数论问题巅峰对决

ysr

风花飘飘 发表于 2022-6-4 20:39
【本帖申请加为精华】求方程 a^2-b^2-2ab=0 的整数解。
http://www.mathchina.com/bbs/forum.php?mod=view ...

可以在理论上证明无整数解，简单的方法专家不认可，复杂的用到高级知识的一般人不会。难呢！

ysr

ysr 发表于 2022-6-4 13:33
Private Sub Command1_Click()
Dim a, b, q
Dim t As Double

17, 47,,5,17,2n=102
37, 67,,5,13,2n=102
67, 97,,5,7,2n=102
11, 41,,3,31,2n=104
17, 47,,3,29,2n=104
53, 83,,3,17,2n=104
71, 101,,3,11,2n=104
83, 113,,3,7,2n=104
13, 43,,3,31,2n=106
37, 67,,3,23,2n=106
67, 97,,3,13,2n=106
73, 103,,3,11,2n=106
97, 127,,3,3,2n=106
13, 43,,5,19,2n=108
23, 53,,5,17,2n=108
43, 73,,5,13,2n=108
53, 83,,5,11,2n=108
73, 103,,5,7,2n=108
83, 113,,5,5,2n=108
17, 47,,3,31,2n=110
23, 53,,3,29,2n=110
41, 71,,3,23,2n=110
53, 83,,3,19,2n=110
59, 89,,3,17,2n=110
71, 101,,3,13,2n=110
101, 131,,3,3,2n=110
43, 73,,3,23,2n=112
73, 103,,3,13,2n=112
79, 109,,3,11,2n=112
97, 127,,3,5,2n=112
29, 59,,5,17,2n=114
59, 89,,5,11,2n=114
79, 109,,5,7,2n=114
23, 53,,3,31,2n=116
29, 59,,3,29,2n=116
59, 89,,3,19,2n=116
83, 113,,3,11,2n=116
101, 131,,3,5,2n=116
107, 137,,3,3,2n=116
7, 37,,3,37,2n=118
31, 61,,3,29,2n=118
67, 97,,3,17,2n=118
79, 109,,3,13,2n=118
97, 127,,3,7,2n=118
109, 139,,3,3,2n=118
29, 59,,7,13,2n=120
43, 73,,7,11,2n=120
71, 101,,7,7,2n=120
11, 41,,3,37,2n=122
29, 59,,3,31,2n=122
53, 83,,3,23,2n=122
71, 101,,3,17,2n=122
83, 113,,3,13,2n=122
101, 131,,3,7,2n=122
107, 137,,3,5,2n=122
13, 43,,3,37,2n=124
31, 61,,3,31,2n=124
37, 67,,3,29,2n=124
67, 97,,3,19,2n=124
73, 103,,3,17,2n=124
109, 139,,3,5,2n=124
11, 41,,5,23,2n=126
31, 61,,5,19,2n=126
41, 71,,5,17,2n=126
71, 101,,5,11,2n=126
101, 131,,5,5,2n=126
17, 47,,3,37,2n=128
41, 71,,3,29,2n=128
59, 89,,3,23,2n=128
71, 101,,3,19,2n=128
107, 137,,3,7,2n=128
7, 37,,3,41,2n=130
37, 67,,3,31,2n=130
43, 73,,3,29,2n=130
73, 103,,3,19,2n=130
79, 109,,3,17,2n=130
97, 127,,3,11,2n=130
109, 139,,3,7,2n=130
17, 47,,5,23,2n=132
37, 67,,5,19,2n=132
67, 97,,5,13,2n=132
97, 127,,5,7,2n=132
107, 137,,5,5,2n=132
11, 41,,3,41,2n=134
23, 53,,3,37,2n=134
41, 71,,3,31,2n=134
83, 113,,3,17,2n=134
101, 131,,3,11,2n=134
7, 37,,3,43,2n=136
13, 43,,3,41,2n=136
43, 73,,3,31,2n=136
67, 97,,3,23,2n=136
79, 109,,3,19,2n=136
97, 127,,3,13,2n=136
127, 157,,3,3,2n=136
23, 53,,5,23,2n=138
43, 73,,5,19,2n=138
53, 83,,5,17,2n=138
73, 103,,5,13,2n=138
83, 113,,5,11,2n=138
11, 41,,3,43,2n=140
17, 47,,3,41,2n=140
29, 59,,3,37,2n=140
53, 83,,3,29,2n=140
71, 101,,3,23,2n=140
83, 113,,3,19,2n=140
101, 131,,3,13,2n=140
107, 137,,3,11,2n=140
13, 43,,3,43,2n=142
31, 61,,3,37,2n=142
73, 103,,3,23,2n=142
109, 139,,3,11,2n=142
127, 157,,3,5,2n=142
29, 59,,5,23,2n=144
59, 89,,5,17,2n=144
79, 109,,5,13,2n=144
109, 139,,5,7,2n=144
17, 47,,3,43,2n=146
23, 53,,3,41,2n=146
53, 83,,3,31,2n=146
59, 89,,3,29,2n=146
107, 137,,3,13,2n=146
137, 167,,3,3,2n=146
