================================================== Joe K. Crump's database of 2^x-3 factorizations ================================================== Email: qqqjoecr@carolina.rr.com [remove the three qs] ================================================== Notes: This list only includes factorizations that could be used to find new 2^n = 3 (mod n) solutions. Updates: 03/08/01 - Rich Brown (rich.brown@mediaone.net) found some factors for exponents 1277 and 1229 using ECM. 03/01/06 - Max Alekseyev contributed factorization of 2^643-3 11/20/06 - Max Alekseyev contributed factorization of 2^557-3 Feb 2007: - Start maintaining the file again after leaving it for 5+ years. :) - Added Max's "Most Wanted" list. Those factorizations are predicted to have a better-than-average chance of creating another 2^n = 3 (mod n) solution. - Max fully factored exponents 677 and 10805 and found factors for exponents 1843 and 2045, as well as pointing out some unnecessary exponents (now eliminated). - Thanks much to my brother John, and friends Mark Miller, and Curtis Krumel for contributing numerous 20-40 digit factors via ECMNet. Mar 2007: - 20+ new factors found with ECMNet (mostly ~30 digits) PS: See the following page for latest factors which may not be in the table yet: http://www.immortaltheory.com/NumberTheory/ecm_report.htm ----------------------------------- 2^5-3 = 29 2^19-3 = 5 * 23 * 47 * 97 2^23-3 = 5 * 1677721 2^29-3 = 536870909 2^47-3 = 5 * 5 * 5629499534213 2^53-3 = 2203 * 5741 * 712176643 2^71-3 = 5 * p21 2^95-3 = 5 * p28 2^97-3 = 1624171 * 10669579483 * 9143880139733 2^101-3 = 163 * p29 2^115-3 = 5 * 97 * p32 2^139-3 = 5 * 19 * 19 * 19 * 1873 * 16964327 * p27 2^145-3 = 29 * 51690246276649 * p29 2^149-3 = 525241 * 288952921 * p31 2^163-3 = 5 * 97 * 106703 * 5245973 * 52188051944423 * p21 2^167-3 = 5 * 5 * 239 * 1061 * 15263 * p40 2^173-3 = 23 * 29 * 53 * 627763583 * 65068129448961907 * p22 2^191-3 = 5 * 71 * 745733 * 83641031 * 5731892549 * 160793976328607 * 153790767336881231 2^197-3 = 1123 * 67079 * 693475275429886459 * p34 2^211-3 = 5 * 19 * 97 * 17597 * 372067 * 1079386087513 * p38 2^235-3 = 5 * 149 * 89476337239123097381 * 679964581067488744201 * p28 2^239-3 = 5 * 23 * 73823 * 945293 * 16262545781347514679638489731 * p31 2^263-3 = 5 * 163 * 1301 * 1438973 * 6074209 * 353928429346379538312047 * p37 2^265-3 = 19 * 28169668365514824313533817 * p54 2^269-3 = 101 * 4852089454632376114362058674979 * p49 2^293-3 = 1218222829032475271 * p71 2^311-3 = 5 * 43992702287946853060181 * 425911341025211590220887061323 * p41 2^317-3 = 139 * 499 * 174702901260684479 * 1106960888484257247788540139339889 * p41 2^355-3 = 5 * 19 * 97 * 181831343461147 * 18125258135569469 * 7616559214204794954263 * 317279016334085989947049976467374306911430432882139 2^359-3 = 5 * 227977 * 208158139901678900639 * p82 2^379-3 = 5 * 203549 * 6157469 * 32889892726645367017218258349811427072016633 * p58 