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O praštevilih

O praštevilih

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Da je praštevil, tj. števil, ki imajo zgolj dva pozitivna cela delitelja - sebe in ena, neskončno mnogo, je že pred več kot dva tisoč leti dokazal Evklid in za njim še mnogo matematikov. Zgodba o praštevilih pa se s tem še ne konča, saj matematiki poleg tega, da iščejo čim večja praštevila, poskušajo dokazati tudi nekaj zanimivih domnev.

Zanimiva podmnožica praštevil so Mersennova praštevila, ki jih lahko zapišemo kot 2p - 1, pri čemer je p tudi praštevilo. Z njihovim iskanjem se ukvarja projekt GIMPS, ki deluje na načelih distributiranega računanja (podobno kot SETI in Folding). Pred dobrima dvema tednoma je Josh Findley odkril enainštiridesto Mersennovo praštevilo 224 036 583 - 1, ki z 7.235.733 znaki v desetiškem zapisu velja za največje praštevilo. Ta petek so uradno potrdili, da gre resnično za praštevilo. Klik!

Še bolj zanimive kot iskanje praštevil pa so domneve o njih. Goldbach je leta 1742 v pismu Eulerju postavil domnevo, da lahko vsako naravno število večje od pet zapišemo kot vsoto treh praštevil. Euler je trditev še zaostril s predpostavko, da lahko vsako sodo število zapišemo kot vsoto dveh praštevil. Ta predpostavka, znana kot Goldbachova domneva, je za zdaj še nedokazana in eden izmed večjih izzivov matematike.

Še bolj zanimiva pa je (bila) predpostavka o praštevilskih dvojčkih. Če pogledamo praštevila, opazimo zanimivo lastnost, da obstaja mnogo praštevilskih dvojčkov, tj. praštevil, katerih razlika znaša dve (naprimer 11 in 13, 281 in 283 ...). Matematike je begalo vprašanje, ali je tudi praštevilskih dvojčkov neskončno mnogo. Očitno jih je, saj je R. F. Arenstorf z nashvillske univerze to trditev dokazal na osemintridesetih straneh.

52 komentarjev

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1
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undefined ::

Wau. Kaj takega. Kdo bi si mislil. :)

Thomas ::

Presenetljivo je, kako so začeli padati težki matematični problemi, ki jim prej ni bilo videti rešitve tudi stoletja.

Še nobenega sledu o Goedlovih nedokazljivih izrekih. Zanimivo!
Man muss immer generalisieren - Carl Jacobi

mchaber ::

>Zanimiva podmnožica praštevil so Mersennova praštevila, ki jih lahko zapišemo kot 2^p - 1, pri čemer je p tudi praštevilo.

Hmm. Kaj pa to:
(neko praštevilo)^2 -2 = neko drugo praštevilo:))

Edit: ne velja, če je "neko praštevilo" =2
.

Thomas ::

Ne velja še za VEČINO drugih praštevil. Poanta GIMPSa je, da gledajo katera so tista redka.
Man muss immer generalisieren - Carl Jacobi

nicnevem ::

>Še nobenega sledu o Goedlovih nedokazljivih izrekih. Zanimivo!


Kaj je pa to za ena huda žvau? :))

CaqKa ::

v bistvu bi blo zlo fajn če bi dali še link na ono staro novico..
pšotem bi sedaj vsaj lahko primerjal če smo prišli kam dalje s temi praštevili al še vedno ne :)

lambda ::

Dobra novica ... skratka fascinating :D

edit.
Zdele sem ravno v Math maturitetnem vzdušju - morem it spat, da bomo zjutraj lahk razmišljal.

BigWhale ::

Fascinating? Ce je prastevil neskoncno mnogo, potem je tudi taksnih 'fint' neskoncno mnogo...

Nekako tako kot PI, ne bo se zacel ponavljati... ... pa, ce se, se tako zelo trudis... Prej ponucas vse svincnike na svetu.

Na nek nacin zavidam matematikom, nucajo samo svincnik in papir, pa nek zadosti tezek racun, pa se zabavajo do nezavesti. ;)

Thomas ::

Seveda. Tudi matematiki so samo ljudje in iščejo p/p optimum. Razen tega so nekoliko sprijeni, pa jim težavnost abstrahiranja zapletenih sistemov povzroča prej pleasure, kot pain. Plus pleasure ob (svetovni) slavi, pa lahko prinese skupaj kar precej.
Man muss immer generalisieren - Carl Jacobi

Jeronimo ::

Zanima me kaj bi se zgodilo, ko bi odkrili vse skrivnosti praštevil? :\
JURIŠ !!!
Preko vode do slobode!

