mersenneforum.org - theory on Mersenne primes ?

mersenneforum.org (https://www.mersenneforum.org/index.php)

- science_man_88 (https://www.mersenneforum.org/forumdisplay.php?f=140)

- - theory on Mersenne primes ? (https://www.mersenneforum.org/showthread.php?t=14151)

I think i found the error I realize why this format helps it helps you go command by command. the error is in the 3 % 4 code in the last section. it doesn't alter the coordinates. fixed:

[CODE](19:17)>crap()
[COLOR="red"]1[/COLOR],[COLOR="red"]0[/COLOR]:1;
[COLOR="red"]1[/COLOR],[COLOR="Red"]1[/COLOR]:[COLOR="lime"]2[/COLOR];
[COLOR="red"]-1[/COLOR],[COLOR="red"]1[/COLOR]:3;
[COLOR="red"]-1[/COLOR],[COLOR="red"]-1[/COLOR]:[COLOR="Lime"]4[/COLOR];
2,[COLOR="red"]2[/COLOR]:[COLOR="lime"]5[/COLOR];
[COLOR="red"]-2[/COLOR],[COLOR="red"]2[/COLOR]:[COLOR="lime"]6[/COLOR];
[COLOR="red"]-2[/COLOR],[COLOR="Red"]0[/COLOR]:7;
[COLOR="red"]3[/COLOR],[COLOR="Red"]3[/COLOR]:[COLOR="lime"]8[/COLOR];
[COLOR="Red"]0[/COLOR],4:[COLOR="lime"]9[/COLOR];
5,[COLOR="red"]3[/COLOR]:10;
-5,[COLOR="red"]-1[/COLOR]:11;
6,[COLOR="Red"]0[/COLOR]:[COLOR="Lime"]12[/COLOR];
[COLOR="red"]3[/COLOR],-11:13;
-6,-12:[COLOR="lime"]14[/COLOR];
[COLOR="red"]0[/COLOR],[COLOR="red"]18[/COLOR]:15;
17,-23:16;
[COLOR="Red"]0[/COLOR],24:17;
-4,[COLOR="Red"]-28[/COLOR]:18;
33,-5:19;
-33,-33:[COLOR="lime"]20[/COLOR];
-49,-35:21;
10,50:22;
-29,53:23;
71,-15:24;
74,[COLOR="red"]18[/COLOR]:25;
-76,[COLOR="red"]-28[/COLOR]:26;
81,-105:27;
47,147:28;
-166,-112:29;
182,98:30;
98,-232:31;
-373,435:32;
464,-360:33;
537,561:34;
-591,-553:35;
863,-267:36;
-869,137:37;
-968,-1320:38;
149,1835:39;
-2291,1005:40;[/CODE]

red = horizontal and verticals relations I've found. note I only named it as a function so I could copy the code and use it fast. green are the easy diagonals ((1,1) or (-1,1) is what I'm counting.) I've found.

[QUOTE=science_man_88;251190]I think i found the error I realize why this format helps it helps you go command by command. the error is in the 3 % 4 code in the last section. it doesn't alter the coordinates. fixed:

[CODE](19:17)>crap()
[COLOR="red"]1[/COLOR],[COLOR="red"]0[/COLOR]:1;
[COLOR="red"]1[/COLOR],[COLOR="Red"]1[/COLOR]:[COLOR="lime"]2[/COLOR];
[COLOR="red"]-1[/COLOR],[COLOR="red"]1[/COLOR]:3;
[COLOR="red"]-1[/COLOR],[COLOR="red"]-1[/COLOR]:[COLOR="Lime"]4[/COLOR];
2,[COLOR="red"]2[/COLOR]:[COLOR="lime"]5[/COLOR];
[COLOR="red"]-2[/COLOR],[COLOR="red"]2[/COLOR]:[COLOR="lime"]6[/COLOR];
[COLOR="red"]-2[/COLOR],[COLOR="Red"]0[/COLOR]:[COLOR="Lime"]7[/COLOR];
[COLOR="red"]3[/COLOR],[COLOR="Red"]3[/COLOR]:[COLOR="lime"]8[/COLOR];
[COLOR="Red"]0[/COLOR],4:[COLOR="lime"]9[/COLOR];
5,[COLOR="red"]3[/COLOR]:10;
-5,[COLOR="red"]-1[/COLOR]:[COLOR="lime"]11[/COLOR];
6,[COLOR="Red"]0[/COLOR]:[COLOR="Lime"]12[/COLOR];
[COLOR="red"]3[/COLOR],-11:13;
-6,-12:[COLOR="lime"]14[/COLOR];
[COLOR="red"]0[/COLOR],[COLOR="red"]18[/COLOR]:15;
17,-23:16;
[COLOR="Red"]0[/COLOR],24:[COLOR="lime"]17[/COLOR];
-4,[COLOR="Red"]-28[/COLOR]:18;
33,-5:19;
-33,-33:[COLOR="lime"]20[/COLOR];
-49,-35:21;
10,50:22;
-29,53:23;
71,-15:24;
74,[COLOR="red"]18[/COLOR]:25;
-76,[COLOR="red"]-28[/COLOR]:26;
81,-105:27;
47,147:28;
-166,-112:29;
182,98:30;
98,-232:31;
-373,435:32;
464,-360:33;
537,561:[COLOR="lime"]34[/COLOR];
-591,-553:35;
863,-267:36;
-869,137:37;
-968,-1320:38;
149,1835:39;
-2291,1005:40;[/CODE]

red = horizontal and verticals relations I've found. note I only named it as a function so I could copy the code and use it fast. green are the easy diagonals ((1,1) or (-1,1) is what I'm counting.) I've found.[/QUOTE]

this is all my formula for diagonals finds stupid law of small numbers by the looks of it. just what you'd expect almost exactly 50% have a relation.

my codes for checking straight lines and diagonals of 1,1 and -1,1 :

[CODE](21:12)>?crap1
crap1()=for(x=1,#mersennex,for(y=1,#mersenney,if(x!=y&&(mersennex[x]-mersennex[y]==mersenney[x]-mersenney[y]),print(x","y))))
(21:14)>?crap2
crap2()=for(x=1,#mersennex,for(y=1,#mersenney,if(x!=y&&(mersennex[x]-mersennex[y]==0||mersenney[x]-mersenney[y]==0),print(x","y))))[/CODE]

[URL="http://wackymania.com/image/2011/1/exam/exam-06.jpg"][B]c[/B][/URL]?

°JC

I would like to know the odds of something before I post this next idea. That thing is the odds of a even composites prime factors adding up to a prime, I would like to know the odds of this because my next idea is based on thinking this might not be as common as it sounds. if not I have an idea that appears to work for all x-th Mersenne exponents, for 39>x>6

[QUOTE=science_man_88;256304]I would like to know the odds of something before I post this next idea. That thing is the odds of a even composites prime factors adding up to a prime, I would like to know the odds of this because my next idea is based on thinking this might not be as common as it sounds. if not I have an idea that appears to work for all x-th Mersenne exponents, for 39>x>6[/QUOTE]

The sum of all its prime factors or the sum of some of its prime factors?

[QUOTE=CRGreathouse;256307]The sum of all its prime factors or the sum of some of its prime factors?[/QUOTE]

all of the prime factors including all muliples like in pari I've done this for all the differences between the exponents 39>x>7 as Me(x)-Me(x-1) and continued the resaon I didn't go lower is because 4 ( the difference I would have got next) doesn't work because 2+2 = 4 =2*2=4=2+2 etc. so I know 4 doesn't work but all the differences until x=39 from x=7 work out that the repetitive sum and factor process eventually seems to lead to a prime ( if the difference is 2 then it works just as a sum of all of the prime factors of 2).