2^383-3 = 5 * 149 * 4561556144335781 * p97 2^389-3 = 643 * 24267450450167 * 19135823522009296186537 * p79 2^409-3 = 19 * 13560842578291 * 53935347980316917 * 3299006856200945021104591373196989 * p59 2^431-3 = 5 * 4089005252863463137901 * 977608633336035856173804530590583 * p75 2^437-3 = 23 * 41399 * 17266493 * 52858911166661 * 5597110599246409 * 605408840557818653 * 294689338316802614763325388117 * p42 2^461-3 = 67801 * p134 2^479-3 = 5 * 47 * 677 * 613493 * 59309996879049282649291084690926487561219 * p93 2^485-3 = 53 * 53 * 2285189 * 5351237 * 130166407115741105132742556824466265630151260716462422214638983 * p68 2^499-3 = 5 * 19 * 97 * 1609130740656289029777350457697 * 884727121 * p108 2^503-3 = 5 * 23 * 479 * 8377 * 660105527 * 18278316868406623407055508189 * 387994463448418142847532674398868744648576913 * p62 2^505-3 = 1397909 * 169721480268840587061180050902372583 * p111 2^509-3 = 29 * p152 2^551-3 = 5 * 1319 * p163 2^557-3 = 311 * 1382972877684884284017852193920677515781 * 1873756573036794605530493226302479106157539654182725341588309 * p66 2^599-3 = 5 * 987029 * 1919032810412525548118999 * 845117759958425601120766612405509769 * p114 2^643-3 = 5 * 19 * 97 * 239 * 401953 * 1739009003318072377 * 348517228735983022805545699 * 170890017904359186727050262036995846555074348155602036792130577971 * p72 2^647-3 = 5 * 5 * 839 * 3146279 * 358407211 * 12369036481436138187121 * p154 2^653-3 = 1637 * 97151 * 5270249981 * 80710547851 * p168 2^667-3 = 5 * 5 * 1439 * 148732613 * 3394201831522114177 * p170 2^677-3 = 29 * 24630249848220893670087394833066510974813623660335490350497021468946096161433804081191 * p117 2^695-3 = 5 * 414920445770216261 * p191 2^701-3 = 23 * 431 * 26161 * 1305018093967517333947518389 * p176 2^719-3 = 5 * 12743 * 677709317 * 719098005778492503967243219204847 * p170 2^743-3 = 5 * 919915458916081 * c209 2^745-3 = 23 * 53 * 59063 * 8186919763171 * 30496310582253756085729536479 * c176 2^773-3 = 599 * 38950493 * 3336407089 * c213 2^797-3 = 53 * 8329 * 47717 * c230 2^815-3 = 5 * 499 * c242 2^821-3 = 71 * 49757 * 91297 * 105172422786152309328456367940148953731747 * p195 2^835-3 = 5 * 97 * 225988148773967 * c235 2^839-3 = 5 * 9239 * 204557 * p243 2^859-3 = 5 * 19 * 1151 * 1201 * 22147 * 6463945762921541 * c231 2^863-3 = 5 * c260 2^865-3 = 345751297 * 165806317225385752173989 * c229 2^887-3 = 5 * 5 * 173 * 6263 * 13151 * 15237413 * 253989776423 * 673533828182573 * c223 2^893-3 = 47 * 379 * 5263229 * 7406981 * 9121897477178907327001 * c230 2^907-3 = 5 * 5 * 5 * 16309939 * 513861818978021 * c250 2^941-3 = 10007 * 20639 * 60413 * 3700573033 * 7032048076129 * 221409850666905937061393351431661 * c216 2^955-3 = 5 * 1373 * 17333 * 17837 * 34469 * 192796809477050231 * 95599474194972701519 * c234 2^983-3 = 5 * 5323 * 99020986442194065154716623 * 2666141 * c260 2^985-3 = 19 * 29 * 47 * 2310720226856117 * c277 2^1007-3 = 5 * 5 * 5 * 139 * 13963531 * 198263927754264752587 * c272 2^1031-3 = 5 * 23 * 47 * 71 * 409 * 461 * 643 * 245851 * 12795697 * 79879727 * 328792370525255203804253 * 1977125414711120909035891 * c229 2^1055-3 = 5 * 15217 * c313 2^1061-3 = 180416744015381 * 227973736908020785502084958875289217 * c270 2^1081-3 = 368059 * 213849660960413 * 7965119236534174160731 * 15152527850255989589711737 * p259 2^1109-3 = 53 * 2487671 * 6209090071967 * 24488096635273 * 17394642851369471 * 11243762648342047753 * p265 2^1123-3 = 5 * 47 * 97 * 149 * c332 2^1129-3 = 19 * 701 * 3307 * 124764807661061481824046793 * c307 2^1151-3 = 5 * 2237 * 19622549 * 3081248344670429 * 89703081305561807 * c303 2^1195-3 = 5 * 90197 * c355 2^1201-3 = 19 * 907 * 50180233 * c350 2^1219-3 = 5 * 19 * 97 * 15527 * 617107 * 214321651 * 28588227493561 * 7076611290527441431157 * 5697135732614324497998413 * p285 2^1229-3 = 23 * 191 * 36341 * 55216127 * 7286494479541729 * 1586965042648405323349954267 * c311 2^1277-3 = 51877253 * 17917883397505207057 * 44479799191248721 * c341 2^1291-3 = 5 * 19 * 119611 * 11320318222177 * c369 2^1301-3 = 719 * c389 2^1315-3 = 5 * 97 * 24598443139 * 26385371549 * c373 2^1319-3 = 5 * 1123 * 4987242863 * 6535350279078929000279 * p362 2^1345-3 = 19 * 639839 * 85126881787 * c387 2^1349-3 = 29 * 29 * 8040793 * 87829753207052099017 * 2115343803532069719907 * c355 2^1363-3 = 5 * 19 * 97 * 213621647 * 27495157483 * 8986776470438965498324290112819993733 * c351 2^1367-3 = 5 * 5 * 1291 * 2267648823916601159 * c389 2^1373-3 = 493012571353 * 368664482138647421 * p385 2^1439-3 = 5 * 409 * 6577 * 614219181529 * 555147346177 * 2362465376261 * c391 2^1465-3 = 3690909105964727 * c426 2^1483-3 = 5 * 1058896292203 * 3258601329639529 * c419 2^1487-3 = 5 * 5 * 311 * 163151 * c439 2^1489-3 = 19 * 29 * 2309 * 4919 * 183697 * 1977535149937 * 232115184186112849 * c404 2^1493-3 = 23 * 660557 * 69524333 * 3029912471 * 47219391718172884799 * 105391419802106953 * c389 2^1537-3 = 23 * 47 * 2377 * 20327 * 1575415651 * c443 2^1555-3 = 5 * 97 * c466 2^1559-3 = 5 * 23 * 139 * 163 * 464999 * p458 2^1585-3 = c478 2^1607-3 = 5 * 5 * 5 * 5 * c481 2^1633-3 = 19 * 6537547 * 1266795503 * 1689663001594578239 * 576475121707 * c445 2^1637-3 = 823746481 * 269107703237 * 221737754692678621291626989 * c447 2^1733-3 = 53 * 431 * p518 2^1747-3 = 5 * 5 * 97 * 173 * 653 * 543232801 * 124762477525426053369309155286427027 * p474 2^1795-3 = 5 * 19 * 97 * 53861 * 3824323 * 548758921 * 1619114716441 * 3950618117093215882664998523107 * c474 2^1823-3 = 5 * 23 * 389 * 66721 * 2060215632308931913061 * 131931114257556282951974075230358663 * c483 2^1843-3 = 5 * 97 * 280711259898210652765343 * 748923757901435055742290186738619 * 1799107932565336322239321783163446985375662307231 * c448 2^1847-3 = 5 * 5 * 409 * 686722907166013363 * 236797449199701974093994814439 * c505 2^1873-3 = 26951 * p560 2^1895-3 = 5 * 42787 * 8287976103773 * 20539088230061171587346303497 * c524 2^1901-3 = 7963 * 46189044029 * p558 2^1915-3 = 5 * 4114512988871 * 281462610900311 * 11218867375119403 * c533 2^1919-3 = 5 * 173 * 1733 * 37773581 * 1016559534235848341 * 524949029569368844871 * 1505563 * c520 2^1945-3 = 197 * 1277 * 45077 * 305342318016278947081 * 11184959 * c548 2^1949-3 = 72379921 * 7750803026717 * c566 2^1973-3 = 139 * 86399 * 180023 * 30127061 * 26998679 * c567 2^1987-3 = 5 * 5 * 97 * 32231569 * 1120629947519 * c576 2^1997-3 = 47 * 1145976034515732587597 * c579 2^2039-3 = 5 * 300623 * 43867897 * c600 2^2045-3 = 53 * 163 * 61561 * 2341133 * 79231846941047 * 1736178472938907195169351 * 2359517151754520155989889799 * 14410887136275556006651345177901903 * c501 2^2063-3 = 5 * c621 2^2069-3 = 101 * 59256859 * 2601035737277 * c601 2^2083-3 = 5 * 19 * 97 * 119072827 * 164239133 * 1097689009 * c598 2^2087-3 = 5 * 5 * 23 * 1141323389 * p617 2^2141-3 = 197 * 19853 * 3944251 * 12870247249 * 819972505644437 * 87490661554821877050231863233 * c578 2^2155-3 = 5 * 19 * 797 * 113591 * 1130813 * 5475422802106802231 * c614 2^2185-3 = 372769 * 273578749969261092887 * c632 2^2203-3 = 5 * c663 2^2207-3 = 5 * 5 * 5 * 163 * 13727413446619 * 2006151459787 * c635 2^2213-3 = 22013 * 24754323715490986288487 * 9081481512435295511498877234048666647 * c603 2^2231-3 = 5 * 25153 * 8184019 * 105774070757 * c649 2^2237-3 = 18701 * p670 2^2305-3 = 53 * 1968384410138324933 * p674 2^2309-3 = 239 * 499 * 11981 * 59879 * 10144227001 * c672 2^2323-3 = 5 * 97 * 839 * 477736507 * 438713535417187 * 7390688635320764417007589151 * c643 2^2333-3 = 4297 * 314303816350529318253481459 * c673 2^2357-3 = 29 * 53 * 248203 * 51317074169463669952421 * p679 2^2371-3 = 5 * 19 * 97 * 128833984198511590176264481 * 3748698170941511665670707 * c660 2^2377-3 = 6142949 * 20202961 * 20631514751 * c692 2^2395-3 = 5 * 23 * 36307 * 65213 * c710 2^2399-3 = 5 * 653 * 293543 * 10449701 * p707 2^2423-3 = 5 * 30936207876697835114802293 * c704 2^2447-3 = 5 * 5 * 492216293 * 3184190244241 * c715 2^2467-3 = 5 * 5 * 97 * 135635702518486140602292029 * c714 2^2473-3 = 1019899 * 17261546884151 * 388266421787674831969 * 3601314020621311864696353389 * c678 2^2477-3 = 35227 * 583249 * c736 2^2491-3 = 5 * 74903 * 29004359 * 653859461 * c729 2^2495-3 = 5 * 1171 * c748 2^2515-3 = 5 * 19 * 97 * 23310811 * 66212989867 * c735 2^2539-3 = 5 * 49891503766099 * 7732037004911 * c738 2^2543-3 = 5 * 38371 * c761 2^2545-3 = 7343874518693 * 10028481983710321 * 271196985416234601527 * 12664991 * 1041980079435056545979685506279 * 354983154391345758205114157312364413 * c645 2^2549-3 = 23 * 47 * 509 * 20282209 * 19234430473 * c744 2^2659-3 = 5 * 19 * 23 * 97 * 557 * 1201 * 47017 * 8500652689 * c775 2^2663-3 = 5 * 139 * 409 * 28181 * 3103546177 * 7584361603321 * 9462509032258507203199537 * c745 2^2683-3 = 5 * c807 2^2693-3 = 29 * 163 * 859 * 15334255118105073532471009 * c779 2^2707-3 = 5 * 5 * 5 * 97 * 743 * p808 2^2711-3 = 5 * 71 * c814 2^2741-3 = 220776424387 * 