三角数不定方程的通解公式的验证

Treenewbee

1楼8楼的解可以统一为：

\[(x,y,z)=(n,\frac{1}{2}n(n+1)-1,\frac{1}{2}n(n+1))\]

ysr

前面的解不全还有（6,9,11）（8,10,13）（9,13,16）（9,21,23）等解呢

Treenewbee

不定方程
\[x (x + 1) + y (y + 1) = z (z + 1)\]
的通解为：
\[(x,y,z)=(pq, \frac{q^2 - p + q - 1}{2},\frac{q^2 + p + q - 1}{2})\]

Treenewbee

当\[p=1,q=n\]时，可得到11楼的特解

ysr

试试这个总公式，代码如下：
Private Sub Command1_Click()
Dim p, q, x, y, z
a = Val(Text1)
p = 1
Do While Val(p) <= a
q = p + 1
Do While q <= a
x = p * q
y = (q ^ 2 - p + q - 1) / 2
z = (q ^ 2 + p + q - 1) / 2

s2 = 2
s1 = s1 & "x=" & x & "  y=" & y & "  z=" & z & vbCrLf

q = q + 1
Loop
p = p + 1
Loop

If Val(s2) > 0 Then
Text2 = s1
Else
Text2 = "wu   jie"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub

ysr

输入20，得到如下结果：（看来答案是全的）

x=2  y=2  z=3
x=3  y=5  z=6
x=4  y=9  z=10
x=5  y=14  z=15
x=6  y=20  z=21
x=7  y=27  z=28
x=8  y=35  z=36
x=9  y=44  z=45
x=10  y=54  z=55
x=11  y=65  z=66
x=12  y=77  z=78
x=13  y=90  z=91
x=14  y=104  z=105
x=15  y=119  z=120
x=16  y=135  z=136
x=17  y=152  z=153
x=18  y=170  z=171
x=19  y=189  z=190
x=20  y=209  z=210
x=6  y=4.5  z=6.5
x=8  y=8.5  z=10.5
x=10  y=13.5  z=15.5
x=12  y=19.5  z=21.5
x=14  y=26.5  z=28.5
x=16  y=34.5  z=36.5
x=18  y=43.5  z=45.5
x=20  y=53.5  z=55.5
x=22  y=64.5  z=66.5
x=24  y=76.5  z=78.5
x=26  y=89.5  z=91.5
x=28  y=103.5  z=105.5
x=30  y=118.5  z=120.5
x=32  y=134.5  z=136.5
x=34  y=151.5  z=153.5
x=36  y=169.5  z=171.5
x=38  y=188.5  z=190.5
x=40  y=208.5  z=210.5
x=12  y=8  z=11
x=15  y=13  z=16
x=18  y=19  z=22
x=21  y=26  z=29
x=24  y=34  z=37
x=27  y=43  z=46
x=30  y=53  z=56
x=33  y=64  z=67
x=36  y=76  z=79
x=39  y=89  z=92
x=42  y=103  z=106
x=45  y=118  z=121
x=48  y=134  z=137
x=51  y=151  z=154
x=54  y=169  z=172
x=57  y=188  z=191
x=60  y=208  z=211
x=20  y=12.5  z=16.5
x=24  y=18.5  z=22.5
x=28  y=25.5  z=29.5
x=32  y=33.5  z=37.5
x=36  y=42.5  z=46.5
x=40  y=52.5  z=56.5
x=44  y=63.5  z=67.5
x=48  y=75.5  z=79.5
x=52  y=88.5  z=92.5
x=56  y=102.5  z=106.5
x=60  y=117.5  z=121.5
x=64  y=133.5  z=137.5
x=68  y=150.5  z=154.5
x=72  y=168.5  z=172.5
x=76  y=187.5  z=191.5
x=80  y=207.5  z=211.5
x=30  y=18  z=23
x=35  y=25  z=30
x=40  y=33  z=38
x=45  y=42  z=47
x=50  y=52  z=57
x=55  y=63  z=68
x=60  y=75  z=80
x=65  y=88  z=93
x=70  y=102  z=107
x=75  y=117  z=122
x=80  y=133  z=138
x=85  y=150  z=155
x=90  y=168  z=173
x=95  y=187  z=192
