[QUOTE=science_man_88;265895]oops sorry mixed up [TEX]16*f+ 2^{p-1}-3 \equiv 0 \text { mod } 2^p-1[/TEX][/QUOTE]

okay I suck point is we know [TEX]\text {the rest } \equiv 0 mod 16[/TEX]

[CODE]for(x=1,39,s=4;until(s>2^Me(x)-1,s=s^2-2);print(s%4*x+3","x))[/CODE]

looks pretty ordinary but appears to yield a pattern that might help if it's not trivial the results I get for the mod of the index into the Mersenne primes all follow 2x+3 where x is the value of the index.

never mind I see why it becomes 2x+3 because for [TEX]{s>=194} \equiv 2 \text { mod } 4[/TEX] I messed up the code.

I've finally realized that:

[TEX]S_0=4=P+1 ; P=2^2-1[/TEX]

we know that:

[TEX]M_(p+x)=(2^x)M_p +M_x[/TEX]

so we've related both back to P. going through [TEX](P+1)^2-2[/TEX] we show the first exponent will always be 2^x where x is the index into the sequence. outside of giving the coefficients follow 1 or multiples of 2^x I can't see what else to do outside of determine[TEX] (P+1)^{2y} \text { mod } M_(p+y)[/TEX]

[QUOTE=science_man_88;270465]outside of determine[TEX] (P+1)^{2y} \text { mod } M_(p+y)[/TEX][/QUOTE]

