, 1831, 1831, 3621,
258, 669, 669, 691, 4346,
259, 669, 1974, 2821, 2821,
260, 670, 1219, 3027, 3027,
261, 675, 675, 1763, 4030,
262, 685, 843, 843, 4286,
263, 685, 938, 3075, 3075,
264, 685, 1155, 1155, 4138,
265, 685, 1874, 2853, 2853,
266, 691, 1971, 1971, 3466,
267, 691, 2186, 2739, 2739,
268, 693, 693, 1714, 4045,
269, 693, 693, 2995, 3214,
270, 697, 697, 1755, 4026,
271, 697, 697, 2325, 3726,
272, 707, 707, 1539, 4110,
273, 714, 2093, 2093, 3315,
274, 715, 777, 777, 4306,
275, 715, 2055, 2055, 3362,
276, 717, 717, 2554, 3565,
277, 717, 717, 3046, 3155,
278, 723, 723, 1469, 4130,
279, 723, 723, 1750, 4019,
280, 730, 2427, 2427, 2819,
281, 730, 2563, 2565, 2565,
282, 733, 733, 954, 4275,
283, 733, 733, 1170, 4221,
284, 750, 1413, 2975, 2975,
285, 750, 2165, 2165, 3213,
286, 753, 753, 2330, 3701,
287, 759, 759, 1406, 4139,
288, 761, 761, 1374, 4149,
289, 761, 761, 1701, 4026,
290, 762, 1985, 1985, 3435,
291, 770, 1197, 1197, 4099,
292, 770, 1931, 2823, 2823,
293, 771, 810, 3083, 3083,
294, 771, 1067, 1067, 4170,
295, 771, 1949, 1949, 3474,
296, 771, 2154, 2741, 2741,
297, 777, 777, 1306, 4165,
298, 797, 797, 1974, 3885,
299, 797, 797, 2499, 3570,
300, 801, 801, 2309, 3694,
301, 805, 805, 1638, 4035,
302, 810, 1779, 2867, 2867,
303, 810, 2515, 2515, 2637,
304, 811, 1239, 1239, 4066,
305, 811, 1510, 2943, 2943,
306, 814, 849, 849, 4261,
307, 819, 819, 1469, 4094,
308, 821, 1090, 3033, 3033,
309, 821, 1383, 1383, 3970,
310, 821, 2186, 2721, 2721,
311, 823, 823, 1419, 4110,
312, 826, 1835, 2847, 2847,
313, 831, 831, 2539, 3526,
314, 838, 1215, 1215, 4075,
315, 838, 1545, 1545, 3845,
316, 838, 1955, 2805, 2805,
317, 843, 843, 1114, 4195,
318, 843, 843, 1661, 4010,
319, 849, 849, 2506, 3541,
320, 850, 1953, 1953, 3451,
321, 851, 851, 1059, 4206,
322, 861, 861, 1981, 3854,
323, 861, 1507, 1507, 3870,
324, 862, 1845, 1845, 3565,
325, 862, 2395, 2625, 2625,
326, 869, 869, 966, 4221,
327, 869, 869, 1134, 4179,
328, 869, 869, 1869, 3906,
329, 869, 1239, 1239, 4054,
330, 873, 873, 2555, 3494,
331, 874, 921, 921, 4219,
332, 874, 1953, 1953, 3445,
333, 875, 993, 993, 4186,
334, 885, 885, 1213, 4150,
335, 887, 887, 891, 4230,
336, 887, 887, 2970, 3141,
337, 890, 1069, 3027, 3027,
338, 891, 891, 1549, 4034,
339, 891, 891, 2749, 3334,
340, 891, 1206, 3001, 3001,
341, 891, 1854, 2831, 2831,
342, 894, 2173, 2173, 3165,
343, 899, 1190, 3003, 3003,
344, 905, 905, 2262, 3675,
