A Personal Prime Number

We created a big prime number for Manindra Agrawal's talk in our Faculty Colloquium. Its binary representation can be interpreted as an ASCII (or Latin-1) string, and it starts with "Manindra Agrawal: PRIMES is in P, Stuttgart, 23.02.2005."

Here it is:

Latin-1
(C syntax)
"Manindra Agrawal: PRIMES is in P, Stuttgart, 23.02.2005.\005\0024c\
\234{wNƷ4amͷ\037s\236@=Ʋ>a\030CJ\225\177\000 0\2117o\
\003\tQ\013\017\033"
hex 4d616e696e647261204167726177616c3a205052494d455320697320696e20502c205374
757474676172742c2032332e30322e323030352e050234cf63e59c7bdb77a14ec6b7ed34
616dcdb71f739e40aee83dc6b23e6118b7434a957f002030d689f2b7b637db6ffefaa303
09b0510b0fdc1bf1
decimal 685848279037904143878006271392269766267359585052693539865282407369399677
747874617101756917649347901318174432475966271394444966869914576914460510
675240212743387221614225407276918395648745917506428573540768672974979434
751171108479897062825276599507610237684754240379651593934216177

The proof

Using Pocklington's Theorem, we can conclude the following fact:

Suppose n-1 = kq, where k<q and q is prime. If there is an integer a>1 such that

then n is prime.

You can easily check that all the qi below are primes by checking these two conditions (with a = 2).

