# Primality Proof of phi(11971,89)

## OpenPFGW

```Primality testing Phi(11971,89) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling N-1 BLS with factored part 25.64% and helper 0.32% (77.25% proof)
Phi(11971,89) is Fermat and Lucas PRP! (394.8081s+0.0009s)
```

## CHG

```                  GP/PARI CALCULATOR Version 2.3.4 (released)
amd64 running linux (x86-64/GMP-4.3.0 kernel) 64-bit version
compiled: Apr 17 2009, gcc-4.1.2 (Gentoo 4.1.2 p1.0.2)
(readline v5.2 enabled, extended help not available)

Copyright (C) 2000-2006 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and
comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000

Testing a PRP called "89_11971.in".

Pol[1, 1] with [h, u]=[5, 1] has ratio=1.6276345495500688194 E-1380 at X, ratio=3.580073548520162205 E-1380 at Y, witness=2.
Pol[2, 1] with [h, u]=[5, 1] has ratio=0.08660704902544096269 at X, ratio=3.3936549920888186994 E-658 at Y, witness=3.
Pol[3, 1] with [h, u]=[5, 1] has ratio=0.007060644288023561680 at X, ratio=1.3527880275180646789 E-329 at Y, witness=3.
Pol[4, 1] with [h, u]=[4, 1] has ratio=7.694101170290470078 E-112 at X, ratio=1.0465645168878783589 E-275 at Y, witness=3.
Pol[5, 1] with [h, u]=[4, 1] has ratio=1.0407484864493000166 E-193 at X, ratio=1.1030445043210059088 E-275 at Y, witness=5.
Pol[6, 1] with [h, u]=[4, 1] has ratio=4.511090965074270439 E-235 at X, ratio=1.0791938965290461405 E-275 at Y, witness=2.
Pol[7, 1] with [h, u]=[4, 1] has ratio=3.110122511856801657 E-256 at X, ratio=3.3832484827740377300 E-276 at Y, witness=7.
Pol[8, 1] with [h, u]=[4, 1] has ratio=8.872749154206530550 E-266 at X, ratio=5.580498414015664906 E-276 at Y, witness=5.
Pol[9, 1] with [h, u]=[7, 2] has ratio=1.1262493967659725278 E-158 at X, ratio=1.8186119493212746442 E-647 at Y, witness=3.
Pol[10, 1] with [h, u]=[7, 2] has ratio=1.2971375494992113004 E-333 at X, ratio=7.730393532825986227 E-496 at Y, witness=2.
Pol[11, 1] with [h, u]=[7, 2] has ratio=3.475208283184315273 E-224 at X, ratio=1.6776840328877040217 E-332 at Y, witness=3.
Pol[12, 1] with [h, u]=[8, 2] has ratio=0.17191352918397914197 at X, ratio=1.3501666378789559442 E-297 at Y, witness=7.
Pol[13, 1] with [h, u]=[8, 2] has ratio=0.08685480477240545768 at X, ratio=1.1604250092500893465 E-170 at Y, witness=2.
Pol[14, 1] with [h, u]=[8, 2] has ratio=0.09090047494910361583 at X, ratio=5.103062575433676584 E-98 at Y, witness=3.
Pol[15, 1] with [h, u]=[9, 3] has ratio=0.10401045455876733599 at X, ratio=6.498699574866059618 E-331 at Y, witness=19.
Pol[16, 1] with [h, u]=[10, 3] has ratio=0.008117400135258075392 at X, ratio=5.082494391432160645 E-495 at Y, witness=5.
Pol[17, 1] with [h, u]=[10, 3] has ratio=0.08635824519284576177 at X, ratio=2.0267658172977691788 E-330 at Y, witness=3.
Pol[18, 1] with [h, u]=[10, 3] has ratio=0.012674748137336487208 at X, ratio=7.608149990471885206 E-221 at Y, witness=5.
Pol[19, 1] with [h, u]=[10, 3] has ratio=0.05172879542200615169 at X, ratio=1.0562498093053093778 E-147 at Y, witness=11.
Pol[20, 1] with [h, u]=[12, 4] has ratio=0.019235200054676021500 at X, ratio=2.0816038266543071378 E-554 at Y, witness=3.
Pol[21, 1] with [h, u]=[12, 4] has ratio=0.07774206369888864278 at X, ratio=1.0657878408348948562 E-403 at Y, witness=7.
Pol[22, 1] with [h, u]=[12, 4] has ratio=0.03437644411117730445 at X, ratio=5.701514329638646543 E-294 at Y, witness=5.
Pol[23, 1] with [h, u]=[12, 4] has ratio=0.007574084119533420700 at X, ratio=2.204123948583350118 E-214 at Y, witness=3.