345, 910, 975, 975, 4187,
346, 914, 1659, 1659, 3731,
347, 915, 915, 1363, 4090,
348, 915, 915, 2275, 3662,
349, 915, 915, 2387, 3590,
350, 915, 915, 2938, 3155,
351, 915, 1926, 2803, 2803,
352, 921, 921, 2206, 3701,
353, 921, 921, 3019, 3074,
354, 926, 1689, 1689, 3701,
355, 927, 927, 1030, 4181,
356, 930, 2093, 2093, 3261,
357, 930, 2425, 2425, 2763,
358, 931, 931, 1389, 4074,
359, 931, 931, 3021, 3066,
360, 931, 1310, 2973, 2973,
361, 931, 1581, 1581, 3794,
362, 938, 2535, 2535, 2555,
363, 946, 1341, 1341, 3971,
364, 946, 2289, 2289, 2981,
365, 954, 1059, 3019, 3019,
366, 954, 2259, 2669, 2669,
367, 954, 2301, 2651, 2651,
368, 957, 957, 1939, 3830,
369, 959, 959, 1266, 4101,
370, 977, 977, 981, 4170,
371, 977, 977, 2619, 3390,
372, 981, 1134, 3001, 3001,
373, 981, 2191, 2191, 3114,
374, 981, 2327, 2327, 2910,
375, 985, 985, 2085, 3738,
376, 985, 985, 2613, 3390,
377, 986, 1499, 2919, 2919,
378, 986, 1521, 1521, 3829,
379, 987, 987, 1634, 3955,
380, 987, 987, 2765, 3266,
381, 994, 2259, 2259, 3011,
382, 995, 1677, 1677, 3694,
383, 995, 2246, 2667, 2667,
384, 1005, 1005, 1763, 3890,
385, 1006, 2019, 2019, 3331,
386, 1014, 1821, 2821, 2821,
387, 1014, 1949, 1949, 3411,
388, 1017, 1017, 1846, 3845,
389, 1017, 1017, 2645, 3346,
390, 1029, 1183, 1183, 4050,
391, 1029, 2310, 2633, 2633,
392, 1034, 1137, 1137, 4075,
393, 1034, 1341, 1341, 3949,
394, 1035, 1615, 1615, 3738,
395, 1035, 2345, 2345, 2862,
396, 1039, 1039, 1941, 3786,
397, 1039, 1039, 2154, 3669,
398, 1041, 1041, 1349, 4034,
399, 1059, 1059, 2381, 3514,
400, 1062, 1245, 2965, 2965,
401, 1062, 2425, 2425, 2715,
402, 1067, 1067, 2595, 3354,
403, 1070, 1339, 2943, 2943,
404, 1070, 1545, 1545, 3787,
405, 1070, 1743, 1743, 3611,
406, 1083, 1083, 1246, 4045,
407, 1083, 1083, 1955, 3754,
408, 1086, 1779, 2821, 2821,
409, 1089, 1089, 1154, 4069,
410, 1090, 1181, 2973, 2973,
411, 1090, 1257, 1257, 3989,
412, 1094, 1931, 2769, 2769,
413, 1095, 1095, 1787, 3830,
414, 1095, 1095, 2987, 2990,
415, 1099, 1449, 1449, 3854,
416, 1113, 1113, 2290, 3541,
417, 1113, 1113, 2891, 3070,
418, 1114, 2079, 2079, 3221,
419, 1129, 1129, 2214, 3579,
420, 1131, 1131, 2771, 3166,
421, 1134, 2219, 2219, 3021,
422, 1134, 2263, 2263, 2955,
423, 1137, 1137, 2459, 3410,
424, 1137, 1137, 2830, 3109,
425, 1139, 2181, 2181, 3074,
426, 1141, 1239, 1239, 3986,
427, 1154, 1981, 2739, 2739,
428, 1155, 1155, 2045, 3662,
429, 1155, 1283, 1283, 3954,