[CODE](18:46)>Me(39)-Me(38)
%8 = 6494324
(18:46)>factor(%)
%9 =
[2 2]

[787 1]

[2063 1]

(18:47)>2+2+787+2063
%10 = 2854
(18:47)>2+8+5+4
%11 = 19
(18:47)>factor(2854)
%12 =
[2 1]

[1427 1]

(18:47)>2+1427
%13 = 1429
(18:48)>isprime(%)
%14 = 1
(18:48)>Me(38)-Me(37)
%15 = 3951216
(19:00)>factor(%)
%16 =
[2 4]

[3 2]

[23 1]

[1193 1]

(19:00)>2+2+2+2+3+3+23+1193
%17 = 1230
(19:00)>factor(%)
%18 =
[2 1]

[3 1]

[5 1]

[41 1]

(19:00)>2+3+5+41
%19 = 51
(19:00)>factor(51)
%20 =
[3 1]

[17 1]

(19:01)>3+17
%21 = 20
(19:01)>factor(20)
%22 =
[2 2]

[5 1]

(19:01)>2+5
%23 = 7
(19:01)>Me(37)-Me(36)
%24 = 45156
(19:01)>factor(%)
%25 =
[2 2]

[3 1]

[53 1]

[71 1]

(19:01)>2+2+3+53+71
%26 = 131
(19:01)>isprime(%)
%27 = 1
(19:02)>isprime(129)
%28 = 0
(19:02)>isprime(133)
%29 = 0
(19:02)>Me(36)-Me(35)
%30 = 1577952
(19:02)>factor(%)
%31 =
[2 5]

[3 2]

[5479 1]

(19:02)>2+2+2+2+2+3+3+5479
%32 = 5495
(19:03)>factor(%)
%33 =
[5 1]

[7 1]

[157 1]

(19:03)>5+7+157
%34 = 169
(19:03)>factor(%)
%35 =
[13 2]

(19:03)>13+13
%36 = 26
(19:03)>factor(%)
%37 =
[2 1]

[13 1]

(19:03)>2+13
%38 = 15
(19:03)>factor(%)
%39 =
[3 1]

[5 1]

(19:03)>3+5
%40 = 8
(19:03)>factor(%)
%41 =
[2 3]

(19:03)>2+2+2
%42 = 6
(19:03)>factor(%)
%43 =
[2 1]

[3 1]

(19:03)>2+3
%44 = 5
(19:03)>Me(35)-Me(34)
%45 = 140482
(19:04)>factor(%)
%46 =
[2 1]

[70241 1]

(19:04)>2+70241
%47 = 70243
(19:04)>isprime(%)
%48 = 0
(19:05)>factor(%47)
%49 =
[19 1]

[3697 1]

(19:05)>19+3697
%50 = 3716
(19:05)>factor(%)
%51 =
[2 2]

[929 1]

(19:05)>2+929
%52 = 931
(19:05)>factor(%)
%53 =
[7 2]

[19 1]

(19:05)>7+19
%54 = 26
(19:05)>factor(%)
%55 =
[2 1]

[13 1]

(19:05)>7+7+19
%56 = 33
(19:05)>factor(%)
%57 =
[3 1]

[11 1]

(19:06)>3+11
%58 = 14
(19:06)>factor(%)
%59 =
[2 1]

[7 1]

(19:06)>2+7
%60 = 9
(19:06)>factor(%)
%61 =
[3 2]

(19:06)>3+3
%62 = 6
(19:06)>factor(%)
%63 =
[2 1]

[3 1]

(19:06)>2+3
%64 = 5
(19:06)>Me(34)-Me(33)
%65 = 398354
(19:06)>factor(%)
%66 =
[2 1]

[11 1]

[19 1]

[953 1]

(19:06)>2+11+19+953
%67 = 985
(19:06)>factor(%)
%68 =
[5 1]

[197 1]

(19:07)>5+197
%69 = 202
(19:07)>factor(%)
%70 =
[2 1]

[101 1]

(19:07)>2+101
%71 = 103
(19:07)>factor(%)
%72 =
[103 1]

(19:07)>c=factor(24)
%73 =
[2 3]

[3 1]

(19:08)>c[1,1]+c[1,2]
%74 = 5
(19:08)>c=factor(24)
%75 =
[2 3]

[3 1]

(19:09)>c=factor(24);for(x=1,#c,print(c[x,1]))
2
3
(19:10)>c=factor(120)
%76 =
[2 3]