Pol[24, 1] with [h, u]=[12, 4] has ratio=0.021448809055287624244 at X, ratio=3.210665225648335482 E-156 at Y, witness=7.
Pol[25, 1] with [h, u]=[12, 4] has ratio=0.005545263614259560663 at X, ratio=4.661708018943123030 E-114 at Y, witness=3.
Pol[26, 1] with [h, u]=[14, 5] has ratio=0.04013590208351762291 at X, ratio=6.798598290223572616 E-475 at Y, witness=5.
Pol[27, 1] with [h, u]=[14, 5] has ratio=0.05205791299013295463 at X, ratio=1.0932528898866788670 E-365 at Y, witness=5.
Pol[28, 1] with [h, u]=[14, 5] has ratio=0.03684700897030245628 at X, ratio=8.469295359499301626 E-282 at Y, witness=3.
Pol[29, 1] with [h, u]=[14, 5] has ratio=0.03211802344930026146 at X, ratio=4.169519063187898225 E-217 at Y, witness=3.
Pol[30, 1] with [h, u]=[14, 5] has ratio=0.0007829063801177717266 at X, ratio=1.9660298739276459676 E-167 at Y, witness=13.
Pol[31, 1] with [h, u]=[14, 5] has ratio=0.04102403050727998453 at X, ratio=4.120768035721435585 E-129 at Y, witness=5.
Pol[32, 1] with [h, u]=[16, 6] has ratio=0.023264787203660780686 at X, ratio=1.8498731517983751678 E-450 at Y, witness=19.
Pol[33, 1] with [h, u]=[16, 6] has ratio=0.007429526928510799737 at X, ratio=1.0177015446773477022 E-360 at Y, witness=5.
Pol[34, 1] with [h, u]=[16, 6] has ratio=0.009595965114034005882 at X, ratio=6.055686060340420475 E-289 at Y, witness=7.
Pol[35, 1] with [h, u]=[16, 6] has ratio=0.03190182764590156245 at X, ratio=1.5113376503765830509 E-231 at Y, witness=47.
Pol[36, 1] with [h, u]=[16, 6] has ratio=0.04802602954892544437 at X, ratio=1.3934160344400962711 E-185 at Y, witness=11.
Pol[37, 1] with [h, u]=[18, 7] has ratio=0.025307651942113461604 at X, ratio=4.835552820002924950 E-472 at Y, witness=3.
Pol[38, 1] with [h, u]=[16, 6] has ratio=0.005901196380147038511 at X, ratio=3.087130380024557399 E-68 at Y, witness=11.
Pol[39, 1] with [h, u]=[18, 7] has ratio=0.006885237989511493483 at X, ratio=1.9434038354803348134 E-375 at Y, witness=19.
Pol[40, 1] with [h, u]=[18, 7] has ratio=0.003432980526624849114 at X, ratio=1.6905937002582526076 E-309 at Y, witness=11.
Pol[41, 1] with [h, u]=[18, 7] has ratio=0.004825496687717025156 at X, ratio=2.886113119406868059 E-255 at Y, witness=7.
Pol[42, 1] with [h, u]=[18, 7] has ratio=0.005157919956925831649 at X, ratio=1.7027458713703474762 E-210 at Y, witness=2.
Pol[43, 1] with [h, u]=[18, 7] has ratio=0.010884061269051163250 at X, ratio=9.988863015495347215 E-174 at Y, witness=2.
Pol[44, 1] with [h, u]=[20, 8] has ratio=0.002000921912743121300 at X, ratio=2.503009890787826333 E-445 at Y, witness=7.
Pol[45, 1] with [h, u]=[20, 8] has ratio=0.00003338263496929236844 at X, ratio=2.5039445216267157700 E-375 at Y, witness=5.
Pol[46, 1] with [h, u]=[20, 8] has ratio=0.012950616094528861736 at X, ratio=2.2979276231726596000 E-316 at Y, witness=7.
Pol[47, 1] with [h, u]=[20, 8] has ratio=0.014983565001885331023 at X, ratio=9.086248814015717698 E-267 at Y, witness=5.
Pol[48, 1] with [h, u]=[20, 8] has ratio=0.002869955519565375994 at X, ratio=5.457590251717533512 E-225 at Y, witness=7.
Pol[49, 1] with [h, u]=[20, 8] has ratio=0.0017577071234049536226 at X, ratio=8.294742153627984320 E-190 at Y, witness=3.
Pol[50, 1] with [h, u]=[20, 8] has ratio=0.011777831171316476521 at X, ratio=3.707089653771678045 E-160 at Y, witness=5.
Pol[51, 1] with [h, u]=[20, 8] has ratio=0.007277373214562809450 at X, ratio=3.377144517245425541 E-135 at Y, witness=2.