430, 1155, 1885, 1885, 3438,
431, 1173, 1173, 1906, 3725,
432, 1181, 1851, 1851, 3466,
433, 1189, 1874, 2769, 2769,
434, 1190, 1287, 1287, 3941,
435, 1190, 1763, 2805, 2805,
436, 1190, 1875, 1875, 3437,
437, 1190, 1891, 2763, 2763,
438, 1190, 1953, 1953, 3349,
439, 1190, 2403, 2403, 2701,
440, 1193, 1193, 1755, 3786,
441, 1197, 1197, 1651, 3830,
442, 1201, 1201, 2091, 3606,
443, 1201, 1201, 2946, 2949,
444, 1205, 1694, 2823, 2823,
445, 1206, 2115, 2677, 2677,
446, 1219, 1221, 1221, 3974,
447, 1219, 1490, 2877, 2877,
448, 1226, 2481, 2481, 2539,
449, 1227, 1227, 1510, 3869,
450, 1227, 1227, 2819, 3050,
451, 1229, 1229, 1731, 3774,
452, 1229, 1229, 2589, 3246,
453, 1237, 1237, 1491, 3870,
454, 1237, 1425, 1425, 3830,
455, 1238, 1785, 1785, 3515,
456, 1239, 1239, 2629, 3206,
457, 1246, 1291, 2919, 2919,
458, 1251, 1251, 1981, 3634,
459, 1251, 1251, 2899, 2954,
460, 1251, 2154, 2651, 2651,
461, 1259, 2049, 2049, 3206,
462, 1261, 1330, 2907, 2907,
463, 1261, 2181, 2181, 3026,
464, 1261, 2486, 2499, 2499,
465, 1266, 1741, 1741, 3549,
466, 1274, 1947, 1947, 3325,
467, 1290, 1437, 2875, 2875,
468, 1291, 1449, 1449, 3794,
469, 1309, 1309, 1581, 3786,
470, 1311, 1311, 2389, 3334,
471, 1315, 1406, 2877, 2877,
472, 1330, 1353, 1353, 3851,
473, 1330, 1923, 1923, 3331,
474, 1330, 2141, 2637, 2637,
475, 1331, 1971, 1971, 3274,
476, 1331, 2427, 2427, 2590,
477, 1339, 1479, 1479, 3754,
478, 1346, 1719, 1719, 3541,
479, 1346, 2101, 2649, 2649,
480, 1349, 1906, 2721, 2721,
481, 1353, 1353, 2626, 3115,
482, 1355, 1355, 2787, 2970,
483, 1355, 1743, 1743, 3514,
484, 1355, 2254, 2583, 2583,
485, 1363, 2290, 2565, 2565,
486, 1370, 2175, 2175, 2987,
487, 1371, 2119, 2119, 3066,
488, 1377, 1377, 2869, 2870,
489, 1386, 2325, 2543, 2543,
490, 1389, 1389, 2669, 3046,
491, 1389, 1389, 2749, 2974,
492, 1389, 1554, 2821, 2821,
493, 1413, 2425, 2425, 2550,
494, 1417, 1417, 1770, 3621,
495, 1417, 1417, 2235, 3354,
496, 1425, 1425, 2725, 2962,
497, 1426, 1661, 2781, 2781,
498, 1430, 1503, 1503, 3701,
499, 1430, 2187, 2187, 2941,
500, 1437, 1437, 2659, 3010,
501, 1437, 1885, 1885, 3330,
502, 1437, 2165, 2165, 2970,
503, 1438, 2415, 2415, 2555,
504, 1449, 1449, 1954, 3499,
505, 1454, 2301, 2301, 2749,
506, 1463, 1463, 2565, 3066,
507, 1463, 1463, 2715, 2934,
508, 1467, 1467, 1790, 3571,
509, 1479, 1479, 1994, 3451,
510, 1491, 1974, 2659, 2659,
511, 1514, 2179, 2571, 2571,