191621861419959336180643 * 6789793 * 2168200950168720456486379822925501 * c751 2^2755-3 = 5 * 97 * 161831 * 219843679709 * 116743827878663 * 6503312556155852017658452007440921 * c763 2^2785-3 = 19 * 239 * 293 * 3818030657923 * 594630137296279297 * 220103320606082159572915830889 * c773 2^2789-3 = 269 * 18792839408392891 * c821 2^2813-3 = 23 * c846 2^2831-3 = 5 * 4666616889415458011207 * c830 2^2837-3 = 1182300566893430027884119710087 * c824 2^2861-3 = 29 * 5336659 * 180422147226438578933 * 631018045040941480847 * c813 2^2879-3 = 5 * 23 * 1901 * c862 2^2903-3 = 5 * 199081 * 190607957 * c860 2^2909-3 = 274033 * 117597414859929601343 * 17004376033 * 111471610561022569607 * c820 2^2929-3 = 19 * 53 * 5246707 * 12943579792132201 * c856 2^2957-3 = 643 * 2179760386560324941 * 10132957147616232443288990707 * c841 2^2995-3 = 5 * 97 * 5676195685618388641219 * c878 2^2999-3 = 5 * 1266692281 * 2332714610310196666253206975679 * c863 2^3023-3 = 5 * 239 * 431 * 1367 * 483671 * c896 2^3119-3 = 5 * 167 * 9817212409 * 854414731 * 79286111713 * 882222460453629252553459 * c883 2^3167-3 = 5 * 5 * 6697160235501767 * 1743203099666257 * c921 2^3173-3 = 51461 * 4289823887 * 1256254989169 * 63774993792649959817 * c909 2^3197-3 = 29 * 60565725579742459 * c945 2^3215-3 = 5 * 139 * 557 * 216475627 * c954 2^3221-3 = 1277 * 115621048177 * 6764339594699761301 * 181914601403108312381 * 30566040370991774336172119 * c891 2^3235-3 = 5 * 19 * 97 * 104947 * 446858257 * c957 2^3265-3 = 25873 * 30891155316662471 * 526971973331350304143901 * c939 2^3287-3 = 5 * 5 * 22343 * 24677 * 44278027 * c972 2^3307-3 = 5 * 5 * 5 * 19 * 15330197544574104538393 * c970 2^3335-3 = 5 * 9297481 * c997 2^3337-3 = 29 * 977381563603 * 29289391253134339 * 51807289181656029899951 * c952 2^3359-3 = 5 * 13633751 * 8581679610929827 * c988 2^3385-3 = 23 * 643 * 827857 * 8421187 * c1002 2^3407-3 = 5 * 5 * 5 * 23 * 4297 * 40376251 * 195656546045491 * 6291855361025953 * 33422256287264483781987815261663 * c950 2^3413-3 = 26588624327 * c1017 2^3427-3 = 5 * 5 * 97 * 47903 * 28314114721 * c1014 2^3433-3 = 19 * 1056217 * 68367457795727 * 12803270443238824948407143 * c988 2^3461-3 = 1571929 * 22044962985697 * 429533645527690958485510629614227 * 6124362808548660596652023 * p965 2^3505-3 = 19 * 29 * 20161 * 30411097 * 138784155449875801 * 10612642999749041734369 * c1002 2^3527-3 = 5 * 5 * 93887 * 11428297 * 1159725398429233 * 25571256584809029632268337 * 6113064035797620784892179 * c984 2^3533-3 = 29 * 3401929 * 963820541241269 * c1041 2^3547-3 = 5 * 5 * 31177 * 402383 * 12594236329 * c1047 2^3557-3 = 3407 * 16874413469990159 * c1052 2^3571-3 = 5 * 97 * 7541 * c1069 2^3581-3 = 4648373 * 3809809 * 484027991 * c1057 2^3595-3 = 5 * 19 * c1081 2^3623-3 = 5 * 90163 * 114409856643321427 * 383291840416033 * c1054 2^3629-3 = 139 * 159769 * 183959 * 8526326265551 * 266653160987953 * 29495964430486752760759330229 * c1025 2^3643-3 = 5 * 135479 * 1728091 * c1085 2^3671-3 = 5 * 23 * 1747 * c1100 2^3677-3 = 1515994897920655398529 * 6597678854553101900289142051 * 48709722012767246903251849 * c1033 2^3701-3 = 29 * 2103149 * 26241097645897 * 4428964997909956157 * c1075 2^3715-3 = 5 * 23 * 97 * 1826977 * 40859584379000248489746941 * 18502493552807699435966453 * c1058 2^3719-3 = 5 * 49883778304949 * c1106 2^3743-3 = 5 * 77528529047 * 6844448394562198156973 * 20331765676462569481969 * 211689739143119358310800811 * c1045 2^3749-3 = 6551 * 10657375250539243 * 159028015952874002063 * c1089 2^3763-3 = 5 * 97 * 389 * 599 * 19087415305373 * 389726706506912269357921 * c1088 2^3767-3 = 5 * 5 * 139 * 146206666897 * 34853528643289 * 33669837198793 * 2057738307350142727805597 * 530284793987409423918236750251 * 192143053489588445115855774293 * c1009 2^3793-3 = 19 * 154849 * 29099981 * c1128 2^3797-3 = 431 * 2309 * 4679 * c1134 2^3841-3 = 29 * c1155 2^3863-3 = 5 * 279551 * c1157 2^3865-3 = 19 * 53 * 2357 * 775019394940489 * 67553631241176147639053182483 * c1114 2^3911-3 = 5 * 1375823 * 253863714977401349760263 * 9674253559530693690538081 * c1123 2^3917-3 = 53 * 379 * 190997 * 450361 * 22601886547 * 7496792547079133933 * c1135 2^3931-3 = 5 * 44257 * 75619 * 18225797853311 * 10051934120022527 * c1144 2^3979-3 = 5 * 23 * 1670852789 * 661806048117808213157 * 100906882710884698271 * 586683495824694445711948845461 * c1116 2^3985-3 = 9954787 * 75457220203 * c1182 2^3989-3 = 163 * 404595793679641 * c1184 2^4003-3 = 5 * 97 * 150523 * 6295287946980816001 * c1179 2^4007-3 = 5 * 5 * 5 * 359 * 2539 * c1199 2^4009-3 = 19 * 29 * 3131861 * 3258371521469 * 7408124215579 * c1173 2^4013-3 = 263 * 5827 * 19699 * c1198 2^4031-3 = 5 * 14423 * 200403113984807 * c1195 2^4079-3 = 5 * 191 * c1225 2^4105-3 = 207121 * 6491373763681 * c1218 2^4127-3 = 5 * 5 * 44203 * 2513976019 * p1227 2^4133-3 = 23 * 1002241 * 6196878749 * 2590244121550238053432575352873 * c1197 2^4157-3 = c1252 2^4195-3 = 5 * 97 * 383 * 2909 * 1093319977 * c1246 2^4219-3 = 5 * 20707 * 127493 * 183503 * 20256359 * c1248 2^4229-3 = 53 * 624017803 * 7737570793 * c1253 2^4243-3 = 5 * 19 * 19 * 23 * 97 * 211 * 15259 * c1265 2^4253-3 = c1281 2^4295-3 = 5 * 379 * 409 * 451377313 * 42672744547 * 433532412811 * 70873636995488377 * c1240 2^4297-3 = 19 * 47 * 197 * 165721 * 175628722301 * 1724312925001 * c1260 2^4315-3 = 5 * 19 * 2151968999 * 21214623372856271 * c1272 2^4321-3 = 71 * 32117 * 841541 * 10083817 * 12408489909843749 * c1266 2^4349-3 = 33457 * 82013 * 5571149 * 704977258201 * 28422676935074421716329373 * c1256 2^4363-3 = 5 * 1124687 * c1307 2^4373-3 = 29 * 37675751 * 8505035340866685316897 * 51908261590300613300711 * c1263 2^4391-3 = 5 * 71 * 36241 * 165612893 * 113878906561852032533 * c1287 2^4393-3 = 3361150011898099 * c1307 2^4397-3 = 23 * 269 * 503 * 414643 * 73991888045174597 * c1295 2^4435-3 = 5 * 47 * 97 * 3307 * 13244710613 * 116175474629 * 1341297878540759 * 538516058490370483 * 