[3 1]

[5 1]

(19:10)>c=factor(24);for(x=1,#c,print(c[x,1]))
2
3
(19:10)>c=factor(120);for(x=1,#c,print(c[x,1]))
2
3
(19:10)>c=factor(120);for(x=1,#c,print(c[x,1]))
2
3
(19:10)>Me(33)-Me(32)
%77 = 102594
(19:42)>factor(%)
%78 =
[2 1]

[3 1]

[17099 1]

(19:42)>2+3+17099
%79 = 17104
(19:42)>factor(%)
%80 =
[2 4]

[1069 1]

(19:42)>2+2+2+2+1069
%81 = 1077
(19:42)>factor(%)
%82 =
[3 1]

[359 1]

(19:42)>3+359
%83 = 362
(19:42)>factor(%)
%84 =
[2 1]

[181 1]

(19:42)>2+181
%85 = 183
(19:42)>factor(%)
%86 =
[3 1]

[61 1]

(19:42)>3+61
%87 = 64
(19:42)>factor(%)
%88 =
[2 6]

(19:42)>2+2+2+2+2+2
%89 = 12
(19:42)>factor(%)
%90 =
[2 2]

[3 1]

(19:42)>2+2+3
%91 = 7
(19:43)>Me(32)-Me(31)
%92 = 540748
(19:45)>factor(%)
%93 =
[2 2]

[13 1]

[10399 1]

(19:45)>2+2+13+10399
%94 = 10416
(19:46)>factor(%)
%95 =
[2 4]

[3 1]

[7 1]

[31 1]

(19:46)>2=2=2=2=3=7=31
*** unused characters: 2=2=2=2=3=7=31
^-------------
(19:46)>2+2+2+2+3+7+31
%96 = 49
(19:46)>factor(%)
%97 =
[7 2]

(19:46)>7+7
%98 = 14
(19:46)>factor(%)
%99 =
[2 1]

[7 1]

(19:46)>2+7
%100 = 9
(19:46)>factor(%)
%101 =
[3 2]

(19:46)>3+3
%102 = 6
(19:46)>factor(%)
%103 =
[2 1]

[3 1]

(19:46)>2+3
%104 = 5
(19:46)>Me(31)-Me(30)
%105 = 84042
(19:47)>factor(%)
%106 =
[2 1]

[3 2]

[7 1]

[23 1]

[29 1]

(19:47)>2+3+3+7+23+29
%107 = 67
(19:47)>factor(%)
%108 =
[67 1]

(19:47)>Me(30)-Me(29)
%109 = 21546
(19:48)>factor(%)
%110 =
[2 1]

[3 4]

[7 1]

[19 1]

(19:48)>2+3+3+3+3+7+19
%111 = 40
(19:48)>factor(%)
%112 =
[2 3]

[5 1]

(19:48)>2+2+2+5
%113 = 11
(19:48)>Me(29)-Me(28)
%114 = 24260
(19:48)>2+2+2+5
%115 = 11
(19:48)>Me(29)-Me(28)
%116 = 24260
(19:49)>factor(%)
%117 =
[2 2]

[5 1]

[1213 1]

(19:49)>2+2+5+1213
%118 = 1222
(19:49)>factor(%)
%119 =
[2 1]

[13 1]

[47 1]

(19:49)>2+13+47
%120 = 62
(19:49)>factor(%)
%121 =
[2 1]

[31 1]

(19:49)>2+31
%122 = 33
(19:49)>factor(%)
%123 =
[3 1]

[11 1]

(19:49)>3+11
%124 = 14
(19:49)>factor(%)
%125 =
[2 1]

[7 1]

(19:49)>2+7
%126 = 9
(19:49)>factor(%)
%127 =
[3 2]

(19:49)>3+3
%128 = 6
(19:49)>Me(28)-Me(27)
%129 = 41746
(19:50)>factor(%)
%130 =
[2 1]

[20873 1]

(19:50)>2+20873
%131 = 20875
(19:51)>factor(%)
%132 =
[5 3]

[167 1]