7, 37,,3,47,2n=148
37, 67,,3,37,2n=148
79, 109,,3,23,2n=148
97, 127,,3,17,2n=148
109, 139,,3,13,2n=148
127, 157,,3,7,2n=148
17, 47,,7,19,2n=150
31, 61,,7,17,2n=150
59, 89,,7,13,2n=150
73, 103,,7,11,2n=150
101, 131,,7,7,2n=150
11, 41,,3,47,2n=152
23, 53,,3,43,2n=152
29, 59,,3,41,2n=152
41, 71,,3,37,2n=152
59, 89,,3,31,2n=152
83, 113,,3,23,2n=152
101, 131,,3,17,2n=152
137, 167,,3,5,2n=152
13, 43,,3,47,2n=154
31, 61,,3,41,2n=154
43, 73,,3,37,2n=154
67, 97,,3,29,2n=154
97, 127,,3,19,2n=154
11, 41,,5,29,2n=156
41, 71,,5,23,2n=156
71, 101,,5,17,2n=156
101, 131,,5,11,2n=156
17, 47,,3,47,2n=158
29, 59,,3,43,2n=158
71, 101,,3,29,2n=158
101, 131,,3,19,2n=158
107, 137,,3,17,2n=158
137, 167,,3,7,2n=158
149, 179,,3,3,2n=158
31, 61,,3,43,2n=160
37, 67,,3,41,2n=160
67, 97,,3,31,2n=160
73, 103,,3,29,2n=160
109, 139,,3,17,2n=160
127, 157,,3,11,2n=160
151, 181,,3,3,2n=160
7, 37,,5,31,2n=162
17, 47,,5,29,2n=162
67, 97,,5,19,2n=162
97, 127,,5,13,2n=162
107, 137,,5,11,2n=162
127, 157,,5,7,2n=162
137, 167,,5,5,2n=162
23, 53,,3,47,2n=164
41, 71,,3,41,2n=164
53, 83,,3,37,2n=164
71, 101,,3,31,2n=164
107, 137,,3,19,2n=164
149, 179,,3,5,2n=164
7, 37,,3,53,2n=166
37, 67,,3,43,2n=166
43, 73,,3,41,2n=166
73, 103,,3,31,2n=166
79, 109,,3,29,2n=166
97, 127,,3,23,2n=166
109, 139,,3,19,2n=166
127, 157,,3,13,2n=166
151, 181,,3,5,2n=166
13, 43,,5,31,2n=168
23, 53,,5,29,2n=168
53, 83,,5,23,2n=168
73, 103,,5,19,2n=168
83, 113,,5,17,2n=168
11, 41,,3,53,2n=170
29, 59,,3,47,2n=170
41, 71,,3,43,2n=170
59, 89,,3,37,2n=170
83, 113,,3,29,2n=170
101, 131,,3,23,2n=170
137, 167,,3,11,2n=170
149, 179,,3,7,2n=170
13, 43,,3,53,2n=172
31, 61,,3,47,2n=172
43, 73,,3,43,2n=172
79, 109,,3,31,2n=172
151, 181,,3,7,2n=172
163, 193,,3,3,2n=172
29, 59,,5,29,2n=174
59, 89,,5,23,2n=174
79, 109,,5,19,2n=174
109, 139,,5,13,2n=174
149, 179,,5,5,2n=174
17, 47,,3,53,2n=176
53, 83,,3,41,2n=176
83, 113,,3,31,2n=176
107, 137,,3,23,2n=176
137, 167,,3,13,2n=176
167, 197,,3,3,2n=176
37, 67,,3,47,2n=178
67, 97,,3,37,2n=178

ysr

500内有893组蔡氏素数:
702/704/706/708/710/712/714/716/718/720/722/724/726/728/730/732/734/736/738/740/
742/744/746/748/750/752/754/756/758/760/762/764/766/768/770/772/774/776/778/780/
782/784/786/788/790/792/794/796/798/800/802/804/806/808/810/812/814/816/818/820/
822/824/826/828/830/832/834/836/838/840/842/844/846/848/850/852/854/856/858/860/
862/864/866/868/870/872/874/876/878/880/882/884/886/888/890/892/894/896/898/900/
902/904/906/908/910/912/914/916/918/920/922/924/926/928/930/932/934/936/938/940/
942/944/946/948/950/952/954/956/958/960/962/964/966/968/970/972/974/976/978/980/
982/984/986/988/990/992/994/996/998/1000/用时32.4970000000008秒

ysr

ysr 发表于 2022-6-5 02:46
500内有893组蔡氏素数:
702/704/706/708/710/712/714/716/718/720/722/724/726/728/730/732/734/736/738/7 ...

