题目261：平方和支点

欧拉计划

Pivotal Square Sums

Let us call a positive integer k a square-pivot, if there is a pair of integers m > 0 and n ≥ k, such that the sum of the (m+1) consecutive squares up to k equals the sum of the m consecutive squares from (n+1) on:

(k-m)² + ... + k² = (n+1)² + ... + (n+m)²

Some small square-pivots are

• 4: 3² + 4² = 5²
• 21: 20² + 21² = 29²
• 24: 21² + 22² + 23² + 24² = 25² + 26² + 27²
• 110: 108² + 109² + 110² = 133² + 134²
  

Find the sum of all distinct square-pivots ≤ 10¹⁰.

题目：

我就称正整数 k 为平方支点，如果存在一对整数 m > 0 和 n ≥ k，使得到 k 为止的 (m+1) 个连续的平方的和等于从 (n+1) 开始的 m 个连续的平方的和：

(k-m)² + ... + k² = (n+1)² + ... + (n+m)².

一些小的平方支点：

• 4：3² + 4² = 5²
• 21：20² + 21² = 29²
• 24：21² + 22² + 23² + 24² = 25² + 26² + 27²
• 110：108² + 109² + 110² = 133² + 134²
  

求所有 ≤ 10¹⁰ 的不同的平方支点的和。

不二如是

跟欧拉学了一招

加了引号，无法被复制～

99592938

1. ##平方支点等式化简为：(m+1)*k**2=m*n**2+m*(m+1)(n+k)
   
2. ##由条件可分析：0<m<=(k-2)/2)**0.5
   
3. ##循环带入k、m，可求出n值，若n>=k且为整数，则k为平方支点。
   
4. def value_n(k,m):
   
5.     l_n=[0,0]
   
6.     a=m
   
7.     b=m*(m+1)
   
8.     c=(m+1)*m*k-(m+1)*k*k
   
9.     if b*b-4*a*c<0:
   
10.         return l_n
    
11.     else:
    
12.         d=(b*b-4*a*c)**0.5
    
13.         n1=(-b+d)/(2*a)
    
14.         n2=(-b-d)/(2*a)
    
15.         if n1>=k and (n1-int(n1))==0:
    
16.             l_n[0]=int(n1)
    
17.         if n2>=k and (n2-int(n2))==0:
    
18.             l_n[1]=int(n2)
    
19.         return l_n
    
20. 
    
21. num = int(input('请输入范围上限：'))
    
22. dic_k={}
    
23. for k in range(2,num+1):
    
24.     for m in range(1,int(((k-2)/2)**0.5)+1):
    
25.         l_n=value_n(k,m)
    
26.         for each in l_n:
    
27.             if each:
    
28.                 dic_k[k]=[m,each]
    
29. print(sum(dic_k.keys()))
    
30. #print(sorted(list(dic_k.keys())))

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拿小范围数来试，程序是正确的，但拿100亿来跑~~~已经半个多小时了
>>>
请输入范围上限：110
355
[4, 12, 21, 24, 40, 60, 84, 110]
>>> ================================ RESTART ================================
>>>
请输入范围上限：10000000000

99592938

不二如是 发表于 2017-1-13 21:52

然而，这是网页加了限制，引号外面也不能复制，哈哈哈

Python_meng

怎么停更了呢- -

去project euler官网去做吧，虽然是英文的(ω)

JAY饭

100亿的话，俨然需要借助数学的方法了，这个真的不会了。

1. def filt1(num,twi):
   
2.     m = num**2
   
3.     for i in range(1,twi+1):
   
4.         m += (num-i)**2
   
5.     return m
   
6. 
   
7. def filt2(num,twi):
   
8.     m = num**2
   
9.     for i in range(1,twi):
   
10.         m += (num+i)**2
    
11.     return m
    
12. 
    
13. def choice(num):
    
14.    
    
15.     flag = int(((num**2) + (num-1)**2)**0.5)
    
16.     for i in range(num+1,flag+1):
    
17.         T = True
    
18.         for j in range(1,num):
    
19.             if filt1(num,j) == filt2(i,j):
    
20.                 print(num,i,j)
    
21.                 return True
    
22.             elif filt1(num,j) < filt2(i,j):
    
23.                 break
    
24.     return False
    
25. 
    