CCfly ::

Seveda. Tudi matematiki so samo ljudje in iščejo p/p optimum.

Fool. Oni samo ekstreme zračunajo.

Tic ::

Jeronimo: Armagedon.
persona civitas ;>

Jeebs ::

Peesda, pa že rajši iščem male zelene, kot pa neka brezvezna praštevila.:\
Press any key to continue... RESET. Didn't you say ANY key?!?!?!

ender ::

mchaber: pa glih za 2 velja >:D
There are only two hard things in Computer Science:
cache invalidation, naming things and off-by-one errors.

gabl136 ::

Nekateri ljudje imajo pa res preveč prostega časa>:D

Thomas ::

> nekateri ljudje imajo pa res preveč prostega časa

Kako naj filozof upraviči svoj obstoj na oslovski farmi?

Ali povedano nič bolj vljudno: Misliš, da ti posvečaš svoje življenje čemu pametnejšemu, kot je raziskovanje praštevil?

Misliš da če ti ne razumeš, potem je pa to kraja časa bogu in denarja ljudem?

Cool your head!
Man muss immer generalisieren - Carl Jacobi

Thomas ::

>> Še nobenega sledu o Goedlovih nedokazljivih izrekih. Zanimivo!

> Kaj je pa to za ena huda žvau?

Vemo, zahvaljujoč se Goedlu, da so v teoriji števil tako težki problemi, da sploh niso rešljivi. Celo največ naj bi bilo takih. Ali pa da je ta teorija protislovna, kar je pa še huje.

Toda vedno ko kakšen problem dodobra naskočijo, se izkaže za rešljivega. To sem mislil, da je zanimivo.
Man muss immer generalisieren - Carl Jacobi

mchaber ::

>mchaber: pa glih za 2 velja

Ne ker potem ni rezultat "neko drugo praštevilo" ampak "neko praštevilo".

>Ne velja še za VEČINO drugih praštevil.

Napiši eno tako število...
.

Sergio ::

>>Peesda, pa že rajši iščem male zelene, kot pa neka brezvezna praštevila.

Yes, enjoy.
Tako grem jaz, tako gre vsak, kdor čuti cilj v daljavi:
če usoda ustavi mu korak,
on se ji zoperstavi.

trs ::

Prastevila niso tolko brezveze kot zveni ... so osnova vecini danasnjih public key/private key kriptirnih algoritmov(RSA ni izjema). Ce kdo izumi nacin kako hitro iskat prastevila(oz. hiter postopek za faktoriziranje stevil) ima v rokah orozje da razbije katerikoli SSH connection ... in se marsikaj drugega ;)

Dr_M ::

Z njihovim iskanjem se ukvarja projekt GIMPS, ki deluje na načelih distributiranega računanja (podobno kot SETI in Folding).

ojoj....se vecja potrata procesorskega casa, kot pa seti...mnogo vecja... :\ :\
The reason why most of society hates conservatives and
loves liberals is because conservatives hurt you with
the truth and liberals comfort you with lies.

Thomas ::

> Napiši eno tako število...

11.
Man muss immer generalisieren - Carl Jacobi

Thomas ::

> ojoj....se vecja potrata procesorskega casa, kot pa seti...mnogo vecja...

Vsekakor manjša, kot uporaba CPU časa za tvoje veleumne doneske tukaj. Trust me!
Man muss immer generalisieren - Carl Jacobi

IceIceBaby ::

A so praštevila še za kej drugega uporabna ? Verjetno jih nebi raziskovali, če nebi blo nobene koristi od tega. Ampak glede na to da nisem v najboljših odnosih z matematiko ne najdem pametnega načina za uporabo praštevil :)

Btw...Thomas. O tistih nerešljivih problemih itd. Jaz sem v gimnaziji dostkrat na kontrolni dobil take nerešljive zadeve, pa mi profesorca ni hotla verjet :D

Dr_M ::

Vsekakor manjša, kot uporaba CPU časa za tvoje veleumne doneske tukaj. Trust me!

zakaj pa bi verjel nekomu, ki cele dneve blodi tuki in klati tpraparije??
The reason why most of society hates conservatives and
loves liberals is because conservatives hurt you with
the truth and liberals comfort you with lies.