Pol[52, 1] with [h, u]=[21, 9] has ratio=0.010633997254611386123 at X, ratio=2.1931960197392222864 E-274 at Y, witness=3.
Pol[53, 1] with [h, u]=[21, 9] has ratio=0.004428920219803951330 at X, ratio=4.128480859888360236 E-247 at Y, witness=2.
Pol[54, 1] with [h, u]=[22, 9] has ratio=0.002818988029597297585 at X, ratio=3.539477340343346077 E-326 at Y, witness=11.
Pol[55, 1] with [h, u]=[22, 9] has ratio=0.0004912077514066266543 at X, ratio=7.081795190279813147 E-280 at Y, witness=2.
Pol[56, 1] with [h, u]=[22, 9] has ratio=0.010611671817143623172 at X, ratio=3.472728168072166020 E-240 at Y, witness=3.
Pol[57, 1] with [h, u]=[22, 9] has ratio=0.0008040264706712018540 at X, ratio=3.258416001069933594 E-206 at Y, witness=7.
Pol[58, 1] with [h, u]=[22, 9] has ratio=0.013223525788128237961 at X, ratio=4.867682533951676213 E-177 at Y, witness=3.
Pol[59, 1] with [h, u]=[23, 10] has ratio=0.0005347855881484648913 at X, ratio=1.6889875708337374280 E-339 at Y, witness=19.
Pol[60, 1] with [h, u]=[23, 10] has ratio=0.003674233742184270100 at X, ratio=1.0366688395432904620 E-308 at Y, witness=7.
Pol[61, 1] with [h, u]=[23, 10] has ratio=0.002444672206245324286 at X, ratio=5.537525022646524968 E-281 at Y, witness=19.
Pol[62, 1] with [h, u]=[23, 10] has ratio=0.000000016707758966911152969 at X, ratio=6.891974076249926785 E-261 at Y, witness=7.
Pol[63, 1] with [h, u]=[23, 10] has ratio=8.246841472787701256 E-36 at X, ratio=1.4662405334570236994 E-241 at Y, witness=13.
Pol[64, 1] with [h, u]=[23, 10] has ratio=0.006348679351796856524 at X, ratio=1.3296521143600452526 E-211 at Y, witness=5.
Pol[65, 1] with [h, u]=[23, 10] has ratio=0.00004002614647898101914 at X, ratio=1.3633237411147619888 E-192 at Y, witness=2.
Pol[66, 1] with [h, u]=[23, 10] has ratio=0.0013340556701777180632 at X, ratio=3.036919221985604575 E-175 at Y, witness=5.
Pol[67, 1] with [h, u]=[23, 10] has ratio=0.010797511837541458393 at X, ratio=1.3739753020432144047 E-159 at Y, witness=5.
Pol[68, 1] with [h, u]=[25, 11] has ratio=0.002128908579847056864 at X, ratio=8.599530773021368685 E-381 at Y, witness=2.
Pol[69, 1] with [h, u]=[25, 11] has ratio=0.011191012683457016786 at X, ratio=3.140237578183987821 E-349 at Y, witness=2.
Pol[70, 1] with [h, u]=[25, 11] has ratio=0.0005368598477042684036 at X, ratio=2.495590292440575163 E-320 at Y, witness=2.
Pol[71, 1] with [h, u]=[25, 11] has ratio=0.0011776063530132000897 at X, ratio=7.681719546194277147 E-294 at Y, witness=5.
Pol[72, 1] with [h, u]=[25, 11] has ratio=0.003733037701423788068 at X, ratio=1.6655008162302403898 E-269 at Y, witness=7.
Pol[73, 1] with [h, u]=[25, 11] has ratio=0.0004716144036731915163 at X, ratio=3.136490935730841646 E-247 at Y, witness=7.
Pol[74, 1] with [h, u]=[25, 11] has ratio=0.006654280787176095130 at X, ratio=6.933035192446414584 E-227 at Y, witness=11.
Pol[75, 1] with [h, u]=[25, 11] has ratio=0.002628071824628631658 at X, ratio=3.926919189753072506 E-208 at Y, witness=2.
Pol[76, 1] with [h, u]=[27, 12] has ratio=0.002140989207660000089 at X, ratio=4.494284462617937037 E-424 at Y, witness=3.
Pol[77, 1] with [h, u]=[25, 11] has ratio=0.004535446123679673568 at X, ratio=8.835352388838363855 E-159 at Y, witness=7.
Pol[78, 1] with [h, u]=[27, 12] has ratio=0.002085062630903842715 at X, ratio=1.3773141862296258476 E-378 at Y, witness=11.
Pol[79, 1] with [h, u]=[27, 12] has ratio=0.002107514042204827413 at X, ratio=1.2382927131375664639 E-349 at Y, witness=3.