512, 1525, 1593, 1593, 3586,
513, 1525, 2038, 2625, 2625,
514, 1551, 1551, 1706, 3541,
515, 1554, 1995, 2633, 2633,
516, 1555, 1797, 1797, 3374,
517, 1579, 1579, 2154, 3261,
518, 1590, 2335, 2335, 2613,
519, 1599, 1599, 2021, 3326,
520, 1610, 1647, 1647, 3499,
521, 1610, 2373, 2373, 2531,
522, 1615, 1615, 1890, 3387,
523, 1630, 2221, 2517, 2517,
524, 1630, 2415, 2415, 2437,
525, 1635, 1635, 2438, 2995,
526, 1638, 2345, 2345, 2565,
527, 1647, 1647, 1874, 3365,
528, 1651, 2246, 2499, 2499,
529, 1653, 1653, 2674, 2765,
530, 1659, 1769, 1769, 3354,
531, 1661, 1661, 1869, 3354,
532, 1666, 2151, 2151, 2869,
533, 1669, 1669, 2589, 2826,
534, 1683, 1683, 2630, 2771,
535, 1694, 2289, 2289, 2629,
536, 1697, 1697, 1926, 3285,
537, 1705, 1705, 2610, 2763,
538, 1706, 1923, 1923, 3155,
539, 1710, 1787, 1787, 3309,
540, 1713, 1713, 2309, 3010,
541, 1719, 1719, 2491, 2854,
542, 1731, 1731, 2086, 3149,
543, 1733, 1733, 1779, 3330,
544, 1738, 1965, 1965, 3085,
545, 1749, 2046, 2551, 2551,
546, 1749, 2047, 2047, 2970,
547, 1750, 1947, 1947, 3101,
548, 1755, 1895, 1895, 3162,
549, 1755, 2263, 2263, 2634,
550, 1765, 1887, 1887, 3166,
551, 1767, 1767, 2171, 3050,
552, 1773, 1773, 1981, 3170,
553, 1786, 2181, 2181, 2749,
554, 1787, 1787, 2115, 3066,
555, 1790, 1861, 2607, 2607,
556, 1811, 2229, 2229, 2654,
557, 1813, 2030, 2535, 2535,
558, 1814, 1911, 1911, 3109,
559, 1814, 2307, 2307, 2515,
560, 1815, 1815, 2245, 2938,
561, 1815, 1815, 2485, 2738,
562, 1829, 1954, 2559, 2559,
563, 1846, 2045, 2517, 2517,
564, 1857, 1857, 2539, 2630,
565, 1861, 2271, 2271, 2546,
566, 1869, 2011, 2011, 2946,
567, 1885, 1885, 2262, 2835,
568, 1885, 1885, 2490, 2637,
569, 1885, 1994, 2523, 2523,
570, 1894, 1939, 2541, 2541,
571, 1899, 1899, 2434, 2669,
572, 1923, 1923, 1981, 2990,
573, 1926, 2219, 2219, 2589,
574, 1934, 2271, 2271, 2491,
575, 1941, 1941, 2186, 2819,
576, 1953, 1953, 2374, 2645,
577, 1953, 1953, 2395, 2626,
578, 1954, 2061, 2061, 2819,
579, 1966, 2021, 2481, 2481,
580, 1985, 1985, 2238, 2715,
581, 1987, 1995, 1995, 2890,
582, 1994, 2291, 2349, 2349,
583, 2007, 2007, 2186, 2725,
584, 2049, 2049, 2171, 2674,
585, 2055, 2055, 2213, 2630,
586, 2085, 2238, 2335, 2335,
587, 2086, 2121, 2121, 2629,
588, 2093, 2093, 2190, 2589,
589, 2106, 2235, 2327, 2327,
590, 2147, 2147, 2190, 2499,
591, 2149, 2241, 2241, 2366,
592, 2154, 2209, 2209, 2421,
593, 2189, 2259, 2259, 2294,
594, 2191, 2191, 2286, 2331,

