Revisions to Is $\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}$ composite for $n\geq 4$?

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edited Oct 15, 2023 at 22:35

Conor Grogan

39
2

I just ran this independently for n=2017; confirmed that this number is in fact prime! So an is composite for 4≤n≤2016

I ran sympy.factorint on a i9-13900k for the 5789 digit number n=2017 and it only returned 1 and the number

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edited Oct 14, 2023 at 20:49

Conor Grogan

39
2

I just ran this independently for n=2017; confirmed that this number is in fact prime! So an is composite for 4≤n≤2016

deleted 5884 characters in body

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edited Oct 14, 2023 at 20:44

Conor Grogan

39
2

I just ran this for n=2017; it turns outconfirmed that this number is not prime. It has a prime factor: 63223146539444127264331417315104100312684325863600957186878024413253611745182928064406479939188054490132838660596269813696794373617032276703128975601578719164280703498091677495948394952244165911912638667915756859591780521327406515972764980651208198563816477600424926750952665631491803393437510388581053994962655860989695201315191890867024868102789729899587692343250256426647530944284504721877101124092377208455771671487978724365783266395245737404593594391104030210821486998296501252292979387009256992973986015196211880486651646015422607964753630348198921148338150159368733827180720900153630110100412467451274192581550192125352192247956585363389004536722158557841611124065033779536896086071913448562402366568780044362046700771846167808671762618778756986448556674995065032716662282340654024722693423272876200631655231222965622535566006628286761842194207857846055319877244093440696424877698828042230919785899114556824258064757187326298495345644902604113462500591721262738357325993748832248295498639763170141245441158393041177198297560157185762004265632710564366267440678487164003542889760131405944054490508900270973676944021122783909698737051220650715805886449888495086376355789401486403226574552684061000444557639102923722566267812983405678541855530701649029196287560874998996010060423709329455795801788933034542548171791557767815053201392768583406315240935858067190468148397420275411256597848820248982778417903598244626158227483392931814033866007246525953799172315651461400951995632705584679693176232511732772841693600766824113975451704478057514220756327533363461956685366850210970676733442265423418875891966487619371513640360162408377696260146455451821655763125925731977370319523275999924164178913588323394804691459758229065229038464316271995893778174198127734939422545526608052481142510075576939916851556394030609226099617151969141395197065338961117946022048419658381689527833502581269906885191781124275195317348695963517938872829385950744684689738007279973393478060548658730794310431640719239373268744260794695395513102080896203043853210349533286768317886680305679753665266403449282420310295551604805130656513090376102223516445877976089018236366864144654338179251050984337874323963734632034207227730192782307857094665283767264302710205195385213148149013937947972967811536634529356728669999481349183140804891988987739550049673563219837423937270279743728567226951895127042459067078595294455606182608186925286118777272413215177174227123805663403780085611404045995159300831556415304654859950504640354098406959755258743978835503474394572565802888892012977973365122829040249082299594680303836660568775986633674042281317830360733369749060436748503875933256309338908332288550824093903929486948834217140945595115516277755986997395207196706466475446453813429447539178124926050320290288909240754164041114010531546193430711027458612666287238101095876667750722558422554387859065892222847151779700185736966121470278579869161320032091360963691466147514436185673632011144497552490222296598606268054476532078299395272165377340898287673012639080031852215840468246582481598781564556545178861068025677036157760724634601106268638910643186476225684775402852590148428066528459205385547888811548994895615278575145956398125608333341118172219853445565909890245754082382863991011917553998772851218564144253573532264039129006769806406632840776007795428044958759134773337535004476753316754336368838888641816220164539654488148472755491537112433737534994828925807928014294074966467439839574577943145973992114580224284405319153670703510282614039442853776202353781580667357547367109796055879023399790503749534199733604777687094685533585616969699398258597078694723495083280716891564629998125713431698012442753317614061727420507909187465176710581032294717606934024218287789241997751257013433549022315095612401113722335413027043319287488681557442042470172994083480088909478915835126616544565481786221372787273021543059435204508678668082442390946262887838473901332105020868112483777363343376747486023403341438627031415159977535175145288226499181486826558144765024836235809199335481456527414413106011148238529912077712075900602763762579851588747319129144062965193058594897134617483333095677885028202849740042100884211747345316791245170912812249471676444465925171545260029525810562943668565528974581372250399395455793932392410091054540567585577164299093904156539161004380410016530647650459763463910179549802531232456396086797234069221281793246164892485343703455635633898995278070862276512650093687688292683829393804188934433724531530498100297639431221167599915182060391893110426386240451574216262154661490623064743917609339817839315290292668277397110680002945669508054000468903401967373577884961848919962332133780475121433808770453763800052611841130101216693657636616611797534825207678238125826524052364194419145203231458830049901062795676017567264861185043619666087545032244948747464345346033972602142664258991030583419287203858018623589583826546763909727322189068634344877726034140424991689373359461823952459309461853106508924657555193831116161958792338487044800261883456177723944627371006874748949064252397249940771019048829945143589552042298387390573196551548176565512564349114521168621738019102633156984445018956102197300623115375255728336920875497268777216852430330401661366154704246620746503971399599021551905873224693761212236571750842446686208441381189149229882322519783808863835479330744188798172582835530048256465183409254943018250719142641706018803460604249233680243802957772345598316999073027404272833456933470908362381011994440500170931085286410426253781506469752570078902343029977928867186814153266451690184035520694771829019822508993058613615270055509653546785252761545064836539489410568354090277105305743244175751471582262147410566728968719108210569494637859046005379483998543617390911237504764165648477785217, Exponent: 1

So, it is composite through at least 2017!

