Let
n
be a 5-digit positive integer.
Given the infinitely nested radical expression below:
Y
= sqrt(n+ sqrt(n+ sqrt(n+… sqrt(n+… ,-
What is the maximal value of
n, such that
Y
is a positive integer?
y^2 = n + y
y^2 - y - n = 0
y = (1 +/- sqrt(1 + 4*n)) / 2
sqrt(1 + 4*n) must be an odd integer for y to have a positive integer value.
Every y will have an n value: n = y^2 - y
The y^2 term will predominate. For n = 100000 (one larger than the largest 5-digit number, y would be around sqrt(100000) or 316.227766016838. So try y = 316, leading to n = 316^2 - 316 = 99540, a 5-digit number. Just to check, y=317 yields 100172, a 6-digit number.
Of course we could do it the computer way:
Check all 5-digit n's:
For n = 10000 To 99999
disc = 1 + 4 * n
sr = Int(Sqr(disc) + 0.5)
If sr * sr = disc Then
If sr Mod 2 = 1 Then
Text1.Text = Text1.Text & n & Str((1 + sr) / 2) & crlf
End If
End If
DoEvents
Next
n y
10100 101
10302 102
10506 103
10712 104
10920 105
11130 106
11342 107
11556 108
11772 109
11990 110
12210 111
12432 112
12656 113
12882 114
13110 115
13340 116
13572 117
13806 118
14042 119
14280 120
14520 121
14762 122
15006 123
15252 124
15500 125
15750 126
16002 127
16256 128
16512 129
16770 130
17030 131
17292 132
17556 133
17822 134
18090 135
18360 136
18632 137
18906 138
19182 139
19460 140
19740 141
20022 142
20306 143
20592 144
20880 145
21170 146
21462 147
21756 148
22052 149
22350 150
22650 151
22952 152
23256 153
23562 154
23870 155
24180 156
24492 157
24806 158
25122 159
25440 160
25760 161
26082 162
26406 163
26732 164
27060 165
27390 166
27722 167
28056 168
28392 169
28730 170
29070 171
29412 172
29756 173
30102 174
30450 175
30800 176
31152 177
31506 178
31862 179
32220 180
32580 181
32942 182
33306 183
33672 184
34040 185
34410 186
34782 187
35156 188
35532 189
35910 190
36290 191
36672 192
37056 193
37442 194
37830 195
38220 196
38612 197
39006 198
39402 199
39800 200
40200 201
40602 202
41006 203
41412 204
41820 205
42230 206
42642 207
43056 208
43472 209
43890 210
44310 211
44732 212
45156 213
45582 214
46010 215
46440 216
46872 217
47306 218
47742 219
48180 220
48620 221
49062 222
49506 223
49952 224
50400 225
50850 226
51302 227
51756 228
52212 229
52670 230
53130 231
53592 232
54056 233
54522 234
54990 235
55460 236
55932 237
56406 238
56882 239
57360 240
57840 241
58322 242
58806 243
59292 244
59780 245
60270 246
60762 247
61256 248
61752 249
62250 250
62750 251
63252 252
63756 253
64262 254
64770 255
65280 256
65792 257
66306 258
66822 259
67340 260
67860 261
68382 262
68906 263
69432 264
69960 265
70490 266
71022 267
71556 268
72092 269
72630 270
73170 271
73712 272
74256 273
74802 274
75350 275
75900 276
76452 277
77006 278
77562 279
78120 280
78680 281
79242 282
79806 283
80372 284
80940 285
81510 286
82082 287
82656 288
83232 289
83810 290
84390 291
84972 292
85556 293
86142 294
86730 295
87320 296
87912 297
88506 298
89102 299
89700 300
90300 301
90902 302
91506 303
92112 304
92720 305
93330 306
93942 307
94556 308
95172 309
95790 310
96410 311
97032 312
97656 313
98282 314
98910 315
99540 316
|
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Posted by Charlie
on 2016-11-15 10:39:36 |
solution