[CODE](11:42)>P=3;for(y=1,200,print(((P+1)^(2*y))%((2^y)*P+(2^y-1))","y))
2,1
1,2
4,3
16,4
64,5
1,6
2,7
4,8
8,9
16,10
32,11
64,12
128,13
256,14
512,15
1024,16
2048,17
4096,18
8192,19
16384,20
32768,21
65536,22
131072,23
262144,24
524288,25
1048576,26
2097152,27
4194304,28
8388608,29
16777216,30
33554432,31
67108864,32
134217728,33
268435456,34
536870912,35
1073741824,36
2147483648,37
4294967296,38
8589934592,39
17179869184,40
34359738368,41
68719476736,42
137438953472,43
274877906944,44
549755813888,45
1099511627776,46
2199023255552,47
4398046511104,48
8796093022208,49
17592186044416,50
35184372088832,51
70368744177664,52
140737488355328,53
281474976710656,54
562949953421312,55
1125899906842624,56
2251799813685248,57
4503599627370496,58
9007199254740992,59
18014398509481984,60
36028797018963968,61
72057594037927936,62
144115188075855872,63
288230376151711744,64
576460752303423488,65
1152921504606846976,66
2305843009213693952,67
4611686018427387904,68
9223372036854775808,69
18446744073709551616,70
36893488147419103232,71
73786976294838206464,72
147573952589676412928,73
295147905179352825856,74
590295810358705651712,75
1180591620717411303424,76
2361183241434822606848,77
4722366482869645213696,78
9444732965739290427392,79
18889465931478580854784,80
37778931862957161709568,81
75557863725914323419136,82
151115727451828646838272,83
302231454903657293676544,84
604462909807314587353088,85
1208925819614629174706176,86
2417851639229258349412352,87
4835703278458516698824704,88
9671406556917033397649408,89
19342813113834066795298816,90
38685626227668133590597632,91
77371252455336267181195264,92
154742504910672534362390528,93
309485009821345068724781056,94
618970019642690137449562112,95
1237940039285380274899124224,96
2475880078570760549798248448,97
4951760157141521099596496896,98
9903520314283042199192993792,99
19807040628566084398385987584,100
39614081257132168796771975168,101
79228162514264337593543950336,102
158456325028528675187087900672,103
316912650057057350374175801344,104
633825300114114700748351602688,105
1267650600228229401496703205376,106
2535301200456458802993406410752,107
5070602400912917605986812821504,108
10141204801825835211973625643008,109
20282409603651670423947251286016,110
40564819207303340847894502572032,111
81129638414606681695789005144064,112
162259276829213363391578010288128,113
324518553658426726783156020576256,114
649037107316853453566312041152512,115
1298074214633706907132624082305024,116
2596148429267413814265248164610048,117
5192296858534827628530496329220096,118
10384593717069655257060992658440192,119
20769187434139310514121985316880384,120
41538374868278621028243970633760768,121
83076749736557242056487941267521536,122
166153499473114484112975882535043072,123
332306998946228968225951765070086144,124
664613997892457936451903530140172288,125
1329227995784915872903807060280344576,126
2658455991569831745807614120560689152,127
5316911983139663491615228241121378304,128
10633823966279326983230456482242756608,129
21267647932558653966460912964485513216,130
42535295865117307932921825928971026432,131
85070591730234615865843651857942052864,132
170141183460469231731687303715884105728,133
340282366920938463463374607431768211456,134
680564733841876926926749214863536422912,135
1361129467683753853853498429727072845824,136
2722258935367507707706996859454145691648,137
5444517870735015415413993718908291383296,138
10889035741470030830827987437816582766592,139
21778071482940061661655974875633165533184,140
43556142965880123323311949751266331066368,141
87112285931760246646623899502532662132736,142
174224571863520493293247799005065324265472,143
348449143727040986586495598010130648530944,144
696898287454081973172991196020261297061888,145
1393796574908163946345982392040522594123776,146
2787593149816327892691964784081045188247552,147
5575186299632655785383929568162090376495104,148
11150372599265311570767859136324180752990208,149
22300745198530623141535718272648361505980416,150
44601490397061246283071436545296723011960832,151
89202980794122492566142873090593446023921664,152
178405961588244985132285746181186892047843328,153
356811923176489970264571492362373784095686656,154
713623846352979940529142984724747568191373312,155
1427247692705959881058285969449495136382746624,156
2854495385411919762116571938898990272765493248,157
5708990770823839524233143877797980545530986496,158
11417981541647679048466287755595961091061972992,159
22835963083295358096932575511191922182123945984,160
45671926166590716193865151022383844364247891968,161
91343852333181432387730302044767688728495783936,162
182687704666362864775460604089535377456991567872,163
365375409332725729550921208179070754913983135744,164
730750818665451459101842416358141509827966271488,165
1461501637330902918203684832716283019655932542976,166
2923003274661805836407369665432566039311865085952,167
5846006549323611672814739330865132078623730171904,168
11692013098647223345629478661730264157247460343808,169
23384026197294446691258957323460528314494920687616,170
46768052394588893382517914646921056628989841375232,171
93536104789177786765035829293842113257979682750464,172
187072209578355573530071658587684226515959365500928,173
374144419156711147060143317175368453031918731001856,174
748288838313422294120286634350736906063837462003712,175
1496577676626844588240573268701473812127674924007424,176
2993155353253689176481146537402947624255349848014848,177
5986310706507378352962293074805895248510699696029696,178
11972621413014756705924586149611790497021399392059392,179
23945242826029513411849172299223580994042798784118784,180
47890485652059026823698344598447161988085597568237568,181
95780971304118053647396689196894323976171195136475136,182
191561942608236107294793378393788647952342390272950272,183
383123885216472214589586756787577295904684780545900544,184
766247770432944429179173513575154591809369561091801088,185
1532495540865888858358347027150309183618739122183602176,186
3064991081731777716716694054300618367237478244367204352,187
6129982163463555433433388108601236734474956488734408704,188
12259964326927110866866776217202473468949912977468817408,189
24519928653854221733733552434404946937899825954937634816,190
49039857307708443467467104868809893875799651909875269632,191
98079714615416886934934209737619787751599303819750539264,192
196159429230833773869868419475239575503198607639501078528,193
392318858461667547739736838950479151006397215279002157056,194
784637716923335095479473677900958302012794430558004314112,195
1569275433846670190958947355801916604025588861116008628224,196
3138550867693340381917894711603833208051177722232017256448,197
6277101735386680763835789423207666416102355444464034512896,198
12554203470773361527671578846415332832204710888928069025792,199
25108406941546723055343157692830665664409421777856138051584,200[/CODE]

so it looks that after y=5 it's 2^(y-6) but that's a vast generalization I can't back up. I might have messed up somewhere.

[QUOTE=science_man_88;270468][CODE](11:42)>P=3;for(y=1,200,print(((P+1)^(2*y))%((2^y)*P+(2^y-1))","y))
[...][/CODE]

so it looks that after y=5 it's 2^(y-6) but that's a vast generalization I can't back up. I might have messed up somewhere.[/QUOTE]

First, if you want to test something like this (up to a finite bound) have GP do it for you:
[CODE]P=3;for(y=6,1e4,if(((P+1)^(2*y))%((2^y)*P+(2^y-1))!=2^(y-6),print(y)))[/CODE]

But this one is easier to prove than to check. Notice that 2^(y+2) = 1 mod ((2^y)*3+(2^y-1)) and so (2^(y+2))^3 = 1^3 = 1 to that modulus. Now what can you conclude?