Private Sub Command1_Click()
Dim a, b, q
Dim t As Double
t = Timer

q = Val(Text1)
m = 702
Do While m <= 1000

p1 = 3
Do While p1 <= q And p1 <= m
p2 = Val(p1 + 30)
p3 = Val(p1 + 60)
p4 = fenjieyinzi1(Val(m - p1))
p5 = Val(m - p1) / Val(p4)
Print p1, p2, p3, p4
a = fenjieyinzi(Val(p1))
b = fenjieyinzi(Val(p2))
c = fenjieyinzi(Val(p3))
d = fenjieyinzi(Val(p4))
f1 = fenjieyinzi(Val(p5))
p7 = 3
m1 = m + 30
m2 = m + 60
Do While zzxc(Val(m), Val(p7)) > 1 Or zzxc(Val(m1), Val(p7)) > 1 Or zzxc(Val(m2), Val(p7)) > 1
p7 = p7 + 2
Loop


If InStr(a, "*") = 0 And InStr(b, "*") = 0 And InStr(c, "*") = 0 And InStr(d, "*") = 0 And InStr(f1, "*") = 0 And p4 = Val(p7) Then
s = s + 1

s101 = s101 & "/" & m

Else
s = s
End If
p1 = Val(p1 + 2)
Loop
m = Val(m + 2)
Loop
s103 = Mid(s101, 1)
Dim i As Integer
Dim ak(), s105, cr(), f
Set f = CreateObject("Scripting.Dictionary")
s105 = Split(s103, "/")
   j1 = UBound(s105)
   Print j1
   For k = 1 To j1

n1 = n1 + 1
       ReDim Preserve ak(1 To n1)
      ak(n1) = s105(n1)
    Next
    Print ak(1)
     n = 0
        For k = 1 To j1
           For i = 1 To j1

n = n + 1
             ReDim Preserve cr(1 To n)
            m = Val(ak(k))
            f(m) = ""
      Next
    Next
      n = 0
      m = f.Keys
      For i = 0 To f.Count - 1
          ReDim Preserve cr(1 To i + 1)
          cr(i + 1) = m(i)
      Next
     For i = 1 To UBound(cr) - 1
        For J = i + 1 To UBound(cr)
            If cr(i) > cr(J) Then
                temp = cr(J)
                cr(J) = cr(i)
                cr(i) = temp  'c数组是排序好的
            End If
        Next J
        
       ' If i Mod 20 = 0 Then
       ' s104 = s104 & temp & "/" & vbCrLf
       ' Else
       ' s104 = s104 & temp & "/"
       ' End If
    Next i
   
      For i = 1 To UBound(cr)
        If i Mod 20 = 0 Then

  s104 = s104 & cr(i) & "/" & vbCrLf
        Else
          s104 = s104 & cr(i) & "/"
        End If

Next
         Print temp
         MsgBox "ok"
     MsgBox s104  '显示数组

Text2 = s104

Combo1 = q & "内有" & s & "组蔡氏素数:" & vbCrLf & Text2 & "用时" & Timer - t & "秒"

End Sub

蔡家雄

【用偶数(1+2)分拆的最小解寻找大素数的方法】

设 2n >=32，且 p1, p2, p3 都是素数，

且 p2 <=p3,  且 p2 是与2n 互素的最小素数，

则 2n=p1+p2*p3 至少有一组素数(p1, p2, p3)解。

则 p1=2n -p2*p3 .

例 p1=100! -101*p3，显然：p2=101 <= p3，

求 最小的 p3= ?

蔡家雄

【求出偶数(1+2)分拆的最小解的解数】

设 2n >=32，且 p1, p2, p3 都是素数，

且 p2 <=p3,  且 p2 是与2n 互素的最小素数，

则 2n=p1+p2*p3 至少有一组素数(p1, p2, p3)解。

例 10! = p1+11*p3，

估计有一千个以上的 p3，使之有解。

蔡家雄

【用偶数(1+2)分拆的最小解寻找大素数的方法】

设 2n >=32，且 p1, p2, p3 都是素数，

且 p2 <=p3,  且 p2 是与2n 互素的最小素数，

则 2n=p1+p2*p3 至少有一组素数(p1, p2, p3)解。

则 p1=2n -p2*p3 .