Pol[80, 1] with [h, u]=[27, 12] has ratio=0.0015134564878487343238 at X, ratio=6.309792873086469274 E-323 at Y, witness=5.
Pol[81, 1] with [h, u]=[27, 12] has ratio=0.0007308393252279501130 at X, ratio=2.994514532690816401 E-298 at Y, witness=7.
Pol[82, 1] with [h, u]=[27, 12] has ratio=0.0012054242470060426216 at X, ratio=1.6774688300833391471 E-275 at Y, witness=7.
Pol[83, 1] with [h, u]=[27, 12] has ratio=0.002583268771034132285 at X, ratio=1.3890359790167784195 E-254 at Y, witness=3.
Pol[84, 1] with [h, u]=[27, 12] has ratio=0.002861581573217274174 at X, ratio=3.891277992305916962 E-235 at Y, witness=17.
Pol[85, 1] with [h, u]=[27, 12] has ratio=0.006274197837916598271 at X, ratio=3.5020106014877371408 E-217 at Y, witness=3.
Pol[86, 1] with [h, u]=[28, 13] has ratio=0.0019258270711376581060 at X, ratio=3.376822899863925413 E-374 at Y, witness=3.
Pol[87, 1] with [h, u]=[28, 13] has ratio=0.002695578795170324074 at X, ratio=1.985155782148016194 E-360 at Y, witness=7.
Pol[88, 1] with [h, u]=[28, 13] has ratio=0.0014469443905976850799 at X, ratio=3.584772088176161542 E-347 at Y, witness=17.
Pol[89, 1] with [h, u]=[28, 13] has ratio=0.002998539410261169340 at X, ratio=2.0748801050215478842 E-334 at Y, witness=7.
Pol[90, 1] with [h, u]=[28, 13] has ratio=0.0017255385572497470788 at X, ratio=4.437743009778393459 E-322 at Y, witness=11.
Pol[91, 1] with [h, u]=[28, 13] has ratio=0.0016225297701974585832 at X, ratio=3.0831289973661454892 E-310 at Y, witness=7.
Pol[1, 2] with [h, u]=[5, 1] has ratio=2.697031699238860107 E-1380 at X, ratio=2.721327373652221996 E-1380 at Y, witness=5.
Pol[2, 2] with [h, u]=[5, 1] has ratio=0.10705508251270807009 at X, ratio=5.206878373572677967 E-658 at Y, witness=5.
Pol[3, 2] with [h, u]=[5, 1] has ratio=0.3333985427166455234 at X, ratio=1.7950098562725369696 E-329 at Y, witness=3.
Pol[4, 2] with [h, u]=[4, 1] has ratio=6.964841614849832521 E-137 at X, ratio=7.883524555351360668 E-301 at Y, witness=2.
Pol[5, 2] with [h, u]=[4, 1] has ratio=3.606724515701922816 E-219 at X, ratio=1.1760902963875158209 E-300 at Y, witness=5.
Pol[6, 2] with [h, u]=[4, 1] has ratio=2.6175421991267143176 E-260 at X, ratio=7.403088177711536565 E-301 at Y, witness=2.
Pol[7, 2] with [h, u]=[4, 1] has ratio=1.0469190416062769626 E-280 at X, ratio=5.441394066784074375 E-301 at Y, witness=2.
Pol[8, 2] with [h, u]=[4, 1] has ratio=9.053314793206067075 E-291 at X, ratio=9.537437480241060494 E-301 at Y, witness=3.
Pol[9, 2] with [h, u]=[7, 2] has ratio=1.4393487668771091203 E-157 at X, ratio=1.3651537649682312076 E-647 at Y, witness=7.
Pol[10, 2] with [h, u]=[7, 2] has ratio=1.1171169349033105514 E-332 at X, ratio=1.0399239762164730378 E-495 at Y, witness=5.
Pol[11, 2] with [h, u]=[7, 2] has ratio=7.307316491849909943 E-224 at X, ratio=2.4788539342068100114 E-332 at Y, witness=3.
Pol[12, 2] with [h, u]=[8, 2] has ratio=0.03210638610606863384 at X, ratio=6.608764615149762445 E-298 at Y, witness=3.
Pol[13, 2] with [h, u]=[8, 2] has ratio=0.06396284508963929494 at X, ratio=1.2287648813926929973 E-170 at Y, witness=13.
Pol[14, 2] with [h, u]=[8, 2] has ratio=0.18867961204082568362 at X, ratio=4.317794622524219145 E-98 at Y, witness=5.
Pol[15, 2] with [h, u]=[9, 3] has ratio=0.03522475081585598996 at X, ratio=6.957849491323385601 E-331 at Y, witness=17.
Pol[16, 2] with [h, u]=[10, 3] has ratio=0.03796485492054546388 at X, ratio=5.482299549367765805 E-495 at Y, witness=7.