[QUOTE=CRGreathouse;270531]First, if you want to test something like this (up to a finite bound) have GP do it for you:
[CODE]P=3;for(y=6,1e4,if(((P+1)^(2*y))%((2^y)*P+(2^y-1))!=2^(y-6),print(y)))[/CODE]

But this one is easier to prove than to check. Notice that 2^(y+2) = 1 mod ((2^y)*3+(2^y-1)) and so (2^(y+2))^3 = 1^3 = 1 to that modulus. Now what can you conclude?[/QUOTE]

my guess is if there's a first, there's a second, so it's likely that it's bad by what I see.

[QUOTE=science_man_88;270559]my guess is if there's a first, there's a second, so it's likely that it's bad by what I see.[/QUOTE]

but so far it hasn't failed ( and yes my windows.old folder had my old copy of PARI).

the reasoning that got me interested in (P+1) to some even power is:

[TEX]S_0=(P+1)[/TEX]
[TEX]S_1=(P+1)^2-2[/TEX]
[TEX]S_2=(P+1)^4-4(P+1)^2+4[/TEX]

lets say (P+1)^4 = 1 mod W and (P+1)^2 =3 mod W then the last equation gets reduced to, 1-(3*4)+4 = 1-12+4 = 1-8 = -7 we've solved for it.

[QUOTE=science_man_88;270588]the reasoning that got me interested in (P+1) to some even power is:

[TEX]S_0=(P+1)[/TEX]
[TEX]S_1=(P+1)^2-2[/TEX]
[TEX]S_2=(P+1)^4-4(P+1)^2+4[/TEX]

lets say (P+1)^4 = 1 mod W and (P+1)^2 =3 mod W then the last equation gets reduced to, 1-(3*4)+4 = 1-12+4 = 1-8 = -7 we've solved for it.[/QUOTE]

so usual error of forgetting to subtract 2.

so the answer would be -9

[QUOTE=science_man_88;270610]so usual error of forgetting to subtract 2.

so the answer would be -9[/QUOTE]

[CODE](21:12)>C=vector(11,n,0);D=vector(11,n,0);C[1]=P;D[1]=P+1;for(x=2,11,C[x]=C[x-1]*2+1;D[x]=D[x-1]^2-2)
(21:12)>##
*** last result computed in 62 ms.
(21:12)>C=vector(11,n,0);D=vector(11,n,0);C[1]=3;D[1]=4;for(x=2,11,C[x]=C[x-1]*2+1;D[x]=D[x-1]^2-2)
(21:13)>##
*** last result computed in 0 ms.[/CODE]

I've tried this ( had to rewrite it as I closed PARI) , and I've tried it with a vector that takes the modular remainder. the problem with that one for me is the algebraic version returns -7/4 for the second one. so obviously I don't know what PARI is doing to it.

[QUOTE=science_man_88;270685][CODE](21:12)>C=vector(11,n,0);D=vector(11,n,0);C[1]=P;D[1]=P+1;for(x=2,11,C[x]=C[x-1]*2+1;D[x]=D[x-1]^2-2)
(21:12)>##
*** last result computed in 62 ms.
(21:12)>C=vector(11,n,0);D=vector(11,n,0);C[1]=3;D[1]=4;for(x=2,11,C[x]=C[x-1]*2+1;D[x]=D[x-1]^2-2)
(21:13)>##
*** last result computed in 0 ms.[/CODE]

I've tried this ( had to rewrite it as I closed PARI) , and I've tried it with a vector that takes the modular remainder. the problem with that one for me is the algebraic version returns -7/4 for the second one. so obviously I don't know what PARI is doing to it.[/QUOTE]

Another line I've thought about is the fact that when I return D:
[CODE]
[[B]P + 1[/B], P^2 + [B]2*P - 1[/B], P^4 + 4*P^3 + 2*P^2 - [B]4*P - 1[/B], P^8 + 8*P^7 + 20*P^6 + 8*P^5 - 30*P^4 - 24*P^3 + 12*P^2 + [B]8*P - 1[/B],[/CODE]

All the highlighted endings are off by 2^x from the one that we check with it. Where x is the index into the sequence starting at x=0. P+1 is 1 over P, (2*p+1)-(2*p-1) =2, (4*p+3)-(4*p-1) = 4 and (8*P+7)-(8*P-1) = 8, so these can be turned into 1 term together.