26. for i in range(1,10000000):
    
27.     choice(i)
    

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永恒的蓝色梦想

不二如是 发表于 2017-1-13 21:52

不是可以复制吗……

代号-K

把公式化简一下，然后进行判断
代码附上

1. 
   
2. typedef unsigned long long ElementType;
   
3. void getSquarePivots(int num){
   
4.     int k, n, m;
   
5.     int flag;
   
6.     ElementType sum = 0, tmp;
   
7.     for(k = 1; k <= num; k++){
   
8.         for(m = 1; m <= k; m++){
   
9.             flag = (int) (pow((m+1)*1.0/m, 0.5) * k);
   
10.             tmp = pow(k-m,2);
    
11.             if(tmp % m == 0){
    
12.             for(n = k; n <= flag - 1; n++){
    
13.                 if(tmp == (ElementType)((n-k+m)*(n+1+k)*m)){
    
14.                     printf("k:%d, m:%d, n:%d\n",k,m,n);
    
15.                     sum+= k;
    
16.                 }
    
17.             }
    
18.             }
    
19.         }
    
20.     }
    
21.     printf("result is follow\n");
    
22.     printf("%lld\n",sum);
    
23. }
    
24. int main(int argc, char *argv[]){
    
25.     //ElementType ret = -1;
    
26.     getSquarePivots(100000);
    
27.     return 0;
    
28. }
    

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代号-K

跑代码，发现个大密秘
当且仅当 n==k 时 m 是递增的

1. 
   
2. k:4, m:1, n:4
   
3. k:12, m:2, n:12
   
4. k:24, m:3, n:24
   
5. k:40, m:4, n:40
   
6. k:60, m:5, n:60
   
7. k:84, m:6, n:84
   
8. k:112, m:7, n:112
   
9. k:144, m:8, n:144
   
10. k:180, m:9, n:180
    
11. k:220, m:10, n:220
    
12. k:264, m:11, n:264
    
13. k:312, m:12, n:312
    
14. k:364, m:13, n:364
    
15. k:420, m:14, n:420
    
16. k:480, m:15, n:480
    
17. k:544, m:16, n:544
    
18. k:612, m:17, n:612
    
19. k:684, m:18, n:684
    
20. k:760, m:19, n:760
    
21. k:840, m:20, n:840
    
22. k:924, m:21, n:924
    
23. k:1012, m:22, n:1012
    
24. k:1104, m:23, n:1104
    
25. k:1200, m:24, n:1200
    
26. k:1300, m:25, n:1300
    
27. k:1404, m:26, n:1404
    
28. k:1512, m:27, n:1512
    
29. k:1624, m:28, n:1624
    
30. k:1740, m:29, n:1740
    
31. k:1860, m:30, n:1860
    
32. k:1984, m:31, n:1984
    
33. k:2112, m:32, n:2112
    
34. k:2244, m:33, n:2244
    
35. k:2380, m:34, n:2380
    
36. k:2520, m:35, n:2520
    
37. k:2664, m:36, n:2664
    
38. k:2812, m:37, n:2812
    
39. k:2964, m:38, n:2964
    
40. k:3120, m:39, n:3120
    
41. k:3280, m:40, n:3280
    
42. k:3444, m:41, n:3444
    
43. k:3612, m:42, n:3612
    
44. k:3784, m:43, n:3784
    
45. k:3960, m:44, n:3960
    
46. k:4140, m:45, n:4140
    
47. k:4324, m:46, n:4324
    
48. k:4512, m:47, n:4512
    
49. k:4704, m:48, n:4704
    
50. k:4900, m:49, n:4900
    
51. k:5100, m:50, n:5100
    
52. k:5304, m:51, n:5304
    
53. k:5512, m:52, n:5512
    
54. k:5724, m:53, n:5724
    
55. k:5940, m:54, n:5940
    
56. k:6160, m:55, n:6160
    
57. k:6384, m:56, n:6384
    
58. k:6612, m:57, n:6612
    
59. k:6844, m:58, n:6844
    
60. k:7080, m:59, n:7080
    
61. k:7320, m:60, n:7320
    
62. k:7564, m:61, n:7564
    
63. k:7812, m:62, n:7812
    
64. k:8064, m:63, n:8064
    
65. k:8320, m:64, n:8320
    
66. k:8580, m:65, n:8580
    
67. k:8844, m:66, n:8844
    
68. k:9112, m:67, n:9112
    
69. k:9384, m:68, n:9384
    
70. k:9660, m:69, n:9660
    
71. k:9940, m:70, n:9940
    
72. k:10224, m:71, n:10224
    
73. k:10512, m:72, n:10512
    
74. k:10804, m:73, n:10804
    
75. k:11100, m:74, n:11100
    
76. k:11400, m:75, n:11400
    
77. k:11704, m:76, n:11704
    
78. k:12012, m:77, n:12012
    
79. k:12324, m:78, n:12324
    
80. k:12640, m:79, n:12640
    
81. k:12960, m:80, n:12960
    
82. k:13284, m:81, n:13284
    
83. k:13612, m:82, n:13612
    
84. k:13944, m:83, n:13944
    
85. k:14280, m:84, n:14280
    
86. k:14620, m:85, n:14620
    
87. k:14964, m:86, n:14964
    
88. k:15312, m:87, n:15312
    
89. k:15664, m:88, n:15664
    
90. k:16020, m:89, n:16020
    
91. k:16380, m:90, n:16380
    
92. k:16744, m:91, n:16744
    
93. k:17112, m:92, n:17112
    
94. k:17484, m:93, n:17484
    
95. k:17860, m:94, n:17860
    
96. k:18240, m:95, n:18240
    
97. k:18624, m:96, n:18624
    
98. k:19012, m:97, n:19012
    
99. k:19404, m:98, n:19404
    
100. k:19800, m:99, n:19800
     
101. k:20200, m:100, n:20200
     
102. k:20604, m:101, n:20604
     
103. k:21012, m:102, n:21012
     
104. k:21424, m:103, n:21424
     
105. k:21840, m:104, n:21840
     
106. k:22260, m:105, n:22260
     
107. k:22684, m:106, n:22684
     
108. k:23112, m:107, n:23112
     
109. k:23544, m:108, n:23544
     
110. k:23980, m:109, n:23980
     
111. k:24420, m:110, n:24420
     
112. k:24864, m:111, n:24864
     
113. k:25312, m:112, n:25312
     
114. k:25764, m:113, n:25764
     
115. k:26220, m:114, n:26220
     
116. k:26680, m:115, n:26680
     
117. k:27144, m:116, n:27144
     
118. k:27612, m:117, n:27612
     
119. k:28084, m:118, n:28084
     
120. k:28560, m:119, n:28560
     
121. k:29040, m:120, n:29040
     
122. k:29524, m:121, n:29524
     
123. k:30012, m:122, n:30012
     
124. k:30504, m:123, n:30504
     
125. k:31000, m:124, n:31000
     
126. k:31500, m:125, n:31500
     
127. k:32004, m:126, n:32004
     
128. k:32512, m:127, n:32512
     
129. k:33024, m:128, n:33024
     
130. k:33540, m:129, n:33540
     
131. k:34060, m:130, n:34060
     
132. k:34584, m:131, n:34584
     
133. k:35112, m:132, n:35112
     
134. k:35644, m:133, n:35644
     
135. k:36180, m:134, n:36180
     
136. k:36720, m:135, n:36720
     
137. k:37264, m:136, n:37264
     
138. k:37812, m:137, n:37812
     
139. k:38364, m:138, n:38364
     
140. k:38920, m:139, n:38920
     
141. k:39480, m:140, n:39480
     
142. k:40044, m:141, n:40044
     
143. k:40612, m:142, n:40612
     
144. k:41184, m:143, n:41184
     
145. k:41760, m:144, n:41760
     
146. k:42340, m:145, n:42340
     
147. k:42924, m:146, n:42924
     
148. k:43512, m:147, n:43512
     
149. k:44104, m:148, n:44104
     
150. k:44700, m:149, n:44700
     
151. k:45300, m:150, n:45300
     
152. k:45904, m:151, n:45904
     

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代号-K

这是正常跑下来的，只能跑到151不知道怎么回事，可能电脑问题吧

1. 
   