Pol[17, 2] with [h, u]=[10, 3] has ratio=0.015523145461949960007 at X, ratio=1.7170153940167003362 E-330 at Y, witness=2.
Pol[18, 2] with [h, u]=[10, 3] has ratio=0.03758488086959789016 at X, ratio=9.143633655548932911 E-221 at Y, witness=5.
Pol[19, 2] with [h, u]=[10, 3] has ratio=0.03631649094835362057 at X, ratio=1.2483296965092361103 E-147 at Y, witness=7.
Pol[20, 2] with [h, u]=[12, 4] has ratio=0.015995873332353288222 at X, ratio=2.1150463838826216342 E-554 at Y, witness=17.
Pol[21, 2] with [h, u]=[12, 4] has ratio=0.09997554883319061183 at X, ratio=1.4754253841572911990 E-403 at Y, witness=3.
Pol[22, 2] with [h, u]=[12, 4] has ratio=0.014120167697435035747 at X, ratio=6.046461389212203122 E-294 at Y, witness=11.
Pol[23, 2] with [h, u]=[12, 4] has ratio=0.05883981114328891747 at X, ratio=3.062201215562837759 E-214 at Y, witness=5.
Pol[24, 2] with [h, u]=[12, 4] has ratio=0.016328964711131802362 at X, ratio=2.966481389077048647 E-156 at Y, witness=2.
Pol[25, 2] with [h, u]=[12, 4] has ratio=0.04855705752101181923 at X, ratio=4.675358260192131547 E-114 at Y, witness=11.
Pol[26, 2] with [h, u]=[14, 5] has ratio=0.018076934976142081032 at X, ratio=5.610344511951583432 E-475 at Y, witness=2.
Pol[27, 2] with [h, u]=[14, 5] has ratio=0.0006609903108622118298 at X, ratio=7.634979847286988220 E-366 at Y, witness=5.
Pol[28, 2] with [h, u]=[14, 5] has ratio=0.028437050422738190064 at X, ratio=1.0541052121745242853 E-281 at Y, witness=2.
Pol[29, 2] with [h, u]=[14, 5] has ratio=0.04470158826602348428 at X, ratio=4.012503755697252436 E-217 at Y, witness=3.
Pol[30, 2] with [h, u]=[14, 5] has ratio=0.0004131272546560228339 at X, ratio=2.2498582625868938484 E-167 at Y, witness=2.
Pol[31, 2] with [h, u]=[14, 5] has ratio=0.04588777333387185553 at X, ratio=3.790561614785791424 E-129 at Y, witness=2.
Pol[32, 2] with [h, u]=[16, 6] has ratio=0.004882646431975632741 at X, ratio=1.8215329800416762258 E-450 at Y, witness=11.
Pol[33, 2] with [h, u]=[16, 6] has ratio=0.014875293411644849198 at X, ratio=8.463120392992621585 E-361 at Y, witness=5.
Pol[34, 2] with [h, u]=[16, 6] has ratio=0.024059458271665445274 at X, ratio=6.115849407490656748 E-289 at Y, witness=2.
Pol[35, 2] with [h, u]=[16, 6] has ratio=0.007267010114379166157 at X, ratio=1.6538968639097066515 E-231 at Y, witness=7.
Pol[36, 2] with [h, u]=[16, 6] has ratio=0.003117787036594418798 at X, ratio=1.4413235532614177300 E-185 at Y, witness=3.
Pol[37, 2] with [h, u]=[18, 7] has ratio=0.019787829462183970350 at X, ratio=5.379118032510874648 E-472 at Y, witness=5.
Pol[38, 2] with [h, u]=[16, 6] has ratio=0.016624020568024547203 at X, ratio=3.213661242229349314 E-68 at Y, witness=3.
Pol[39, 2] with [h, u]=[18, 7] has ratio=0.0016037053762497140487 at X, ratio=2.2982354320309409650 E-375 at Y, witness=17.
Pol[40, 2] with [h, u]=[18, 7] has ratio=0.0016526757568233879183 at X, ratio=1.8271142587485652267 E-309 at Y, witness=11.
Pol[41, 2] with [h, u]=[18, 7] has ratio=0.008828484873134986003 at X, ratio=3.079480441800319633 E-255 at Y, witness=11.
Pol[42, 2] with [h, u]=[18, 7] has ratio=0.007116719469440952554 at X, ratio=1.7271967197390732131 E-210 at Y, witness=2.
Pol[43, 2] with [h, u]=[18, 7] has ratio=0.00012908567117953554950 at X, ratio=9.304691754601338030 E-174 at Y, witness=2.