q0= 23
q1 = 2 * q0 + 1 = 47
q2 = 6 * q1 + 1 = 283
q3 = 6 * q2 + 1 = 1699
q4 = 12 * q3 + 1 = 20389
q5 = 12 * q4 + 1 = 244669
q6 = 30 * q5 + 1 = 7340071
q7 = 28 * q6 + 1 = 205521989
q8 = 2 * q7 + 1 = 411043979
q9 = 12 * q8 + 1 = 4932527749
q10 = 60 * q9 + 1 = 295951664941
q11 = 16 * q10 + 1 = 4735226639057
q12 = 48 * q11 + 1 = 227290878674737
q13 = 54 * q12 + 1 = 12273707448435799
q14 = 72 * q13 + 1 = 883706936287377529
q15 = 28 * q14 + 1 = 24743794216046570813
q16 = 180 * q15 + 1 = 4453882958888382746341
q17 = 102 * q16 + 1 = 454296061806615040126783
q18 = 4 * q17 + 1 = 1817184247226460160507133
q19 = 12 * q18 + 1 = 21806210966717521926085597
q20 = 106 * q19 + 1 = 2311458362472057324165073283
q21 = 50 * q20 + 1 = 115572918123602866208253664151
q22 = 18 * q21 + 1 = 2080312526224851591748565954719
q23 = 48 * q22 + 1 = 99855001258792876403931165826513
q24 = 106 * q23 + 1 = 10584630133432044898816703577610379
q25 = 44 * q24 + 1 = 465723725871009975547934957414856677
q26 = 56 * q25 + 1 = 26080528648776558630684357615231973913
q27 = 2 * q26 + 1 = 52161057297553117261368715230463947827
q28 = 18 * q27 + 1 = 938899031355956110704636874148351060887
q29 = 126 * q28 + 1 = 118301277950850469948784246142692233671763
q30 = 190 * q29 + 1 = 22477242810661589290269006767111524397634971
q31 = 42 * q30 + 1 = 944044198047786750191298284218684024700668783
q32 = 420 * q31 + 1 = 396498563180070435080345279371847290374280888861
q33 = 102 * q32 + 1 = 40442853444367184378195218495928423618176650663823
q34 = 54 * q33 + 1 = 2183914085995827956422541798780134875381539135846443
q35 = 24 * q34 + 1 = 52413938063899870954141003170723237009156939260314633
q36 = 40 * q35 + 1 = 2096557522555994838165640126828929480366277570412585321
q37 = 2 * q36 + 1 = 4193115045111989676331280253657858960732555140825170643
q38 = 56 * q37 + 1 = 234814442526271421874551694204840101801023087886209556009
q39 = 260 * q38 + 1 = 61051755056830569687383440493258426468266002850414484562341
q40 = 56 * q39 + 1 = 3418898283182511902493472667622471882222896159623211135491097
q41 = 60 * q40 + 1 = 205133896990950714149608360057348312933373769577392668129465821
q42 = 42 * q41 + 1 = 8615623673619929994283551122408629143201698322250492061437564483
q43 = 256 * q42 + 1 = 2205599660446702078536589087336609060659634770496125967728016507649
q44 = 24 * q43 + 1 = 52934391850720849884878138096078617455831234491907023225472396183577
q45 = 40 * q44 + 1 = 2117375674028833995395125523843144698233249379676280929018895847343081
q46 = 150 * q45 + 1 = 317606351104325099309268828576471704734987406951442139352834377101462151
q47 = 658 * q46 + 1 = 208984979026645915345498889203318381715621713774048927694165020132762095
359
q48 = 182 * q47 + 1 = 380352661828495565928807978350039454722431519068769048403380336641627013
55339
q49 = 18 * q48 + 1 = 684634791291292018671854361030071018500376734323784287126084605954928624
396103
q50 = 100 * q49 + 1 = 684634791291292018671854361030071018500376734323784287126084605954928624
39610301
q51 = 248 * q50 + 1 = 169789428240240420630619881535457612588093430112298503207268982276822298
85023354649
q52 = 44 * q51 + 1 = 747073484257057850774727478756013495387611092494113414111983522018018114
941027604557
q53 = 206 * q52 + 1 = 153897137756953917259593860623738780049847885053787363307068605535711731
677851686538743
q54 = 110 * q53 + 1 = 169286851532649308985553246686112658054832673559166099637775466089282904
84563685519261731
q55 = 338 * q54 + 1 = 572189558180354664371169973799060784225334436629981416775681075381776218
3782525705510465079
q56 = 354 * q55 + 1 = 202555103595845551187394170724867517615768390567013421538591100685148781
3059014099750704637967
q57 = 138 * q56 + 1 = 279526042962266860638603955600317174309760378982478521723255718945505318
202143945765597240039447
q58 = 54 * q57 + 1 = 150944063199624104744846136024171274127270604650538401730558088230572871
82915773071342250962130139
q59 = 148 * q58 + 1 = 223397213535443675022372281315773485708360494882796834561225970581247850
3071534414558653142395260573
q60 = 50 * q59 + 1 = 111698606767721837511186140657886742854180247441398417280612985290623925
153576720727932657119763028651
q61 = 222 * q60 + 1 = 247970907024342479274833232260508569136280149319904486362960827345185113
84094032001601049880587392360523
q62 = 46 * q61 + 1 = 114066617231197540466423286839833941802688868687156063726961980578785152
3668325472073648294507020048584059
q63 = 14 * q62 + 1 = 159693264123676556652992601575767518523764416162018489217746772810299213
31356556609031076123098280680176827
q64 = 354 * q63 + 1 = 565314154997815010551593809578217015574126033213545451830823575748459215
1300221039597000947576791360782596759
q65 = 402 * q64 + 1 = 227256290309121634241740711450443240260798665351845271635991077450880604
4822688857917994380925870127034603897119
q66 = 4 * q65 + 1 = 909025161236486536966962845801772961043194661407381086543964309803522417
9290755431671977523703480508138415588477
q67 = 270 * q66 + 1 = 245436793533851364981079968366478699481662558579992893366870363646951052
8408503966551433931399939737197372208888791
q68 = 340 * q67 + 1 = 834485098015094640935671892446027578237652699171975837447359236399633579
658891348627487536675979510647106551022188941
q69 = 608 * q68 + 1 = 507366939593177541688888510607184767568492841096561309167994415730977216
432605939965512422298995542473440783021490876129
q70 = 354 * q69 + 1 = 179607896615984849757866532754943407719246465748182703445470023168765934
617142502747791397493844422035598037189607770149667
q71 = 34 * q70 + 1 = 610666848494348489176746211366807586245437983543821191714598078773804177
6982845093424907514790710349210333264446664185088679
q72 = 122 * q71 + 1 = 745013555163105156795630377867505255219434339923461853891809656104041096
791907101397838716804466662603660658262493030580818839
q73 = 402 * q72 + 1 = 299495449175568273031843411902737112598212604649231665264507481753824520
910346654761931164155395598366671584621522198293489173279
q74 = 370 * q73 + 1 = 110813316194960261021782062404012731661338663720215716147867768248915072
736828262261914530737496371395668486309963213368590994113231
q75 = 906 * q74 + 1 = 100396864472633996485734548538035534885172829330515438829968198033517055
899566405609294564848171712484475648596826671311943440666587287
q76 = 60 * q75 + 1 = 602381186835803978914407291228213209311036975983092632979809188201102335
3973984336557673890890302749068538915809600278716606439995237221
q77 = 82 * q76 + 1 = 493952573205359262709813978807134831635050320306135959043443534324903915
025866715597729259053004825423620191096387222854761728079609452123
q78 = 264 * q77 + 1 = 130403479326214845355390890405083595551653284560819893187469093061774633
566828812917800524389993273911835730449446226833657096213016895360473
q79 = 525943236009983468210154244860062602027123012155279980775428662753589024
4648436980051004757283329234764708177817368765476482672199032198512 * q78 + 1
 = 685848279037904143878006271392269766267359585052693539865282407369399677
747874617101756917649347901318174432475966271394444966869914576914460510
675240212743387221614225407276918395648745917506428573540768672974979434
751171108479897062825276599507610237684754240379651593934216177

Note: Although we didn't use any random data to compute this particular prime number it is also possible to use random data in the process, such that different random bitstings will always result in different prime numbers.