x=100  y=207  z=212
x=42  y=24.5  z=30.5
x=48  y=32.5  z=38.5
x=54  y=41.5  z=47.5
x=60  y=51.5  z=57.5
x=66  y=62.5  z=68.5
x=72  y=74.5  z=80.5
x=78  y=87.5  z=93.5
x=84  y=101.5  z=107.5
x=90  y=116.5  z=122.5
x=96  y=132.5  z=138.5
x=102  y=149.5  z=155.5
x=108  y=167.5  z=173.5
x=114  y=186.5  z=192.5
x=120  y=206.5  z=212.5
x=56  y=32  z=39
x=63  y=41  z=48
x=70  y=51  z=58
x=77  y=62  z=69
x=84  y=74  z=81
x=91  y=87  z=94
x=98  y=101  z=108
x=105  y=116  z=123
x=112  y=132  z=139
x=119  y=149  z=156
x=126  y=167  z=174
x=133  y=186  z=193
x=140  y=206  z=213
x=72  y=40.5  z=48.5
x=80  y=50.5  z=58.5
x=88  y=61.5  z=69.5
x=96  y=73.5  z=81.5
x=104  y=86.5  z=94.5
x=112  y=100.5  z=108.5
x=120  y=115.5  z=123.5
x=128  y=131.5  z=139.5
x=136  y=148.5  z=156.5
x=144  y=166.5  z=174.5
x=152  y=185.5  z=193.5
x=160  y=205.5  z=213.5
x=90  y=50  z=59
x=99  y=61  z=70
x=108  y=73  z=82
x=117  y=86  z=95
x=126  y=100  z=109
x=135  y=115  z=124
x=144  y=131  z=140
x=153  y=148  z=157
x=162  y=166  z=175
x=171  y=185  z=194
x=180  y=205  z=214
x=110  y=60.5  z=70.5
x=120  y=72.5  z=82.5
x=130  y=85.5  z=95.5
x=140  y=99.5  z=109.5
x=150  y=114.5  z=124.5
x=160  y=130.5  z=140.5
x=170  y=147.5  z=157.5
x=180  y=165.5  z=175.5
x=190  y=184.5  z=194.5
x=200  y=204.5  z=214.5
x=132  y=72  z=83
x=143  y=85  z=96
x=154  y=99  z=110
x=165  y=114  z=125
x=176  y=130  z=141
x=187  y=147  z=158
x=198  y=165  z=176
x=209  y=184  z=195
x=220  y=204  z=215
x=156  y=84.5  z=96.5
x=168  y=98.5  z=110.5
x=180  y=113.5  z=125.5
x=192  y=129.5  z=141.5
x=204  y=146.5  z=158.5
x=216  y=164.5  z=176.5
x=228  y=183.5  z=195.5
x=240  y=203.5  z=215.5
x=182  y=98  z=111
x=195  y=113  z=126
x=208  y=129  z=142
x=221  y=146  z=159
x=234  y=164  z=177
x=247  y=183  z=196
x=260  y=203  z=216
x=210  y=112.5  z=126.5
x=224  y=128.5  z=142.5
x=238  y=145.5  z=159.5
x=252  y=163.5  z=177.5
x=266  y=182.5  z=196.5
x=280  y=202.5  z=216.5
x=240  y=128  z=143
x=255  y=145  z=160
x=270  y=163  z=178
x=285  y=182  z=197
x=300  y=202  z=217
x=272  y=144.5  z=160.5
x=288  y=162.5  z=178.5
x=304  y=181.5  z=197.5
x=320  y=201.5  z=217.5
x=306  y=162  z=179
x=323  y=181  z=198
x=340  y=201  z=218
x=342  y=180.5  z=198.5
x=360  y=200.5  z=218.5
x=380  y=200  z=219

Treenewbee

\[(x,y,z)=(n, \frac{-m - m^2 + n + n^2}{2m},\frac{-m + m^2 + n + n^2}{2 m})\]