Pol[44, 2] with [h, u]=[20, 8] has ratio=0.006958040387039118401 at X, ratio=2.2329911427725664572 E-445 at Y, witness=5.
Pol[45, 2] with [h, u]=[20, 8] has ratio=0.0008344038514504084139 at X, ratio=2.4261485170209140644 E-375 at Y, witness=5.
Pol[46, 2] with [h, u]=[20, 8] has ratio=0.013129176324140056248 at X, ratio=2.1842985158473835198 E-316 at Y, witness=11.
Pol[47, 2] with [h, u]=[20, 8] has ratio=0.002660044212762904803 at X, ratio=8.336998472527172290 E-267 at Y, witness=5.
Pol[48, 2] with [h, u]=[20, 8] has ratio=0.004036742455543674494 at X, ratio=4.555404022392737834 E-225 at Y, witness=11.
Pol[49, 2] with [h, u]=[20, 8] has ratio=0.005662099927381009773 at X, ratio=1.0202232878533004217 E-189 at Y, witness=23.
Pol[50, 2] with [h, u]=[20, 8] has ratio=0.005192003929348401192 at X, ratio=3.492003093831764079 E-160 at Y, witness=5.
Pol[51, 2] with [h, u]=[20, 8] has ratio=0.023703035309880702168 at X, ratio=3.694043093012294643 E-135 at Y, witness=2.
Pol[52, 2] with [h, u]=[21, 9] has ratio=0.004840187948601853691 at X, ratio=2.2492350014526484224 E-274 at Y, witness=13.
Pol[53, 2] with [h, u]=[21, 9] has ratio=0.0014425549454055985963 at X, ratio=4.010874271667781490 E-247 at Y, witness=3.
Pol[54, 2] with [h, u]=[22, 9] has ratio=0.003722455414957501526 at X, ratio=3.886418251977453318 E-326 at Y, witness=2.
Pol[55, 2] with [h, u]=[22, 9] has ratio=0.009153693835673468553 at X, ratio=7.245121049883145039 E-280 at Y, witness=11.
Pol[56, 2] with [h, u]=[22, 9] has ratio=0.002359372538362088087 at X, ratio=3.178675749642641646 E-240 at Y, witness=5.
Pol[57, 2] with [h, u]=[22, 9] has ratio=0.002695376023557842404 at X, ratio=3.200336164110746866 E-206 at Y, witness=13.
Pol[58, 2] with [h, u]=[22, 9] has ratio=0.010997305024975105532 at X, ratio=5.109492848835602702 E-177 at Y, witness=2.
Pol[59, 2] with [h, u]=[23, 10] has ratio=0.005393935318190739750 at X, ratio=2.484953399330564612 E-339 at Y, witness=13.
Pol[60, 2] with [h, u]=[23, 10] has ratio=0.0007889473569082460419 at X, ratio=8.491328920220983746 E-309 at Y, witness=3.
Pol[61, 2] with [h, u]=[23, 10] has ratio=0.0018738487047188125120 at X, ratio=7.829132133408909538 E-281 at Y, witness=2.
Pol[62, 2] with [h, u]=[23, 10] has ratio=0.00000003954772763435028654 at X, ratio=7.104964084090141535 E-261 at Y, witness=7.
Pol[63, 2] with [h, u]=[23, 10] has ratio=4.504222674499113578 E-35 at X, ratio=1.6026741389573257810 E-241 at Y, witness=7.
Pol[64, 2] with [h, u]=[23, 10] has ratio=0.010391921175709166460 at X, ratio=1.4844679113273990320 E-211 at Y, witness=2.
Pol[65, 2] with [h, u]=[23, 10] has ratio=0.009317534876000226821 at X, ratio=1.1156398985129602949 E-192 at Y, witness=11.
Pol[66, 2] with [h, u]=[23, 10] has ratio=0.003789854486214475990 at X, ratio=2.774612975095634850 E-175 at Y, witness=13.
Pol[67, 2] with [h, u]=[23, 10] has ratio=0.00011194582193241333855 at X, ratio=1.4218695131680493929 E-159 at Y, witness=7.
Pol[68, 2] with [h, u]=[25, 11] has ratio=0.003562508724754614967 at X, ratio=9.001651782077120158 E-381 at Y, witness=2.
Pol[69, 2] with [h, u]=[25, 11] has ratio=0.0011194838483371833177 at X, ratio=3.088200529274584466 E-349 at Y, witness=13.
Pol[70, 2] with [h, u]=[25, 11] has ratio=0.000008212740856607978446 at X, ratio=2.670897365079623067 E-320 at Y, witness=5.
Pol[71, 2] with [h, u]=[25, 11] has ratio=0.003007379124514089538 at X, ratio=9.539856223576833253 E-294 at Y, witness=2.