2. k:4, m:1, n:4
   
3. k:12, m:2, n:12
   
4. k:21, m:1, n:28
   
5. k:24, m:3, n:24
   
6. k:40, m:4, n:40
   
7. k:60, m:5, n:60
   
8. k:84, m:6, n:84
   
9. k:110, m:2, n:132
   
10. k:112, m:7, n:112
    
11. k:120, m:1, n:168
    
12. k:144, m:8, n:144
    
13. k:180, m:9, n:180
    
14. k:220, m:10, n:220
    
15. k:264, m:11, n:264
    
16. k:312, m:12, n:312
    
17. k:315, m:3, n:360
    
18. k:364, m:13, n:364
    
19. k:420, m:14, n:420
    
20. k:480, m:15, n:480
    
21. k:544, m:16, n:544
    
22. k:612, m:17, n:612
    
23. k:684, m:4, n:760
    
24. k:684, m:18, n:684
    
25. k:697, m:1, n:984
    
26. k:760, m:19, n:760
    
27. k:820, m:8, n:861
    
28. k:840, m:20, n:840
    
29. k:924, m:21, n:924
    
30. k:1012, m:22, n:1012
    
31. k:1080, m:2, n:1320
    
32. k:1104, m:23, n:1104
    
33. k:1200, m:24, n:1200
    
34. k:1265, m:5, n:1380
    
35. k:1300, m:25, n:1300
    
36. k:1404, m:26, n:1404
    
37. k:1512, m:27, n:1512
    
38. k:1624, m:28, n:1624
    
39. k:1740, m:29, n:1740
    
40. k:1860, m:30, n:1860
    
41. k:1984, m:31, n:1984
    
42. k:2106, m:6, n:2268
    
43. k:2112, m:32, n:2112
    
44. k:2244, m:33, n:2244
    
45. k:2380, m:34, n:2380
    
46. k:2520, m:35, n:2520
    
47. k:2664, m:36, n:2664
    
48. k:2812, m:37, n:2812
    
49. k:2964, m:38, n:2964
    
50. k:3120, m:39, n:3120
    
51. k:3255, m:7, n:3472
    
52. k:3280, m:40, n:3280
    
53. k:3444, m:41, n:3444
    
54. k:3612, m:42, n:3612
    
55. k:3784, m:43, n:3784
    
56. k:3960, m:44, n:3960
    
57. k:4060, m:1, n:5740
    
58. k:4140, m:45, n:4140
    
59. k:4324, m:46, n:4324
    
60. k:4368, m:3, n:5040
    
61. k:4512, m:47, n:4512
    
62. k:4704, m:48, n:4704
    
63. k:4760, m:8, n:5040
    
64. k:4900, m:49, n:4900
    
65. k:5100, m:50, n:5100
    
66. k:5304, m:51, n:5304
    
67. k:5512, m:52, n:5512
    
68. k:5724, m:53, n:5724
    
69. k:5940, m:54, n:5940
    
70. k:6160, m:55, n:6160
    
71. k:6384, m:56, n:6384
    
72. k:6612, m:57, n:6612
    
73. k:6669, m:9, n:7020
    
74. k:6844, m:58, n:6844
    
75. k:7080, m:59, n:7080
    
76. k:7320, m:60, n:7320
    
77. k:7564, m:61, n:7564
    
78. k:7812, m:62, n:7812
    
79. k:8064, m:63, n:8064
    
80. k:8320, m:64, n:8320
    
81. k:8580, m:65, n:8580
    
82. k:8844, m:66, n:8844
    
83. k:9030, m:10, n:9460
    
84. k:9112, m:67, n:9112
    
85. k:9384, m:68, n:9384
    
86. k:9660, m:69, n:9660
    
87. k:9940, m:70, n:9940
    
88. k:10224, m:71, n:10224
    
89. k:10512, m:72, n:10512
    
90. k:10682, m:2, n:13080
    
91. k:10804, m:73, n:10804
    
92. k:11100, m:74, n:11100
    
93. k:11400, m:75, n:11400
    
94. k:11704, m:76, n:11704
    
95. k:11772, m:24, n:11990
    
96. k:11891, m:11, n:12408
    
97. k:12012, m:77, n:12012
    
98. k:12240, m:4, n:13680
    
99. k:12324, m:78, n:12324
    
100. k:12640, m:79, n:12640
     