Thomas ::

Ja, res je. Verjemi in ne trosi takih neumnosti več.

Zdaj pojdi z bogom in ne vmešavaj se več v stvari, ki jih ne razumeš.
Man muss immer generalisieren - Carl Jacobi

Dr_M ::

bom upostevu tvoj nasvet in se ne bom umesavu...
kar se pa razumevanja tega tice, nikol se nism trudil razumet neumnih, nesmiselnih in neuporabnih stvari :)
The reason why most of society hates conservatives and
loves liberals is because conservatives hurt you with
the truth and liberals comfort you with lies.

CCfly ::

Isknaje praštevil niso ravno traparije. Nekdo je že omenil uporabo v kriptografiji.

Toda vedno ko kakšen problem dodobra naskočijo, se izkaže za rešljivega. To sem mislil, da je zanimivo.

Rešljiv v kakšnem času ?

mchaber ::

>11.

11^2 = 121

121 - 2 = 119

119 je praštevilo.

:\
.

Roadkill ::

"kar se pa razumevanja tega tice, nikol se nism trudil razumet neumnih, nesmiselnih in neuporabnih stvari "
Paradox^3. :)

Microsoft ::

Mogoce banalno vprasanje, ampak vseeno.

Kaj se da hitreje iskati: prastevila ali ne-prastevila?

Pac, mogoce je hitreje priti do prastevila tako, da isces ne-prastevila in potem reces; ...vsa ostala stevila so prastevila...


by Miha
s8eqaWrumatu*h-+r5wre3$ev_pheNeyut#VUbraS@e2$u5ESwE67&uhukuCh3pr

Thomas ::

mchaber!!!

2^11-1=2048-1=2047

2047 pa sam faktoriziraj!
Man muss immer generalisieren - Carl Jacobi

Thomas ::

> Kaj se da hitreje iskati: prastevila ali ne-prastevila?

Praštevila. Če iščeš po kompletnem intervalu.

Če pa iščeš posamezne primerke, pa je lažje poiskati sestavljeno število. To je celo tako lahko, da ne pritegne nobene pozornosti.

Da bi pa na nekem visokem intervalu poiskal vsa sestavljena števila, je pa bolj zamudno, kot da poiščeš vsa praštevila.

Ampak kdo ve. Morda bi se tvojo idejo na kakšen način le dalo porabiti? V resnici je Eratosten prvi prišel do nje in je zumil svoje rešeto. Z njim je mogoče najlažje ločiti VSE kozle od ovnov (praštevila od sestavljenih) na vsakem intervalu, ki se prične na začetku naravnih števil.
Man muss immer generalisieren - Carl Jacobi

Roadkill ::

A obstaja še kak drug način dokazevanja, da je število praštevilo, kot to, da ga probaš delit z vsemi manjšimi praštevili?
Vrjetno je to stvar, ki jo vsi želijo odkrit. :)

Thomas ::

Kot je razvidno iz tega linka se da. Tudi kako se da.
Man muss immer generalisieren - Carl Jacobi

McHusch ::

Še ena fascinantna metoda za preverjanje sestavljenosti števila (sicer v praksi neuporabna) - Wilsonov izrek.

ovdje kokoš ::

enkrat sedim pr nemi pr sosolcu, pa gledam kaj pocne, pa mi pove das e ze neki casa zeza z magicnemu kwadratom, cez neki casa je napisu enacbo za prastevila, ki pa ne dekuje vedno x na 3. minus (x-1) na 3. je najverjetneje prastevilo

(ce se slucajno kermu kej userje pa najde ksno prou velko stevilo, zahtevava provizijo :))

kritizirajte

mchaber ::

>mchaber!!!

>2^11-1=2048-1=2047

>2047 pa sam faktoriziraj!

Aja ti si odgovarjal na formulo za Mersennova praštevila.

Jaz sem imel v mislih "mojo" formulo:
(neko praštevilo)^2 -2 = neko drugo praštevilo
.

JerKoJ ::

no ja pa lih na 11 tvoja teorija pade

11^2-2=119
to pa je 7*17 in torej ni prastevilo

najprej se nauc prastevila racunat :D

Thomas ::

Kaj če bi se ti naučil brat?

Sprašuje me za izjemo. Torej da 2^n-1 NI praštevilo.

