Is this a Perfect Number ?
 

[TEX] 2^{127}*2[/TEX]

(19:33) gp > 2^127*2
%85 = 340282366920938463463374607431768211456
(19:40) gp > factor ( %85)
%86 =
[2 128]

:no:

"In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself" [[URL="https://en.wikipedia.org/wiki/Perfect_number"]wiki[/URL]]

[QUOTE=paulunderwood;441485]:no:

"In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself" [[URL="https://en.wikipedia.org/wiki/Perfect_number"]wiki[/URL]][/QUOTE]

TL;DR No

With which program can I calculate, this number?
-EDIT-
because [B]2178[/B] -8712 = -6534 + [B]4356[/B] = -2178 [U][B]inverted[/B][/U]

[CODE]
(20:24) gp > 2178^2178
%96 = 1950081059793717728316532672097834427941452404788642501980393902363840261425584064190007930301120306924898218450138483264544333931122005262524746435913669408058974896899481357645435956704284530052894198059325188325491138755622647837906235397320449037449779980010417943756039614778076393752524798934801871593505065307193308624862599128095308702851849875543914644322626115534441044021533410524691703674436501014365155935460225829202827144991942396898598698755085084183844164361295224400819181006912848434068165727699239977862176204821802527958004796033877620005800031891359137410290222117423676259695119132229605294491076536984838863914202683213905159052994943220721783407478702119798983879260226863985710404531861234940769685609737661042841387663058327449389408667137256854180708049852648167305999897646452007591249672723708185304598048021957831598901550869903295723645774204654016304904161763814756270325852931124770073960553660400853849404153941864948979388127688070693351310445992951543209921426871029109605090863333818825238461083211425667468931755019497936602869588339741645865869372168938288355468215427389354250606026382715223909057278967605090553985443058255234294467129823107285864014350549692437034434163420331739726092616685011486761254779764941110050291275402457103663051432683089589324342491805552767493176806659642723577504073916331316926284840791062811208013724460393293297018584863469570142860663862287470409101713355014587688737948950583482698256916402461891000456362068645039119598837307587230302025726350829366638361166110748044171017223599655887290237562869860885616266468383360341365676442214884262923513775709207779800391217139205186738211326537645180493897804989878706416715782227176846238579131009858568745956636885966432878781989782619532895199015807655613509687152387290702843839200826312680131681340883925690469382046880461661895308930044832484926508721226820633225897278400017597982571648524957628407408310656106287585649858063612808601866132068926942151804631806696568265862042589040019024809358792870275914413393723801330897209387369772088774949251316204555782920529985632325282309071280227580726737262991180180756824468527057816987608956197625039032650831838110325965225714048031064862903561455223272840296349786474816348022858419058814013790826707953961044301809698620883047503325912579727129351677460136446249321132061699350779888496375450560476965926145230312542051766445621311053069316659009927529205788139678147962691284285355528560758547696003204852035566493435544741085960123422853866506905204455947215488398374927791013166181280936470530615474196336179885773014775976888963227484332702480169447101852330045784450716679416692714395777420545818481071189206613995467430148085862927184528919034341426826492931346975688113013578245817170337105871200833952364136872778127721138114813211513292949888767864657389476941026832278580060720072886984251565097261054949983264636790143557099152178889815069997082807781471417939690071525258030300448398319271341904693[+++]
(20:27) gp > factor ( %96)
%97 =
[ 2 2178]

[ 3 4356]

[11 4356][/CODE]

[QUOTE=Godzilla;441488]With which program can I calculate, this number?

because [B]2178[/B] -8712 = -6534 + [B]4356[/B] = 2178 [U][B]inverted[/B][/U]


[CODE](20:24) gp > 2178^2178
%96 = 1950081059793717728316532672097834427941452404788642501980393902363840261425584064190007930301120306924898218450138483264544333931122005262524746435913669408058974896899481357645435956704284530052894198059325188325491138755622647837906235397320449037449779980010417943756039614778076393752524798934801871593505065307193308624862599128095308702851849875543914644322626115534441044021533410524691703674436501014365155935460225829202827144991942396898598698755085084183844164361295224400819181006912848434068165727699239977862176204821802527958004796033877620005800031891359137410290222117423676259695119132229605294491076536984838863914202683213905159052994943220721783407478702119798983879260226863985710404531861234940769685609737661042841387663058327449389408667137256854180708049852648167305999897646452007591249672723708185304598048021957831598901550869903295723645774204654016304904161763814756270325852931124770073960553660400853849404153941864948979388127688070693351310445992951543209921426871029109605090863333818825238461083211425667468931755019497936602869588339741645865869372168938288355468215427389354250606026382715223909057278967605090553985443058255234294467129823107285864014350549692437034434163420331739726092616685011486761254779764941110050291275402457103663051432683089589324342491805552767493176806659642723577504073916331316926284840791062811208013724460393293297018584863469570142860663862287470409101713355014587688737948950583482698256916402461891000456362068645039119598837307587230302025726350829366638361166110748044171017223599655887290237562869860885616266468383360341365676442214884262923513775709207779800391217139205186738211326537645180493897804989878706416715782227176846238579131009858568745956636885966432878781989782619532895199015807655613509687152387290702843839200826312680131681340883925690469382046880461661895308930044832484926508721226820633225897278400017597982571648524957628407408310656106287585649858063612808601866132068926942151804631806696568265862042589040019024809358792870275914413393723801330897209387369772088774949251316204555782920529985632325282309071280227580726737262991180180756824468527057816987608956197625039032650831838110325965225714048031064862903561455223272840296349786474816348022858419058814013790826707953961044301809698620883047503325912579727129351677460136446249321132061699350779888496375450560476965926145230312542051766445621311053069316659009927529205788139678147962691284285355528560758547696003204852035566493435544741085960123422853866506905204455947215488398374927791013166181280936470530615474196336179885773014775976888963227484332702480169447101852330045784450716679416692714395777420545818481071189206613995467430148085862927184528919034341426826492931346975688113013578245817170337105871200833952364136872778127721138114813211513292949888767864657389476941026832278580060720072886984251565097261054949983264636790143557099152178889815069997082807781471417939690071525258030300448398319271341904693[+++]
(20:27) gp > factor ( %96)
%97 =
[ 2 2178]

