王守恩有理数问题：对于任一有理数 m/n，均可表为 (a^2+b) / (a+b^2) 的形式，

elim · 发表于 2018-5-29 22:51

王守恩有理数问题：对于任一有理数 m/n，均可表为 (a^2+b) / (a+b^2) 的形式。其中 a,b 是有理数.

证：若 m/n = 0,  取 a = 1, b = -1 就有 (a^2+b) / (a+b^2)  = 0.
      若 m/n = -1, 取 a = 0, b = -1 就有 (a^2+b) / (a+b^2)  = 0.
      若 m/n ≠ 0, ≠ -1, 取 a = m/n, b = n/m, 则
      (a^2+b) / (a+b^2) = m/n.

王守恩 · 发表于 2018-5-30 06:12

永远发表于 2018-5-29 23:07
话说王守恩先生是一名中学奥数教师，不知是高中还是初中奥数教师？？？？？？？？？？

原题是这样的：
a，b，n 均为正整数，满足： n = (a^2+b) / (a+b^2) 。
求证：对任意 n 来说，都能找到对应的 a，b。

风花飘飘 · 发表于 2018-5-30 07:31

提示: 作者被禁止或删除内容自动屏蔽

elim · 发表于 2018-5-31 09:16

正整数的相应问题的确不简单。先编程找点感觉：

\\ Given n, The following function is to find [a,b] such that a×a+b = n(a+b×b)
abn(n)= {
my(k=n);
my(nn=n*n);
while((mm=(1+n*(k*k-nn)))&&(m=floor(sqrt(mm))),
if((k=k+2)&&(m*m==mm)&&((m+1)%(2*n)==0),
return([(n+k-2)/2,(m+1)/(2*n)]))
);
return([]);
}
list_abn(m,n)={
for(k=m,n, printf("%5d: %s\n",k,abn(k)));
}

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Reading GPRC: E:\Academy\Pari\/gprc.txt ...Done.
GP/PARI CALCULATOR Version 2.9.2 (released)
i686 running mingw (ix86/GMP-6.0.0 kernel) 32-bit version
compiled: Mar 22 2017, gcc version 4.9.1 (GCC)
threading engine: single
(readline v6.2 enabled, extended help enabled)
PARI/GP is free software, covered by the GNU General Public License,
and comes WITHOUT ANY WARRANTY WHATSOEVER.
Type ? for help, \q to quit.
Type ?15 for how to get moral (and possibly technical) support.
parisize = 4000000, primelimit = 500000
(18:57) gp > list_abn(1,100)
1: [1, 1]
2: [5, 3]
3: [5, 2]
4: [10, 4]
5: [27, 11]
6: [69, 27]
7: [12, 3]
8: [38, 12]
9: [20, 5]
10: [103, 31]
11: [14, 2]
12: [335, 95]
13: [19, 3]
14: [1859, 495]
15: [147, 36]
16: [37, 7]
17: [293, 69]
18: [44, 8]
19: [54, 10]
20: [1043, 231]
21: [89, 17]
22: [50, 8]
23: [38, 5]
24: [404, 80]
25: [36, 4]
26: [33, 3]
27: [367, 68]
28: [263, 47]
29: [45, 5]
30: [77, 11]
31: [84, 12]
32: [147, 23]
33: [12350, 2147]
34: [129, 19]
35: [57, 6]
36: [1962, 324]
37: [49, 4]
38: [37665, 6107]
39: [109, 14]
40: [217, 31]
41: [42185, 6585]
42: [5225, 803]
43: [103, 12]
44: [1110, 164]
45: [119, 14]
46: [101, 11]
47: [65, 5]
48: [1770, 252]
49: [550, 75]
50: [2507, 351]
51: [88, 8]
52: [1180, 160]
53: [153, 17]
54: [9683509, 1317755]
55: [171, 19]
56: [122, 12]
57: [407, 50]
58: [214, 24]
59: [15699, 2040]
60: [44577, 5751]
61: [80, 5]
62: [191015, 24255]
63: [170, 17]
64: [8224, 1024]
65: [4233, 521]
66: [218139, 26847]
67: [75, 3]
68: [300, 32]
69: [5265595, 633899]
70: [2244, 264]
71: [3246, 381]
72: [16455, 1935]
73: [270, 27]
74: [855, 95]
75: [40273, 4646]
76: [118600, 13600]
77: [1250, 138]
78: [174793, 19787]
79: [170, 14]
80: [5192, 576]
81: [101, 5]
82: [6733, 739]
83: [264, 24]
84: [230, 20]
85: [12913, 1396]
86: [297, 27]
87: [138, 9]
88: [123, 7]
89: [22276589, 2361309]
90: [26333, 2771]
91: [307, 27]
92: [197, 15]
93: [1089, 108]
94: [824, 80]
95: [780, 75]
96: [34527, 3519]
97: [2649, 264]
98: [1505, 147]
99: [8109, 810]
100: [114, 4]
(18:58) gp >

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