Zato sem pa rekel, da naj ga faktorizira sam.

Predlagam, da tisti ki nimate pojma NE sodelujete v resnih razpravah.

Vzemi dobronamerno-
Man muss immer generalisieren - Carl Jacobi

Thomas ::

Ma jest sem kreten v odnosu do Jerka!

Se mu raje opravičim, kot da bi popravil post. Naj se vidi, da sem zmotljiv.

Itak pa post velja za Mchasberja.
Man muss immer generalisieren - Carl Jacobi

Thomas ::

boneman je poštudiral, da med dvema zaporednima kuboma je razlika rada praštevilo.

Sem dal to njegovo formulo dal Critticallu, da jo še izboljša.


 
 $declareint i zero upto c cc bad x xi tmp one df
 $DIMENSIONS t[4501] niz[4501]
 $INVAR t(0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,1,
 0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,
 0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,
 0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
 1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,
 0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,
 1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,
 0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
 1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,
 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,
 0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
 0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,
 0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,
 1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,
 0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,
 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,
 0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
 1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,
 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
 1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,
 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
 0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,
 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
 0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
 1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,
 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,
 0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,
 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,1,
 0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,
 0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
 0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,
 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,
 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
 1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,
 0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,
 0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
 1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,
 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
 0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,
 1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
 0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
 0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,
 0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,
 1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,
 0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
 0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
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 0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,
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 0,0,0,0,0,0,1,0,0,0,0,0,0,0)
 $resvar t[]
 $resvar bad zero upto i x
 $penval bad
 $showvar bad niz[]
 $enhancing off
 $weights commands=0
 $minimizelines 26
 bad=40;
 upto=36;
 for (i=zero;i<upto;i++) {
 x=i; 
 $bes 
 c=x*x; 
 c=c*x;
 xi=x;xi++;
 cc=xi*xi;
 cc=cc*xi;
 $ees 
 df=cc-c;
 one=1;zero=0;
 tmp=t[df];
 t[df]=zero;
 niz[i]=df;
 if (tmp==one) {bad--;}
 }


Prišel je ven s precej boljšim zadevanjem praštevil.
Man muss immer generalisieren - Carl Jacobi

ThePlayer ::

>Naj se vidi, da sem zmotljiv.

:) smo lahko kar pomirjeni ostali, da si tudi ti le človek:D ok mal heca mora bit...

Thomas ::

Ja, jest sem človk ja. Zato se Jerku še enkrat opravičujem.

Ampak Critticall pa ni. Našel je tole formulo namesto bonemanove. Med $bes in $ees.

df=one-c;
if (tmp==critticall1) {
one=1;
df|=2;
}
critticall1=tmp^one;
if (df<cc) {
c+=-2;
}
cc=df-c;

Najde tole zaporedje:

2,3,5,7,11,17,23,29,35,43,47,53,59,65,71,79,83,89,95,103,107,113,119,127,131,137,143,151,155,163,167,173,179,185,191,199

Večinsko so praštevila.
Man muss immer generalisieren - Carl Jacobi

CCfly ::

@Thomas: skorajda binarni spam.

undefined ::

> @Thomas: skorajda binarni spam.

Spam is unsolicited e-mail on the Internet. From the sender's point-of-view, it's a form of bulk mail, often to a list obtained from a spambot or to a list obtained by companies that specialize in creating e-mail distribution lists. To the receiver, it usually seems like junk e-mail. It's roughly equivalent to unsolicited telephone marketing calls except that the user pays for part of the message since everyone shares the cost of maintaining the Internet. Spammers typically send a piece of e-mail to a distribution list in the millions, expecting that only a tiny number of readers will respond to their offer. Spam has become a major problem for all Internet users.

Preden ponovno kdo butne kakšno podobno neumnost, naj se najprej pouči kaj posamezen izraz, kot je spam, sploh pomeni.

Thomas ::

O generatorjih praštevil bom odprl novo temo v Z&T.

Critticall mi jih je čez noč našel kar precej. Later.
Man muss immer generalisieren - Carl Jacobi

CCfly ::

humor
n 1: a message whose ingenuity or verbal skill or incongruity has
the power to evoke laughter [syn: wit, humour, witticism,
wittiness]

undefined ::

Humor zaenkrat še ne obsega področja neznanja, čeprav se večkrat nad neznanjem da poustvariti dosti humorja. :)
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