101. k:12960, m:80, n:12960
     
102. k:13284, m:81, n:13284
     
103. k:13612, m:82, n:13612
     
104. k:13944, m:83, n:13944
     
105. k:14280, m:84, n:14280
     
106. k:14620, m:85, n:14620
     
107. k:14964, m:86, n:14964
     
108. k:15300, m:12, n:15912
     
109. k:15312, m:87, n:15312
     
110. k:15664, m:88, n:15664
     
111. k:16020, m:89, n:16020
     
112. k:16380, m:90, n:16380
     
113. k:16744, m:91, n:16744
     
114. k:17112, m:92, n:17112
     
115. k:17484, m:93, n:17484
     
116. k:17860, m:94, n:17860
     
117. k:18240, m:95, n:18240
     
118. k:18624, m:96, n:18624
     
119. k:19012, m:97, n:19012
     
120. k:19305, m:13, n:20020
     
121. k:19404, m:98, n:19404
     
122. k:19800, m:99, n:19800
     
123. k:20200, m:100, n:20200
     
124. k:20604, m:101, n:20604
     
125. k:21012, m:102, n:21012
     
126. k:21424, m:103, n:21424
     
127. k:21840, m:104, n:21840
     
128. k:22260, m:105, n:22260
     
129. k:22684, m:106, n:22684
     
130. k:23112, m:107, n:23112
     
131. k:23544, m:108, n:23544
     
132. k:23661, m:1, n:33460
     
133. k:23954, m:14, n:24780
     
134. k:23980, m:109, n:23980
     
135. k:24420, m:110, n:24420
     
136. k:24864, m:111, n:24864
     
137. k:25312, m:112, n:25312
     
138. k:25764, m:113, n:25764
     
139. k:26220, m:114, n:26220
     
140. k:26680, m:115, n:26680
     
141. k:27144, m:116, n:27144
     
142. k:27612, m:117, n:27612
     
143. k:27720, m:5, n:30360
     
144. k:27724, m:8, n:29397
     
145. k:28084, m:118, n:28084
     
146. k:28441, m:49, n:28680
     
147. k:28560, m:119, n:28560
     
148. k:29040, m:120, n:29040
     
149. k:29295, m:15, n:30240
     
150. k:29524, m:121, n:29524
     
151. k:30012, m:122, n:30012
     
152. k:30504, m:123, n:30504
     
153. k:31000, m:124, n:31000
     
154. k:31500, m:125, n:31500
     
155. k:32004, m:126, n:32004
     
156. k:32512, m:127, n:32512
     
157. k:33024, m:128, n:33024
     
158. k:33540, m:129, n:33540
     
159. k:34060, m:130, n:34060
     
160. k:34584, m:131, n:34584
     
161. k:35112, m:132, n:35112
     
162. k:35376, m:16, n:36448
     
163. k:35644, m:133, n:35644
     
164. k:36180, m:134, n:36180
     
165. k:36720, m:135, n:36720
     
166. k:37264, m:136, n:37264
     
167. k:37812, m:137, n:37812
     
168. k:38364, m:138, n:38364
     
169. k:38920, m:139, n:38920
     
170. k:39480, m:140, n:39480
     
171. k:40044, m:141, n:40044
     
172. k:40612, m:142, n:40612
     
173. k:41184, m:143, n:41184
     
174. k:41760, m:144, n:41760
     
175. k:42245, m:17, n:43452
     
176. k:42340, m:145, n:42340
     
177. k:42924, m:146, n:42924
     
178. k:43512, m:147, n:43512
     
179. k:44104, m:148, n:44104
     
180. k:44700, m:149, n:44700
     
181. k:45300, m:150, n:45300
     
182. k:45904, m:151, n:45904

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