Pol[72, 2] with [h, u]=[25, 11] has ratio=0.005272578169155549923 at X, ratio=1.6774639566648019732 E-269 at Y, witness=5.
Pol[73, 2] with [h, u]=[25, 11] has ratio=0.008402632226402782931 at X, ratio=2.938036110851068221 E-247 at Y, witness=11.
Pol[74, 2] with [h, u]=[25, 11] has ratio=0.0019277241574896004474 at X, ratio=7.069370205923520798 E-227 at Y, witness=3.
Pol[75, 2] with [h, u]=[25, 11] has ratio=0.0013403896476976787768 at X, ratio=4.153001981356500745 E-208 at Y, witness=3.
Pol[76, 2] with [h, u]=[27, 12] has ratio=0.004095921533260436613 at X, ratio=4.659646719994380944 E-424 at Y, witness=11.
Pol[77, 2] with [h, u]=[25, 11] has ratio=0.003206991085056617513 at X, ratio=9.238052409430196440 E-159 at Y, witness=7.
Pol[78, 2] with [h, u]=[27, 12] has ratio=0.0009416439947623540681 at X, ratio=1.6009826376976127056 E-378 at Y, witness=3.
Pol[79, 2] with [h, u]=[27, 12] has ratio=0.005781757920972739952 at X, ratio=1.2016817692575006528 E-349 at Y, witness=23.
Pol[80, 2] with [h, u]=[27, 12] has ratio=0.00010509323230450691724 at X, ratio=6.848542060642634626 E-323 at Y, witness=11.
Pol[81, 2] with [h, u]=[27, 12] has ratio=0.002364674468838728044 at X, ratio=3.289204643040281341 E-298 at Y, witness=5.
Pol[82, 2] with [h, u]=[27, 12] has ratio=0.002573593414136458872 at X, ratio=1.7074160079434692510 E-275 at Y, witness=5.
Pol[83, 2] with [h, u]=[27, 12] has ratio=0.0015489610111122002963 at X, ratio=1.2981774817055430499 E-254 at Y, witness=17.
Pol[84, 2] with [h, u]=[27, 12] has ratio=0.003917670529926937422 at X, ratio=3.668869209763165404 E-235 at Y, witness=2.
Pol[85, 2] with [h, u]=[27, 12] has ratio=0.004640924800463198913 at X, ratio=3.0050576338230114472 E-217 at Y, witness=11.
Pol[86, 2] with [h, u]=[28, 13] has ratio=0.0008403246085377766062 at X, ratio=3.557776796629505378 E-374 at Y, witness=29.
Pol[87, 2] with [h, u]=[28, 13] has ratio=0.002582218320535849434 at X, ratio=2.268274656193541833 E-360 at Y, witness=3.
Pol[88, 2] with [h, u]=[28, 13] has ratio=0.002599970018356594929 at X, ratio=3.819077500527086208 E-347 at Y, witness=7.
Pol[89, 2] with [h, u]=[28, 13] has ratio=0.002306076716086692479 at X, ratio=2.5548637881224193240 E-334 at Y, witness=7.
Pol[90, 2] with [h, u]=[28, 13] has ratio=0.0006424165092147135069 at X, ratio=4.294955958886652571 E-322 at Y, witness=13.
Pol[91, 2] with [h, u]=[28, 13] has ratio=0.0010421579758292881981 at X, ratio=3.0299266831491240734 E-310 at Y, witness=5.

Validated in 250 sec.

Goodbye!
```
A copy of the CHG certificate `89_11971.out` (153MB) is included in: `89_11971.zip` and can be verified by running the Pari-GP script `verify.gp`.