(19:51)>5+5+5+167
%133 = 182
(19:51)>factor(%)
%134 =
[2 1]

[7 1]

[13 1]

(19:51)>2+7+13
%135 = 22
(19:51)>factor(%)
%136 =
[2 1]

[11 1]

(19:51)>2+11
%137 = 13
(19:51)>Me(27)-Me(26)
%138 = 21288
(19:51)>factor(%)
%139 =
[2 3]

[3 1]

[887 1]

(19:51)>2+2+2+3+887
%140 = 896
(19:52)>factor(%)
%141 =
[2 7]

[7 1]

(19:52)>2+2+2+2+2+2+2+7
%142 = 21
(19:52)>factor(%)
%143 =
[3 1]

[7 1]

(19:52)>3+7
%144 = 10
(19:52)>factor(%)
%145 =
[2 1]

[5 1]

(19:52)>2+5
%146 = 7
(19:52)>Me(26)-Me(25)
%147 = 1508
(19:52)>factor(%)
%148 =
[2 2]

[13 1]

[29 1]

(19:52)>2+2+13+29
%149 = 46
(19:52)>factor(%)
%150 =
[2 1]

[23 1]

(19:52)>2+23
%151 = 25
(19:53)>factor(%)
%152 =
[5 2]

(19:53)>5+5
%153 = 10
(19:53)>factor(%)
%154 =
[2 1]

[5 1]

(19:53)>Me(25)-Me(24)
%155 = 1764
(19:53)>factor(%)
%156 =
[2 2]

[3 2]

[7 2]

(19:53)>2+2+3+3+7+7
%157 = 24
(19:53)>factor(%)
%158 =
[2 3]

[3 1]

(19:53)>2+2+2+3
%159 = 9
(19:53)>factor(%)
%160 =
[3 2]

(19:53)>3+3
%161 = 6
(19:53)>factor(%)
%162 =
[2 1]

[3 1]

(19:53)>2+3
%163 = 5
(19:53)>Me(24)-Me(23)
%164 = 8724
(19:53)>factor(%)
%165 =
[2 2]

[3 1]

[727 1]

(19:53)>2+2+3+727
%166 = 734
(19:54)>factor(%)
%167 =
[2 1]

[367 1]

(19:54)>2+367
%168 = 369
(19:54)>factor(%)
%169 =
[3 2]

[41 1]

(19:54)>3+3+41
%170 = 47
(19:54)>factor(%)
%171 =
[47 1]

(19:54)>Me(23)-Me(22)
%172 = 1272
(19:54)>factor(%)
%173 =
[2 3]

[3 1]

[53 1]

(19:54)>2+2+2+3+53
%174 = 62
(19:54)>factor(%)
%175 =
[2 1]

[31 1]

(19:54)>Me(22)-Me(21)
%176 = 252
(19:55)>factor(%)
%177 =
[2 2]

[3 2]

[7 1]

(19:55)>2=2=3=3=7
*** unused characters: 2=2=3=3=7
^--------
(19:55)>2+2+3+3+7
%178 = 17
(19:55)>Me(21)-Me(20)
%179 = 5266
(07:55)>factor(%)
%180 =
[2 1]

[2633 1]

(07:55)>2+2633
%181 = 2635
(07:55)>factor(%)
%182 =
[5 1]

[17 1]

[31 1]

(07:55)>5+17+31
%183 = 53
(07:55)>factor(%)
%184 =
[53 1]

(07:55)>Me(20)-Me(19)
%185 = 170
(07:55)>factor(%)
%186 =
[2 1]

[5 1]

[17 1]

(07:55)>2+5+17
%187 = 24
(07:56)>factor(%)
%188 =
[2 3]

[3 1]

(07:56)>2+2+2+3
%189 = 9
(07:56)>factor(%)
%190 =
[3 2]

(07:56)>Me(19)-Me(18)
%191 = 1036
(07:56)>factor(%)
%192 =
[2 2]

[7 1]

[37 1]

(07:56)>2+2+7+37
%193 = 48
(07:56)>factor(%)
%194 =
[2 4]

[3 1]