例 p1=100! -101*9901 是素数，但不知 p3=9901 是否最小的 ？

ysr

加强筛连乘积公式结果： 偶数2023022488888  其方根内最大素数1422293 方根内的素数个数m=108650 (方根为)1422329.95078076  有108649个区间，其中每个区间哥猜解素数对个数的平均值16016.1344115318  总对数为1740136987.67852方根内的解的个数为1474.51865828173

ysr

如下为根据加强筛拟合的函数得到的偶数哥德巴赫程序解的下限：
2023022488888 的哥德巴赫猜想解的个数的下限是：1776829851.81477

ysr

50000 的哥德巴赫猜想解的个数的下限是：409.914100075607
10000的哥德巴赫猜想解的下限是 88.62461526955
100000的哥德巴赫猜想解的下限是 732.569076768859
1000000的哥德巴赫猜想解的下限是 4943.41147402807
10000000的哥德巴赫猜想解的下限是 32665.4314958269
100000000的哥德巴赫猜想解的下限是 212836.62421646
1000000000的哥德巴赫猜想解的下限是 1789472.55110611
10000000000的哥德巴赫猜想解的下限是 11505183.4023235
100000000000的哥德巴赫猜想解的下限是 73467242.7310537
1000000000000的哥德巴赫猜想解的下限是 1007445003.01865
10000000000000的哥德巴赫猜想解的下限是 6421983741.65592
100000000000000的哥德巴赫猜想解的下限是 40730891038.2436
1E+15的哥德巴赫猜想解的下限是 257272091061.987
1E+16的哥德巴赫猜想解的下限是 1619492212134.68
1E+17的哥德巴赫猜想解的下限是 10165167937428.4
1E+18的哥德巴赫猜想解的下限是 63647489868918.5
1E+19的哥德巴赫猜想解的下限是 397670787930535
1E+20的哥德巴赫猜想解的下限是 2.48004140464722E+15
1E+21的哥德巴赫猜想解的下限是 1.54412765929521E+16
1E+22的哥德巴赫猜想解的下限是 9.60011565439899E+16
1E+23的哥德巴赫猜想解的下限是 5.96081862682551E+17
1E+24的哥德巴赫猜想解的下限是 3.69682188695747E+18
1E+25的哥德巴赫猜想解的下限是 2.29030378799066E+19
1E+26的哥德巴赫猜想解的下限是 1.41756066201421E+20
1E+27的哥德巴赫猜想解的下限是 8.76618688427078E+20
1E+28的哥德巴赫猜想解的下限是 5.41666798066324E+21
1E+29的哥德巴赫猜想解的下限是 3.34452287790109E+22
1E+30的哥德巴赫猜想解的下限是 2.06367533096909E+23
1E+31的哥德巴赫猜想解的下限是 1.27255284838008E+24
1E+32的哥德巴赫猜想解的下限是 7.84254764174065E+24
1E+33的哥德巴赫猜想解的下限是 4.83062107713709E+25
1E+34的哥德巴赫猜想解的下限是 2.97391848157417E+26
1E+35的哥德巴赫猜想解的下限是 1.829994151598E+27
1E+36的哥德巴赫猜想解的下限是 1.125583772393E+28
1E+37的哥德巴赫猜想解的下限是 6.92030390529084E+28
1E+38的哥德巴赫猜想解的下限是 4.25306904345366E+29
1E+39的哥德巴赫猜想解的下限是 2.61287960430241E+30
1E+40的哥德巴赫猜想解的下限是 1.60466737233368E+31
1E+41的哥德巴赫猜想解的下限是 9.85161724546252E+31
1E+42的哥德巴赫猜想解的下限是 6.04636690247899E+32
1E+43的哥德巴赫猜想解的下限是 3.70982062518663E+33
1E+44的哥德巴赫猜想解的下限是 2.27556494098484E+34
1E+45的哥德巴赫猜想解的下限是 1.39543448540019E+35
1E+46的哥德巴赫猜想解的下限是 8.55498239325858E+35
1E+47的哥德巴赫猜想解的下限是 5.24352576885398E+36
1E+48的哥德巴赫猜想解的下限是 3.2131213222756E+37
1E+49的哥德巴赫猜想解的下限是 1.96849702602694E+38
1E+50的哥德巴赫猜想解的下限是 1.20573125709911E+39