I just ran this for n=2017; it turns out that this number is not prime. It has a prime factor: 63223146539444127264331417315104100312684325863600957186878024413253611745182928064406479939188054490132838660596269813696794373617032276703128975601578719164280703498091677495948394952244165911912638667915756859591780521327406515972764980651208198563816477600424926750952665631491803393437510388581053994962655860989695201315191890867024868102789729899587692343250256426647530944284504721877101124092377208455771671487978724365783266395245737404593594391104030210821486998296501252292979387009256992973986015196211880486651646015422607964753630348198921148338150159368733827180720900153630110100412467451274192581550192125352192247956585363389004536722158557841611124065033779536896086071913448562402366568780044362046700771846167808671762618778756986448556674995065032716662282340654024722693423272876200631655231222965622535566006628286761842194207857846055319877244093440696424877698828042230919785899114556824258064757187326298495345644902604113462500591721262738357325993748832248295498639763170141245441158393041177198297560157185762004265632710564366267440678487164003542889760131405944054490508900270973676944021122783909698737051220650715805886449888495086376355789401486403226574552684061000444557639102923722566267812983405678541855530701649029196287560874998996010060423709329455795801788933034542548171791557767815053201392768583406315240935858067190468148397420275411256597848820248982778417903598244626158227483392931814033866007246525953799172315651461400951995632705584679693176232511732772841693600766824113975451704478057514220756327533363461956685366850210970676733442265423418875891966487619371513640360162408377696260146455451821655763125925731977370319523275999924164178913588323394804691459758229065229038464316271995893778174198127734939422545526608052481142510075576939916851556394030609226099617151969141395197065338961117946022048419658381689527833502581269906885191781124275195317348695963517938872829385950744684689738007279973393478060548658730794310431640719239373268744260794695395513102080896203043853210349533286768317886680305679753665266403449282420310295551604805130656513090376102223516445877976089018236366864144654338179251050984337874323963734632034207227730192782307857094665283767264302710205195385213148149013937947972967811536634529356728669999481349183140804891988987739550049673563219837423937270279743728567226951895127042459067078595294455606182608186925286118777272413215177174227123805663403780085611404045995159300831556415304654859950504640354098406959755258743978835503474394572565802888892012977973365122829040249082299594680303836660568775986633674042281317830360733369749060436748503875933256309338908332288550824093903929486948834217140945595115516277755986997395207196706466475446453813429447539178124926050320290288909240754164041114010531546193430711027458612666287238101095876667750722558422554387859065892222847151779700185736966121470278579869161320032091360963691466147514436185673632011144497552490222296598606268054476532078299395272165377340898287673012639080031852215840468246582481598781564556545178861068025677036157760724634601106268638910643186476225684775402852590148428066528459205385547888811548994895615278575145956398125608333341118172219853445565909890245754082382863991011917553998772851218564144253573532264039129006769806406632840776007795428044958759134773337535004476753316754336368838888641816220164539654488148472755491537112433737534994828925807928014294074966467439839574577943145973992114580224284405319153670703510282614039442853776202353781580667357547367109796055879023399790503749534199733604777687094685533585616969699398258597078694723495083280716891564629998125713431698012442753317614061727420507909187465176710581032294717606934024218287789241997751257013433549022315095612401113722335413027043319287488681557442042470172994083480088909478915835126616544565481786221372787273021543059435204508678668082442390946262887838473901332105020868112483777363343376747486023403341438627031415159977535175145288226499181486826558144765024836235809199335481456527414413106011148238529912077712075900602763762579851588747319129144062965193058594897134617483333095677885028202849740042100884211747345316791245170912812249471676444465925171545260029525810562943668565528974581372250399395455793932392410091054540567585577164299093904156539161004380410016530647650459763463910179549802531232456396086797234069221281793246164892485343703455635633898995278070862276512650093687688292683829393804188934433724531530498100297639431221167599915182060391893110426386240451574216262154661490623064743917609339817839315290292668277397110680002945669508054000468903401967373577884961848919962332133780475121433808770453763800052611841130101216693657636616611797534825207678238125826524052364194419145203231458830049901062795676017567264861185043619666087545032244948747464345346033972602142664258991030583419287203858018623589583826546763909727322189068634344877726034140424991689373359461823952459309461853106508924657555193831116161958792338487044800261883456177723944627371006874748949064252397249940771019048829945143589552042298387390573196551548176565512564349114521168621738019102633156984445018956102197300623115375255728336920875497268777216852430330401661366154704246620746503971399599021551905873224693761212236571750842446686208441381189149229882322519783808863835479330744188798172582835530048256465183409254943018250719142641706018803460604249233680243802957772345598316999073027404272833456933470908362381011994440500170931085286410426253781506469752570078902343029977928867186814153266451690184035520694771829019822508993058613615270055509653546785252761545064836539489410568354090277105305743244175751471582262147410566728968719108210569494637859046005379483998543617390911237504764165648477785217, Exponent: 1

So, it is composite through at least 2017!

I just ran this for n=2017; confirmed that this number is prime!

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edited Oct 14, 2023 at 20:32

Conor Grogan

39
2

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created Oct 14, 2023 at 20:31

Conor Grogan

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2

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