(07:56)>2+2+2+2+3
%195 = 11
(07:56)>Me(18)-Me(17)
%196 = 936
(07:57)>factor(%)
%197 =
[2 3]

[3 2]

[13 1]

(07:57)>2+2+2+3+3+13
%198 = 25
(07:57)>factor(%)
%199 =
[5 2]

(07:57)>5+5
%200 = 10
(07:57)>factor(%)
%201 =
[2 1]

[5 1]

(07:57)>Me(17)-Me(16)
%202 = 78
(07:57)>factor(%)
%203 =
[2 1]

[3 1]

[13 1]

(07:57)>2+3+13
%204 = 18
(07:58)>factor(%)
%205 =
[2 1]

[3 2]

(07:58)>2+3+3
%206 = 8
(07:58)>factor(%)
%207 =
[2 3]

(07:58)>2+2+2
%208 = 6
(07:58)>factor(%)
%209 =
[2 1]

[3 1]

(07:58)>Me(16)-Me(15)
%210 = 924
(07:58)>factor(%)
%211 =
[2 2]

[3 1]

[7 1]

[11 1]

(07:58)>2+2+3+7+11
%212 = 25
(07:58)>factor(%)
%213 =
[5 2]

(07:58)>Me(15)-Me(14)
%214 = 672
(07:59)>factor(%)
%215 =
[2 5]

[3 1]

[7 1]

(07:59)>2+2+2+2+2+3+7
%216 = 20
(07:59)>factor(%)
%217 =
[2 2]

[5 1]

(07:59)>Me(14)-Me(13)
%218 = 86
(07:59)>factor(%)
%219 =
[2 1]

[43 1]

(07:59)>2+43
%220 = 45
(07:59)>factor(%)
%221 =
[3 2]

[5 1]

(07:59)>3+3+5
%222 = 11
(07:59)>Me(13)-Me(12)
%223 = 394
(08:00)>factor(%)
%224 =
[2 1]

[197 1]

(08:00)>2+`97
*** syntax error, unexpected '`': 2+`97
^---
(08:00)>2+197
%225 = 199
(08:00)>factor(%)
%226 =
[199 1]

(08:00)>Me(12)-Me(11)
%227 = 20
(08:00)>factor(%)
%228 =
[2 2]

[5 1]

(08:00)>Me(11)-Me(10)
%229 = 18
(08:00)>factor(%)
%230 =
[2 1]

[3 2]

(08:00)>2+3+3
%231 = 8
(08:00)>factor(%)
%232 =
[2 3]

(08:00)>Me(10)-Me(9)
%233 = 28
(08:01)>factor(%)
%234 =
[2 2]

[7 1]

(08:01)>2+2+7
%235 = 11
(08:01)>Me(9)-Me(8)
%236 = 30
(08:01)>factor(%)
%237 =
[2 1]

[3 1]

[5 1]

(08:01)>2+3+5
%238 = 10
(08:01)>factor(%)
%239 =
[2 1]

[5 1]

(08:01)>Me(8)-Me(7)
%240 = 12
(08:01)>factor(%)
%241 =
[2 2]

[3 1]

(08:01)>Me(7)-Me(6)
%242 = 2[/CODE]

heres what I did. I'll have to figure a proof or disproof though which doesn't seem to be easy for me. I found an error looking this over but I know from what else I've done that it didn't affect my result because the new value works out.

[QUOTE=science_man_88;256308]all of the prime factors including all muliples[/QUOTE]

That's sopfr, Sloane's [url=https://oeis.org/A001414]A001414[/url].

With high probability for random n, sopfr(n) is approximately (within a factor of epsilon) equal to the largest prime factor of n. The largest prime factor of n is typically between n^a and n^b for 0 < a < b < 1 -- depending on how much of the time 'typically' has to cover you can shift the values around. Let's say .3 and .9, which gives ~90% coverage. The chance of a random number around n^.3 being prime is 1/log(n^.3) = 1/(.3 * log n) and similarly with .9.

So a decent first guess would be 'somewhere between 10 / 3log n and 10 / 9log n'.