## Helper File

Based on factorization of N-1 and N+1:
```9675530778541454219503428190759722938271960044905609377519757229028379197927415288522445293792032093344476398720590626931437022497768505141052052100281514651567023107981510708750081486445033940608708099739039035170034980463997101908659940819961767202923176435711740001166070807770943744789221684882544017642659386516074063016699052173316582781313156452273065935356347217802769567968750127817737772270251439040433549371633181470603637297742063595254550253398505727012068554563607963443660894151856942902195854918761516730569935651316411144986781130551199796473719337116262195597786318733266278583812916781627857322283976645392745988565032731355266312890287801084379210641101183254836739481766110538637766266132259631509997119707983669109741320977766362381178867000334489034069420001168322672555681056696525213171467826682801966704902322814228018833337915933036901743888312245469884296184736043108063971721056629971168388908525008011466606069668073143630731224530019305400153730161427001115388429217784286591174720239431639955753433718057959522853822177046431212989354762326528535395395064501369677249680126845321494887241162865566987018688548070809330124192648898238282339149493616708271588955007045556833308839445910085697341789132639506128149644509155711641363816204129609840457740002564947852458004837838605332156396945550665654685484026840630878098635712621789019145409809246883291218969804943234610520128557796704564593727135605804464081339189756746050216840278273584961751538782642905474653729770602532126336685135098892452160884001153832966747395487378699942485268509675566335907947016311968712508898481564195434964704770391779377119528211
19577862797667857064523761784905236354241754044573574448688283359164543420071137460040567494145504372201070823741699782787341557019127133006935721316634347778219614973118387354908234026564619092432115346308617718877512487328268373360332235521474643544490856365941362819357958793163583500027185848125783603281569376868325215482326936086239449076950574658880847135441705045141051192273595893674905844280405926486158015334963908618378629422745373747426975231225659116947402397932066388002121571303995849148308817583346715933416666953231049300283210065223857770024586306682119434474163741081157784616379766783822918599867956378376120556212992974225087818843240924981792671685978490322165439093561585699322403439641072190215835718660003303607326561102880985384252955581
31309472062391856992253103812433616569312063224013686919644841793007515341138774035818652009776308379377269437025327916409375526323316355852445678441821225990116827673906435401701501145151692276846595307819644593126542396635306035785305621764281129951950171129373969018273201
858830523723879377829364977102079691635913460543481039435417646486626370198980238412507812316992404379778809547321203791807279060456445861455478043001164759138377562153043009271804546605017340732593183692245936864289790428629145486407801
3663791860452840626194526222768931891148566198162307749934465334395037358514300944434525316291350802542586536043678991977521796908730662593731610964135271895540459317090023309848169388652598648240676825478646220395647199617156773031
20061667895727426041395750885653333225575460390050144010344964458645079488420532190027951692964892304822081732882625661710776907495783
39113591300596515762346196531565350802110661353088953017973898932194552112048375922761195164141455038696710305511
5562790266090328834945166268699029569199863108594559361022538240879154029312567212547573036941455075160545628709
148825386979399622845348614223901542635864398910774470848755154738579192097337495095036928300483964835605499761
142930732537727801627301765041299884820393293196326972842650995257535946922023440378841
27387900588402355895254774843806814357801270132378076312176086938158743660348577634761
9423055447844765546061119036668303899067072236192530130676552317619702390990040003
922374523459419261082559791153538702367603449663133503414501959221306547234631657
23951217836326037378215298615132348447310456341310123229375773418560883
319253404954264973136967362582150853690575242000843930401
43484451779255533149037817043335180264520523510538888623
56446744592436000160410195816610790848272342457180531
337442882972800845950314819977463482463263275747509
292368249098271826360563771388968911082878011
23019663630281523487565428377358097766453991
2469467652296763115318663993936011104869721
56720247909816966079065745811260507729561
2694047972732129661772469500704479955991
2167174701521677009644782562981482674303
506651396724139292973223195238130636461
267214716195146703447437422562520456421
60852507059273073081950769901489509011
1361339262450710449164416674313464213
132826866247838386026847074523570651
110956215666343670301789060637551481
83999238361920629083112156644098941
92957292708831868393140148841149
3019434409417859181869738532979
469005661292895001762371525637
420745812771180172831231619071
304468992684888692702752314551
54307168149603562110475145351
1233454707491164759506195721
38446247824311279000318481
7204894610050427986159381
35663932185978584946691
21062392661862533682541
8376410251871526831391
2858235254856451102639
259589135279367544177
110689786879261483849
52948932423599020031
48929327007211179761
43149950492043667091
9234420785371044761
4322696829498252451
1123157296866691991
795787283527238521
359931808994487031
21367609695414361
15146818079392621
10202255437240651
5091409074798439
4674705441489931
4099137663888793
3892362992828641
2675667780679681
1378730198623051
1062852974906941
350395182714847
108663243300991
77920323619951
41322241509739
28650459490921
8463794159539
5629604449951
502628805631
130602367771
120081279391
64156711609
35885986039
35669522701
26205153553
24698067997
8718957649
8281163311
6815459113
6807972547
5869174411
4234864591
1787867551
711638731
385617541
311995657
308397361
167757157
135551701
134372281
87552571
29465353
24703741
22974421
6714601
4539277
3235261
3139991
2225851
1971901
1726379
1537621
1491001
407741
400291
354047
347131
329617
127681
71821
58711
47881
33151
31333
20161
11971
8443
8011
6301
4789
4327
3221
2927
2851
1901
1597
1051
761
701
691
631
571
419
373
211
191
131
127
89
73
71
43
31
19
19
7
7
5
5
3
3
3
3
2
7137077042934537673236868831
222946072284908512271
39334160917
345848533
41647271
2
2
```

## Prime Factor Certification

Signed Primo certificates for the three largest helper primes, at 1645 digits, 764 digits, and 275 digits, are included in: `89_11971.zip`

Tom Wu