2 个数相同。x2 = a + a + c + d——有 594 种可能。——关键是这个——人工可以有吗？？？

风花飘飘

王守恩 发表于 2026-8-13 13:13
题目。质数 20260319 可以表示为四个非负整数 x, y, z, w 的平方和。求 0 ≤ x ≤ y ≤ z ≤ w 组合(无序) ...

验证了最后一个，我没看明白——


用代码精确计算，结果如下：

精确计算

立方和

2191^3 + 2191^3 + 2286^3 + 2331^3
= 45,647,508,089 不是素数，它可以分解为两个素数的乘积：
其中 6133 和 7,442,933 都是素数，因此这是该数的唯一素因子分解。








除了这三组分拆，谁能找到第四组？

风花飘飘

我把题目的平方看成是立方了，哈哈

风花飘飘

汇总一：末项为4501的四元数组
1}2.15.33.4501
2}6.21.29.4501
3}9.9.34.4501

汇总二：末项为4051的四元数组
1.[411.411.1874.4051]
2.[470.1347.1347.4051]



还有没有？

王守恩

再总结——为自己！——谢谢 风花飘飘！！

原题。质数 20260319 可以表示为四个非负整数 x, y, z, w 的平方和。求该四个非负整数组合数。

节点1——求该四个非负整数组合数 = 求 0 ≤ x ≤ y ≤ z ≤ w 组合数 = 求 0 ≤ x ≤ y ≤ z ≤ w 组合(无序)数。——说成“0 ≤ x ≤ y ≤ z ≤ w 组合(无序)数”太浪费了！

题目。质数 20260319 可以表示为四个非负整数 x, y, z, w 的平方和。求 0 ≤ x ≤ y ≤ z ≤ w 组合(无序)数。

答:  组合数 = 组合(无序)数 = 421793 + 594 = 422387。——答案正确——确认！

4个数不同。x1 = a + b + c + d——有 421793 种可能。——这个——人工可以有。

2个数相同。x2 = a + a + c + d——有 594 种可能。——关键是这个——人工可以有吗？

2对数相同。x3 = a + a + c + c——有 0 种可能。——人工可以有。

3个数相同。x4 = a + a + a + d——有 0 种可能。——人工可以有。

4个数相同。x5 = a + a + a + a——有 0 种可能。——人工可以有。

基本公式——雅可比四平方定理——也就是有符号、有序的四平方和表示数。

8*(20260319 + 1) = 384*x1 + 192*x2 + 96*x3 + 64*x4 + 16*x5。——没问题！

其中:  384, 192, 96, 64, 16————人工可以有。

384=16(符号变化)*24(排列数)。

192=16(符号变化)*12(排列数)。

096=16(符号变化)*04(排列数)。

016=16(符号变化)*01(排列数)。

补充。

a,  b,  c,  d 不可能有 0。

因为 20260319 = 质数。20260319 = 模 8 余 7 的数——模 8 余 7 的数不可能表示为三平方和。

原题也就变成了——质数 20260319 可以表示为四个正整数 x, y, z, w 的平方和。求该四个正整数组合数。

2 个数相同——有 594 种可能。——详见6#
1, 3, 3, 670, 4451,
2, 5, 927, 927, 4306,
3, 5, 1383, 1383, 4054,
4, 5, 1562, 2985, 2985,
5, 5, 1714, 2943, 2943,
6, 9, 9, 34, 4501,
7, 9, 9, 2506, 3739,
8, 13, 1310, 3045, 3045,
9, 13, 2475, 2475, 2830,
10, 19, 579, 579, 4426,
11, 21, 751, 751, 4374,
12, 26, 1797, 1797, 3715,
13, 26, 2189, 2781, 2781,
14, 29, 386, 3171, 3171,
15, 29, 459, 459, 4454,
16, 29, 910, 3117, 3117,
17, 30, 535, 535, 4437,
18, 33, 33, 1246, 4325,
19, 33, 33, 2165, 3946,
20, 35, 2234, 2763, 2763,
......

2 个数相同。x2 = a + a + c + d——有 594 种可能。——关键是这个——人工可以有吗？？——这个人工还真来不了！！！