Treenewbee

只要找到任意一个 n^2+n的因子2m即可

Treenewbee

1. Solve[{x (x + 1) + y (y + 1) == z (z + 1), 0 < x <= y < z < 100}, {x,
   
2.   y, z}, Integers]

复制代码

{{x->2,y->2,z->3},{x->3,y->5,z->6},{x->4,y->9,z->10},{x->5,y->6,z->8},{x->5,y->14,z->15},{x->6,y->9,z->11},{x->6,y->20,z->21},{x->7,y->27,z->28},{x->8,y->10,z->13},{x->8,y->35,z->36},{x->9,y->13,z->16},{x->9,y->21,z->23},{x->9,y->44,z->45},{x->10,y->26,z->28},{x->10,y->54,z->55},{x->11,y->14,z->18},{x->11,y->20,z->23},{x->11,y->65,z->66},{x->12,y->17,z->21},{x->12,y->24,z->27},{x->12,y->77,z->78},{x->13,y->44,z->46},{x->13,y->90,z->91},{x->14,y->14,z->20},{x->14,y->18,z->23},{x->14,y->33,z->36},{x->14,y->51,z->53},{x->15,y->21,z->26},{x->15,y->38,z->41},{x->17,y->22,z->28},{x->17,y->49,z->52},{x->17,y->75,z->77},{x->18,y->25,z->31},{x->18,y->55,z->58},{x->18,y->84,z->86},{x->19,y->35,z->40},{x->19,y->45,z->49},{x->20,y->26,z->33},{x->20,y->39,z->44},{x->20,y->50,z->54},{x->20,y->68,z->71},{x->21,y->29,z->36},{x->21,y->35,z->41},{x->21,y->75,z->78},{x->23,y->30,z->38},{x->23,y->90,z->93},{x->24,y->33,z->41},{x->24,y->57,z->62},{x->25,y->27,z->37},{x->25,y->62,z->67},{x->26,y->34,z->43},{x->26,y->55,z->61},{x->27,y->37,z->46},{x->27,y->50,z->57},{x->27,y->92,z->96},{x->28,y->54,z->61},{x->29,y->38,z->48},{x->29,y->69,z->75},{x->29,y->84,z->89},{x->30,y->41,z->51},{x->30,y->74,z->80},{x->30,y->90,z->95},{x->32,y->42,z->53},{x->33,y->45,z->56},{x->33,y->90,z->96},{x->34,y->35,z->49},{x->34,y->54,z->64},{x->34,y->81,z->88},{x->35,y->46,z->58},{x->35,y->65,z->74},{x->35,y->86,z->93},{x->36,y->49,z->61},{x->36,y->69,z->78},{x->38,y->50,z->63},{x->39,y->44,z->59},{x->39,y->53,z->66},{x->41,y->54,z->68},{x->42,y->57,z->71},{x->43,y->80,z->91},{x->44,y->58,z->73},{x->44,y->76,z->88},{x->44,y->84,z->95},{x->45,y->48,z->66},{x->45,y->61,z->76},{x->47,y->62,z->78},{x->48,y->65,z->81},{x->49,y->80,z->94},{x->50,y->66,z->83},{x->50,y->77,z->92},{x->51,y->69,z->86},{x->53,y->70,z->88},{x->54,y->56,z->78},{x->54,y->73,z->91},{x->56,y->65,z->86},{x->56,y->74,z->93},{x->57,y->77,z->96},{x->59,y->78,z->98},{x->65,y->69,z->95}}

ysr

代码如下：

Private Sub Command1_Click()
Dim p, q, x, y, z
a = Val(Text1)
m = 1
Do While Val(m) <= a
n = n + 1
Do While n <= a
x = n
y = (-m ^ 2 - m + n ^ 2 + n) / (2 * m)
z = (-m ^ 2 + m + n ^ 2 + n) / (2 * m)

s2 = 2
s1 = s1 & "x=" & x & "  y=" & y & "  z=" & z & vbCrLf

n = n + 1
Loop
m = m + 1
Loop

If Val(s2) > 0 Then
Text2 = s1
Else
Text2 = "wu   jie"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub