判定梅森质数的卢卡斯序列

蔡家雄

奇偶( 差 )猜想

设 2n+1 >=1，设 p1, p2, p3=2*p2 -1 都是素数，

则 2n+1=p1 -2*p2 与 2n+2=p1 -p3 有无限多对素数解。

例 23=29 -2*3=37 -2*7=61 -2*19=97 -2*37=181 -2*79=337 -2*157=.........

蔡家雄

奇偶( 差 )猜想A

设 2n+3 >=1，设 p1, p2, p3=2*p2 -1 都是素数，

则 2n+3=p1 -2*p2 与 2n+4=p1 -p3 有无限多对素数解。

奇偶( 差 )猜想B

设 2n+3 >=1，设 p1, p2, p3=2*p2+1 都是素数，

则 2n+3=p1 -2*p2 与 2n+2=p1 -p3 有无限多对素数解。

蔡家雄

若设 2n+3 >=9，且 p1, p2, p3=2*p2+1 都是素数，

或设 2n+3 >=9，且 p1, p2, p3=2*p2 -1 都是素数，

奇偶( 和 )猜想：任意大于8的偶数，

或是 2n+3=p1+2*p2 与 2n+4=p1+p3 至少有一对素数解，

或是 2n+3=p1+2*p2 与 2n+2=p1+p3 至少有一对素数解。

二者必居其一，甚至必居其二，不可能出现完全无解的情况。

蔡家雄

蔡氏奇数分拆

设 2n+15 >=33,

则 2n+15=p1+2*p2 , 2n+45=p3+2*p2 , 2n+75=p4+2*p2 均有素数解。

注：p2 可以等于2，2也是素数。

蔡氏奇数分拆

设 2n+15 >=33,

则 2n+15=p1+2*p2 , 2n+75=p3+2*p2 , 2n+135=p4+2*p2 均有素数解。

蔡家雄

蔡氏偶数分拆

设 2n >=64，且 p1, p2=2n -p1, p3=2n -p1 -30 , p4=p1+30 都是素数，

则 2n -30=p1+p3 , 2n=p1+p2=p3+p4 , 2n+30=p2+p4 至少有一组素数(p1, p2, p3, p4)解。

蔡氏偶数分拆

设 2n >=280，且 p1, p2=2n -p1, p3=2n -p1 -210 , p4=p1+210 都是素数，

则 2n -210=p1+p3 , 2n=p1+p2=p3+p4 , 2n+210=p2+p4 至少有一组素数(p1, p2, p3, p4)解。

蔡家雄

蔡氏偶数(1+2)分拆（最小解）

设 2n >=62，且 p1, p2=p1+210, p3=p1+630, p4, p5 都是素数，

且 p4 <=p5,  且 p4 是与2n, 2n+210, 2n+630 都互素的最小素数，

则 2n=p1+p4*p5 , 2n+210=p2+p4*p5 , 2n+630=p3+p4*p5 至少有一组素数(p1, p2, p3, p4, p5)解。

例 p4=5 是与2022 , 2022+210 , 2022+630 都互素的最小素数，

2022=1787+5*47，
2022=1667+5*71，
2022=1657+5*73，
2022=1277+5*149，
2022=1867+5*157，
2022=1117+5*181，
2022=0677+5*269，
2022=0467+5*311，
2022=0367+5*331，
2022=0337+5*337，
2022=0277+5*349，
2022=0257+5*353，
2022=0157+5*373，
2022=0127+5*379，
2022=0017+5*401.


蔡氏偶数(1+2)分拆（最小解）

设 2n >=62，且 p1, p2=p1+420, p3=p1+840, p4, p5 都是素数，

且 p4 <=p5,  且 p4 是与2n, 2n+420, 2n+840 都互素的最小素数，

则 2n=p1+p4*p5 , 2n+420=p2+p4*p5 , 2n+840=p3+p4*p5 至少有一组素数(p1, p2, p3, p4, p5)解。

例 p4=5 是与2022 , 2022+420 , 2022+840 都互素的最小素数，

2022=1997+5*5，
2022=1867+5*31，
2022=1607+5*83，
2022=1367+5*131，
2022=1187+5*167，
2022=0907+5*223，
2022=0857+5*233，
2022=0487+5*307，
2022=0467+5*311，
2022=0457+5*313，
2022=0257+5*353，
2022=0157+5*373，
2022=0127+5*379，
2022=0037+5*397.

ysr

蔡家雄 发表于 2022-6-12 10:21
【蔡氏偶数分拆】

若 p1=a^2+b^2 与 p2=c^2+d^2 均为素数，

200内有137组蔡氏素数:
37( 1+36),29(4+25),31,4n+2=66
61( 25+36),5(1+4),7,4n+2=66
29( 4+25),41(16+25),43,4n+2=70
41( 16+25),29(4+25),31,4n+2=70
53( 4+49),17(1+16),19,4n+2=70
37( 1+36),41(16+25),43,4n+2=78
61( 25+36),17(1+16),19,4n+2=78
73( 9+64),5(1+4),7,4n+2=78
41( 16+25),41(16+25),43,4n+2=82
53( 4+49),29(4+25),31,4n+2=82
61( 25+36),29(4+25),31,4n+2=90
73( 9+64),17(1+16),19,4n+2=90
53( 4+49),41(16+25),43,4n+2=94
89( 25+64),5(1+4),7,4n+2=94
61( 25+36),41(16+25),43,4n+2=102
73( 9+64),29(4+25),31,4n+2=102
97( 16+81),5(1+4),7,4n+2=102
5( 1+4),101(1+100),103,4n+2=106
89( 25+64),17(1+16),19,4n+2=106
101( 1+100),5(1+4),7,4n+2=106
13( 4+9),101(1+100),103,4n+2=114
73( 9+64),41(16+25),43,4n+2=114
97( 16+81),17(1+16),19,4n+2=114
109( 9+100),5(1+4),7,4n+2=114
17( 1+16),101(1+100),103,4n+2=118
89( 25+64),29(4+25),31,4n+2=118
101( 1+100),17(1+16),19,4n+2=118
113( 49+64),5(1+4),7,4n+2=118
97( 16+81),29(4+25),31,4n+2=126
109( 9+100),17(1+16),19,4n+2=126
29( 4+25),101(1+100),103,4n+2=130
89( 25+64),41(16+25),43,4n+2=130
101( 1+100),29(4+25),31,4n+2=130
113( 49+64),17(1+16),19,4n+2=130
37( 1+36),101(1+100),103,4n+2=138
97( 16+81),41(16+25),43,4n+2=138
109( 9+100),29(4+25),31,4n+2=138
5( 1+4),137(16+121),139,4n+2=142
41( 16+25),101(1+100),103,4n+2=142
101( 1+100),41(16+25),43,4n+2=142
113( 49+64),29(4+25),31,4n+2=142
137( 16+121),5(1+4),7,4n+2=142
13( 4+9),137(16+121),139,4n+2=150
109( 9+100),41(16+25),43,4n+2=150
5( 1+4),149(49+100),151,4n+2=154
17( 1+16),137(16+121),139,4n+2=154
53( 4+49),101(1+100),103,4n+2=154
113( 49+64),41(16+25),43,4n+2=154
137( 16+121),17(1+16),19,4n+2=154
149( 49+100),5(1+4),7,4n+2=154
13( 4+9),149(49+100),151,4n+2=162
61( 25+36),101(1+100),103,4n+2=162
157( 36+121),5(1+4),7,4n+2=162
17( 1+16),149(49+100),151,4n+2=166
29( 4+25),137(16+121),139,4n+2=166
137( 16+121),29(4+25),31,4n+2=166
149( 49+100),17(1+16),19,4n+2=166
37( 1+36),137(16+121),139,4n+2=174
73( 9+64),101(1+100),103,4n+2=174
157( 36+121),17(1+16),19,4n+2=174
29( 4+25),149(49+100),151,4n+2=178
41( 16+25),137(16+121),139,4n+2=178
137( 16+121),41(16+25),43,4n+2=178
149( 49+100),29(4+25),31,4n+2=178
173( 4+169),5(1+4),7,4n+2=178
37( 1+36),149(49+100),151,4n+2=186
157( 36+121),29(4+25),31,4n+2=186
181( 81+100),5(1+4),7,4n+2=186
41( 16+25),149(49+100),151,4n+2=190
53( 4+49),137(16+121),139,4n+2=190
89( 25+64),101(1+100),103,4n+2=190
149( 49+100),41(16+25),43,4n+2=190
173( 4+169),17(1+16),19,4n+2=190
61( 25+36),137(16+121),139,4n+2=198
97( 16+81),101(1+100),103,4n+2=198
157( 36+121),41(16+25),43,4n+2=198
181( 81+100),17(1+16),19,4n+2=198
193( 49+144),5(1+4),7,4n+2=198
5( 1+4),197(1+196),199,4n+2=202
53( 4+49),149(49+100),151,4n+2=202
101( 1+100),101(1+100),103,4n+2=202
173( 4+169),29(4+25),31,4n+2=202
197( 1+196),5(1+4),7,4n+2=202
13( 4+9),197(1+196),199,4n+2=210
61( 25+36),149(49+100),151,4n+2=210
73( 9+64),137(16+121),139,4n+2=210
109( 9+100),101(1+100),103,4n+2=210
181( 81+100),29(4+25),31,4n+2=210
193( 49+144),17(1+16),19,4n+2=210
17( 1+16),197(1+196),199,4n+2=214
113( 49+64),101(1+100),103,4n+2=214
173( 4+169),41(16+25),43,4n+2=214
197( 1+196),17(1+16),19,4n+2=214
73( 9+64),149(49+100),151,4n+2=222
181( 81+100),41(16+25),43,4n+2=222
193( 49+144),29(4+25),31,4n+2=222
29( 4+25),197(1+196),199,4n+2=226
89( 25+64),137(16+121),139,4n+2=226
197( 1+196),29(4+25),31,4n+2=226
37( 1+36),197(1+196),199,4n+2=234
97( 16+81),137(16+121),139,4n+2=234
193( 49+144),41(16+25),43,4n+2=234
41( 16+25),197(1+196),199,4n+2=238
89( 25+64),149(49+100),151,4n+2=238
101( 1+100),137(16+121),139,4n+2=238
137( 16+121),101(1+100),103,4n+2=238
197( 1+196),41(16+25),43,4n+2=238
97( 16+81),149(49+100),151,4n+2=246
109( 9+100),137(16+121),139,4n+2=246
53( 4+49),197(1+196),199,4n+2=250
101( 1+100),149(49+100),151,4n+2=250
113( 49+64),137(16+121),139,4n+2=250
149( 49+100),101(1+100),103,4n+2=250
61( 25+36),197(1+196),199,4n+2=258
109( 9+100),149(49+100),151,4n+2=258
157( 36+121),101(1+100),103,4n+2=258
113( 49+64),149(49+100),151,4n+2=262
73( 9+64),197(1+196),199,4n+2=270
5( 1+4),269(100+169),271,4n+2=274
137( 16+121),137(16+121),139,4n+2=274
173( 4+169),101(1+100),103,4n+2=274
13( 4+9),269(100+169),271,4n+2=282
181( 81+100),101(1+100),103,4n+2=282
5( 1+4),281(25+256),283,4n+2=286
17( 1+16),269(100+169),271,4n+2=286
89( 25+64),197(1+196),199,4n+2=286
137( 16+121),149(49+100),151,4n+2=286
149( 49+100),137(16+121),139,4n+2=286
13( 4+9),281(25+256),283,4n+2=294
97( 16+81),197(1+196),199,4n+2=294
157( 36+121),137(16+121),139,4n+2=294
193( 49+144),101(1+100),103,4n+2=294
17( 1+16),281(25+256),283,4n+2=298
29( 4+25),269(100+169),271,4n+2=298
101( 1+100),197(1+196),199,4n+2=298
149( 49+100),149(49+100),151,4n+2=298
197( 1+196),101(1+100),103,4n+2=298
用时0.253000000003436秒

ysr

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

q = Val(Text1)
m = 66
Do While m <= 300

p1 = 5

Do While p1 <= q And p1 <= m
a1 = 1
B1 = Sqr(Val(p1 - a1 ^ 2))
Do While InStr(B1, ".") <> 0 And p1 > Val(a1 ^ 2)
a1 = a1 + 1
B1 = Sqr(Abs(Val(p1 - a1 ^ 2)))
Loop
p2 = m - p1

C2 = 1
D2 = Sqr(Val(p2 - C2 ^ 2))
Do While InStr(D2, ".") <> 0 And p2 > Val(C2 ^ 2)
C2 = C2 + 1
D2 = Sqr(Abs(Val(p2 - C2 ^ 2)))
Loop

p3 = Val(p2 + 2)


a = fenjieyinzi(Val(p1))
B = fenjieyinzi(Val(p2))
c = fenjieyinzi(Val(p3))


If InStr(a, "*") = 0 And InStr(B, "*") = 0 And InStr(c, "*") = 0 And InStr(B1, ".") = 0 And InStr(D2, ".") = 0 Then
s = s + 1
Print p1, p2, p3, p4
Text2 = Text2 & CStr(p1) & "( " & a1 ^ 2 & "+" & B1 ^ 2 & ")," & CStr(p2) & "(" & C2 ^ 2 & "+" & D2 ^ 2 & ")," & p3 & ",4n+2=" & m & vbCrLf

Else
s = s
End If


p1 = Val(p1 + 2)


Loop
m = Val(m + 4)
Loop
Combo1 = q & "内有" & s & "组蔡氏素数:" & vbCrLf & Text2 & "用时" & Timer - t & "秒"

End Sub

ysr

+66步长为4，p3=p2-2的解：
200内有131组蔡氏素数:
5( 1+4),61(25+36),59,4n+2=66
53( 4+49),13(4+9),11,4n+2=66
61( 25+36),5(1+4),3,4n+2=66
13( 4+9),61(25+36),59,4n+2=74
61( 25+36),13(4+9),11,4n+2=74
5( 1+4),73(9+64),71,4n+2=78
17( 1+16),61(25+36),59,4n+2=78
73( 9+64),5(1+4),3,4n+2=78
13( 4+9),73(9+64),71,4n+2=86
73( 9+64),13(4+9),11,4n+2=86
17( 1+16),73(9+64),71,4n+2=90
29( 4+25),61(25+36),59,4n+2=90
89( 25+64),5(1+4),3,4n+2=94
37( 1+36),61(25+36),59,4n+2=98
29( 4+25),73(9+64),71,4n+2=102
41( 16+25),61(25+36),59,4n+2=102
89( 25+64),13(4+9),11,4n+2=102
97( 16+81),5(1+4),3,4n+2=102
101( 1+100),5(1+4),3,4n+2=106
37( 1+36),73(9+64),71,4n+2=110
97( 16+81),13(4+9),11,4n+2=110
5( 1+4),109(9+100),107,4n+2=114
41( 16+25),73(9+64),71,4n+2=114
53( 4+49),61(25+36),59,4n+2=114
101( 1+100),13(4+9),11,4n+2=114
109( 9+100),5(1+4),3,4n+2=114
113( 49+64),5(1+4),3,4n+2=118
13( 4+9),109(9+100),107,4n+2=122
61( 25+36),61(25+36),59,4n+2=122
109( 9+100),13(4+9),11,4n+2=122
17( 1+16),109(9+100),107,4n+2=126
53( 4+49),73(9+64),71,4n+2=126
113( 49+64),13(4+9),11,4n+2=126
61( 25+36),73(9+64),71,4n+2=134
73( 9+64),61(25+36),59,4n+2=134
29( 4+25),109(9+100),107,4n+2=138
137( 16+121),5(1+4),3,4n+2=142
37( 1+36),109(9+100),107,4n+2=146
73( 9+64),73(9+64),71,4n+2=146
41( 16+25),109(9+100),107,4n+2=150
89( 25+64),61(25+36),59,4n+2=150
137( 16+121),13(4+9),11,4n+2=150
149( 49+100),5(1+4),3,4n+2=154
97( 16+81),61(25+36),59,4n+2=158
53( 4+49),109(9+100),107,4n+2=162
89( 25+64),73(9+64),71,4n+2=162
101( 1+100),61(25+36),59,4n+2=162
149( 49+100),13(4+9),11,4n+2=162
157( 36+121),5(1+4),3,4n+2=162
61( 25+36),109(9+100),107,4n+2=170
97( 16+81),73(9+64),71,4n+2=170
109( 9+100),61(25+36),59,4n+2=170
157( 36+121),13(4+9),11,4n+2=170
101( 1+100),73(9+64),71,4n+2=174
113( 49+64),61(25+36),59,4n+2=174
173( 4+169),5(1+4),3,4n+2=178
73( 9+64),109(9+100),107,4n+2=182
109( 9+100),73(9+64),71,4n+2=182
5( 1+4),181(81+100),179,4n+2=186
113( 49+64),73(9+64),71,4n+2=186
173( 4+169),13(4+9),11,4n+2=186
181( 81+100),5(1+4),3,4n+2=186
13( 4+9),181(81+100),179,4n+2=194
181( 81+100),13(4+9),11,4n+2=194
5( 1+4),193(49+144),191,4n+2=198
17( 1+16),181(81+100),179,4n+2=198
89( 25+64),109(9+100),107,4n+2=198
137( 16+121),61(25+36),59,4n+2=198
193( 49+144),5(1+4),3,4n+2=198
197( 1+196),5(1+4),3,4n+2=202
13( 4+9),193(49+144),191,4n+2=206
97( 16+81),109(9+100),107,4n+2=206
193( 49+144),13(4+9),11,4n+2=206
17( 1+16),193(49+144),191,4n+2=210
29( 4+25),181(81+100),179,4n+2=210
101( 1+100),109(9+100),107,4n+2=210
137( 16+121),73(9+64),71,4n+2=210
149( 49+100),61(25+36),59,4n+2=210
197( 1+196),13(4+9),11,4n+2=210
37( 1+36),181(81+100),179,4n+2=218
109( 9+100),109(9+100),107,4n+2=218
157( 36+121),61(25+36),59,4n+2=218
29( 4+25),193(49+144),191,4n+2=222
41( 16+25),181(81+100),179,4n+2=222
113( 49+64),109(9+100),107,4n+2=222
149( 49+100),73(9+64),71,4n+2=222
37( 1+36),193(49+144),191,4n+2=230
157( 36+121),73(9+64),71,4n+2=230
5( 1+4),229(4+225),227,4n+2=234
41( 16+25),193(49+144),191,4n+2=234
53( 4+49),181(81+100),179,4n+2=234
173( 4+169),61(25+36),59,4n+2=234
13( 4+9),229(4+225),227,4n+2=242
61( 25+36),181(81+100),179,4n+2=242
181( 81+100),61(25+36),59,4n+2=242
5( 1+4),241(16+225),239,4n+2=246
17( 1+16),229(4+225),227,4n+2=246
53( 4+49),193(49+144),191,4n+2=246
137( 16+121),109(9+100),107,4n+2=246
173( 4+169),73(9+64),71,4n+2=246
13( 4+9),241(16+225),239,4n+2=254
61( 25+36),193(49+144),191,4n+2=254
73( 9+64),181(81+100),179,4n+2=254
181( 81+100),73(9+64),71,4n+2=254
193( 49+144),61(25+36),59,4n+2=254
17( 1+16),241(16+225),239,4n+2=258
29( 4+25),229(4+225),227,4n+2=258
149( 49+100),109(9+100),107,4n+2=258
197( 1+196),61(25+36),59,4n+2=258
37( 1+36),229(4+225),227,4n+2=266
73( 9+64),193(49+144),191,4n+2=266
157( 36+121),109(9+100),107,4n+2=266
193( 49+144),73(9+64),71,4n+2=266
29( 4+25),241(16+225),239,4n+2=270
41( 16+25),229(4+225),227,4n+2=270
89( 25+64),181(81+100),179,4n+2=270
197( 1+196),73(9+64),71,4n+2=270
37( 1+36),241(16+225),239,4n+2=278
97( 16+81),181(81+100),179,4n+2=278
41( 16+25),241(16+225),239,4n+2=282
53( 4+49),229(4+225),227,4n+2=282
89( 25+64),193(49+144),191,4n+2=282
101( 1+100),181(81+100),179,4n+2=282
173( 4+169),109(9+100),107,4n+2=282
61( 25+36),229(4+225),227,4n+2=290
97( 16+81),193(49+144),191,4n+2=290
109( 9+100),181(81+100),179,4n+2=290
181( 81+100),109(9+100),107,4n+2=290
53( 4+49),241(16+225),239,4n+2=294
101( 1+100),193(49+144),191,4n+2=294
113( 49+64),181(81+100),179,4n+2=294
用时0.249999999998835秒