13189342416508943 * 34982471706932068341901273 * c1232 2^4463-3 = 5 * 23 * 23 * 544513 * c1335 2^4465-3 = 9049 * 24859 * 3114389 * 78123483259 * 34074288791495949307 * c1299 2^4493-3 = 197 * 3917 * 47087 * 1510067549 * 2598446113 * c1324 2^4507-3 = 5 * 5 * 5 * 23 * 100877351 * 590858126489473 * c1331 2^4517-3 = 45962779 * 79612249 * 418459685969724481 * c1327 2^4531-3 = 5 * 19 * 71 * 97 * 5323 * 25254173227 * 181938612893 * 187023029 * 216290360438611 * c1311 2^4535-3 = 5 * 677 * c1362 2^4541-3 = 29 * 53 * 131251 * 14524343 * c1352 2^4559-3 = 5 * 971904097 * 11642589841 * c1353 2^4583-3 = 5 * 2126441323709 * 820983557 * c1358 2^4603-3 = 5 * 19 * 3642008497 * c1375 2^4637-3 = 163 * 69791965971021878603113 * c1371 2^4679-3 = 5 * 17683 * 51047 * 258241 * 10679636974392241 * 53272438912437036662960275831921 * c1346 2^4703-3 = 5 * 409 * 7134647091418088964541844951 * c1385 2^4705-3 = 23 * 1129 * 67981435505141 * c1399 2^4723-3 = 5 * 97 * 359 * 8008373 * 21450059815708671024893 * 710784776768452728411077 * c1364 2^4727-3 = 5 * 5 * 23 * 6577 * 4202371 * 9215471 * 4240717220847634666450973 * c1379 2^4729-3 = 19 * 284737 * 10252079 * c1410 2^4747-3 = 5 * 5 * 19 * c1427 2^4799-3 = 5 * 163 * 263 * 499 * 160681 * 4242708225527 * 752038629490140843259 * c1398 2^4843-3 = 5 * 173 * 107021 * c1450 2^4853-3 = 53 * 1491961 * 20547931 * p1446 2^4877-3 = 29 * 40411179217 * 979177599643 * 110142989547529 * 233180554192919 * c1416 2^4915-3 = 5 * 97 * 431 * 9883 * 1508251 * 64417007 * 82072897 * c1449 2^4919-3 = 5 * 44851 * 47981 * 131381 * 22803241949 * c1456 2^4943-3 = 5 * 29946875218174035721180614911 * c1459 2^4967-3 = 5 * 5 * 1928621 * 297425477 * 30984788156161079 * c1463 2^4973-3 = 2399 * 22993 * 147792832438060050225241 * p1467 2^4987-3 = 5 * 5 * 47 * 9337 * c1495 2^4997-3 = 197371 * 183606383 * 698712795651179335613 * c1470 ----------------------------------- Some most wanted bigger values: ----------------------------------- 2^5855-3 = 5 * 2161 * 3563029277 * 335410117891 * 21272490352073383243 * c1719 2^9215-3 = 5 * 23 * 431 * 32847695189 * 6160332805922889924031308137779 * 1453809470551961545587934733997027128352123409 * c2683 2^9365-3 = 4801 * 70051 * c2811 2^35587-3 = 5 * 5 * 97 * 311 * 797 * 72481 * 76444103357 * 200219234070611857 * c10671 2^138643-3 = 5 * 97 * 211 * 2467 * 10559 * 31177 * 236105371 * c41711 2^177935-3 = 5 * 112939 * c53559 2^693215-3 = 5 * 139 * c208676 ----------------------------------- Some other factorizations ----------------------------------- 2^10805-3 = 139 * 530103602687 * 35846450981017 * 3515306589614322630174697210644514897 * p3189 ----------------------------------- Notes: For p|2^x-3 where x odd, 6 is a quadratic residue mod p. This means p = 1, 5, 19, 23 (mod 24) Exponents like 1843 and 9215 are wanted candidates is because their order mod 2 is small and divisible by 24. - Written by Joe K. Crump - qqqjoecr@carolina.rr.com (remove the 3 qs)