[ 3 4356]

[11 4356][/CODE][/QUOTE]

you can use the code tags that may decrease the scroll needed for people to look at the number. you've made a sign error in your first calculation I can tell you that even without delving into it really. but here's how you messed up see your first calculation is nothing more than the statement x-4x=-3x+2x=x but -3x+2x = -x. your second is just the fact that x= 2178 = (2^1)(3^2)(11^2) so x^x= (2^x)(3^(2x))(11^(2x))

F.T.A. [[URL="https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic"]wiki[/URL]]

:rolleyes:

[QUOTE=science_man_88;441489]you can use the code tags that may decrease the scroll needed for people to look at the number. you've made a sign error in your first calculation I can tell you that even without delving into it really. but here's how you messed up see your first calculation is nothing more than the statement x-4x=-3x+2x=x but -3x+2x = -x. your second is just the fact that x= 2178 = (2^1)(3^2)(11^2) so x^x= (2^x)(3^(2x))(11^(2x))[/QUOTE]

But the two numbers have the same factors (only the factor 3 is plus one time ). Why ?

[CODE]

%43 = 2178
(23:51) gp > factor(%43)
%44 =
[ 2 1]

[ 3 2]

[11 2]

(23:51) gp > 6534
%45 = 6534
(23:52) gp > factor(%45)
%46 =
[ 2 1]

[ 3 [B]3[/B]]

[11 2]


[/CODE][U][B]The 9's in the middle 65-9..9-34 and 21-9..9-78[/B][/U]


[CODE]
(23:47) gp > 65[B]9[/B]34653465346534653465346534
%39 = 65[B]9[/B]34653465346534653465346534
(23:47) gp > factor(%39)
%40 =
[ 2 1]

[ 3 [B]4[/B]]

[ 11 1]

[ 37 1]

[ 139 1]

[7194315905762589999359 1]

(23:47) gp > 21[B]9[/B]78217821782178217821782178
%41 = 21[B]9[/B]78217821782178217821782178
(23:47) gp > factor(%41)
%42 =
[ 2 1]

[ 3 3]

[ 11 1]

[ 37 1]

[ 139 1]

[7194315905762589999359 1]



[/CODE]

[CODE]

%47 = 21[B]9[/B]7821[B]999[/B]7821782178217821782178
(00:04) gp > factor(%47)
%48 =
[ 2 1]

[ 3 4]

[ 11 3]

[ 37 1]

[2754848512151242424269727 1]

(00:04) gp > 65934659993465346534653465346534
%49 = 65[B]9[/B]3465[B]999[/B]3465346534653465346534
(00:04) gp > factor(%49)
%50 =
[ 2 1]

[ 3 5]

[ 11 3]

[ 37 1]

[2754848512151242424269727 1]


[/CODE]

[QUOTE=Godzilla;441504]But the two numbers have the same factors (only the factor 3 is plus one time ). Why ?
[/QUOTE]

because one of them ( 6534) is 3 times the other (2178) so of course they will have the same factors other than the extra 3.

[QUOTE=Godzilla;441484][TEX] 2^{127}*2[/TEX]
is this a perfect number?
[/QUOTE]
No.
[CODE]
gp > s=2^127*2
%1 = 340282366920938463463374607431768211456
gp > divisors(s)
%2 = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, ...etc... many rows of numbers... 340282366920938463463374607431768211456]
gp > sigma(s)
%3 = 680564733841876926926749214863536422911
gp > sigma(s)-s
%4 = 340282366920938463463374607431768211455
gp > 2^128-1
%5 = 340282366920938463463374607431768211455
gp >
[/CODE]
(why? can you tell?)

Also, you seem to confuse "prime factors" with "proper divisors".
[CODE]gp > factorint(2178)
%6 =
[ 2 1]
[ 3 2]
[11 2]
gp > divisors(2178)
%7 = [1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 121, 198, 242, 363, 726, 1089, 2178]
gp >[/CODE]