[QUOTE=science_man_88;256308]all of the prime factors including all muliples like in pari I've done this for all the differences between the exponents 39>x>7 as Me(x)-Me(x-1) and continued the resaon I didn't go lower is because 4 ( the difference I would have got next) doesn't work because 2+2 = 4 =2*2=4=2+2 etc. so I know 4 doesn't work but all the differences until x=39 from x=7 work out that the repetitive sum and factor process eventually seems to lead to a prime ( if the difference is 2 then it works just as a sum of all of the prime factors of 2).[/QUOTE]

So you're looking at sopfr(Me(n+1) - Me(n)) and seeing if it's prime; if not, you look at dsum(sopfr(Me(n+1) - Me(n))) and see if that's prime; if not, you look at dsum(dsum(sopfr(Me(n+1) - Me(n)))) and see if that's prime. I think you could get a prime from a lot of numbers with that process.

Let's try it with random 9-digit numbers:
[code]is(a,b)=my(n=sopfr(a-b));if(isprime(n),1,n=dsum(n);isprime(n)|isprime(dsum(n)));
sum(i=1,1000,is(random(10^8*9)+10^8,random(10^8*9)+10^8))
%1 = 599[/code]

So this happens about 60% of the time for random 9-digit numbers. Your numbers should be more likely to make this work, since they're going to be closer together than random 9-digit numbers.

[QUOTE=CRGreathouse;256313]That's sopfr, Sloane's [url=https://oeis.org/A001414]A001414[/url].

With high probability for random n, sopfr(n) is approximately (within a factor of epsilon) equal to the largest prime factor of n. The largest prime factor of n is typically between n^a and n^b for 0 < a < b < 1 -- depending on how much of the time 'typically' has to cover you can shift the values around. Let's say .3 and .9, which gives ~90% coverage. The chance of a random number around n^.3 being prime is 1/log(n^.3) = 1/(.3 * log n) and similarly with .9.

So a decent first guess would be 'somewhere between 10 / 3log n and 10 / 9log n'.

So you're looking at sopfr(Me(n+1) - Me(n)) and seeing if it's prime; if not, you look at dsum(sopfr(Me(n+1) - Me(n))) and see if that's prime; if not, you look at dsum(dsum(sopfr(Me(n+1) - Me(n)))) and see if that's prime. I think you could get a prime from a lot of numbers with that process.

Let's try it with random 9-digit numbers:
[code]is(a,b)=my(n=sopfr(a-b));if(isprime(n),1,n=dsum(n);isprime(n)|isprime(dsum(n)));
sum(i=1,1000,is(random(10^8*9)+10^8,random(10^8*9)+10^8))
%1 = 599[/code]

So this happens about 60% of the time for random 9-digit numbers. Your numbers should be more likely to make this work, since they're going to be closer together than random 9-digit numbers.[/QUOTE]

So in other words I have a 60% chance of this happening ? I'm confused a bit.

okay now what's the odds that the last Mersenne exponent + those given even numbers is prime because the even numbers I asked about are the differences between prime exponents that are known. if it's getting rarer from that could this help at all ?

[QUOTE=science_man_88;256317]So in other words I have a 60% chance of this happening ? I'm confused a bit.[/QUOTE]

Roughly 60% odds if you chose two 9-digit numbers at random rather than choosing consecutive Mersenne exponents. The odds improve if you choose them to be close together.

[QUOTE=science_man_88;256317]okay now what's the odds that the last Mersenne exponent + those given even numbers is prime because the even numbers I asked about are the differences between prime exponents that are known. if it's getting rarer from that could this help at all ?[/QUOTE]

What are "those given even numbers", and how do they vary?

[QUOTE=CRGreathouse;256318]Roughly 60% odds if you chose two 9-digit numbers at random rather than choosing consecutive Mersenne exponents. The odds improve if you choose them to be close together.

What are "those given even numbers", and how do they vary?[/QUOTE]

those given even numbers would be the even numbers that this sopfr ? work for, because according to my idea mersenne prime exponent (x) - mersenne prime exponent (x-1) fits into these even numbers for the x>=7.