ysr

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

q = Val(Text1)
m = 66
Do While m <= 300

p1 = 5

Do While p1 <= q And p1 <= m
a1 = 1
B1 = Sqr(Val(p1 - a1 ^ 2))
Do While InStr(B1, ".") <> 0 And p1 > Val(a1 ^ 2)
a1 = a1 + 1
B1 = Sqr(Abs(Val(p1 - a1 ^ 2)))
Loop
p2 = m - p1

C2 = 1
D2 = Sqr(Val(p2 - C2 ^ 2))
Do While InStr(D2, ".") <> 0 And p2 > Val(C2 ^ 2)
C2 = C2 + 1
D2 = Sqr(Abs(Val(p2 - C2 ^ 2)))
Loop

p3 = Abs(Val(p2 - 2))


a = fenjieyinzi(Val(p1))
B = fenjieyinzi(Val(p2))
c = fenjieyinzi(Val(p3))


If InStr(a, "*") = 0 And InStr(B, "*") = 0 And InStr(c, "*") = 0 And InStr(B1, ".") = 0 And InStr(D2, ".") = 0 Then
s = s + 1
Print p1, p2, p3, p4
Text2 = Text2 & CStr(p1) & "( " & a1 ^ 2 & "+" & B1 ^ 2 & ")," & CStr(p2) & "(" & C2 ^ 2 & "+" & D2 ^ 2 & ")," & p3 & ",4n+2=" & m & vbCrLf

Else
s = s
End If


p1 = Val(p1 + 2)

Loop
m = Val(m + 4)
Loop
Combo1 = q & "内有" & s & "组蔡氏素数:" & vbCrLf & Text2 & "用时" & Timer - t & "秒"

End Sub