Točno
28. travnja 2012. 17:40 (12 godine, 2 mjeseci)
Upozorenje: Ovaj zadatak još niste riješili!
Kliknite ovdje kako biste prikazali rješenje.
Kliknite ovdje kako biste prikazali rješenje.
Svaki
za koji ovo vrijedi mora imati u svom rastavu na proste faktore sve proste brojeve
, inače dobivamo
a
pa
sigurno nije oblike
.
Pretpostavimo da postoji
za koji zadatak vrijedi. Tada je
.
Kako je
znamo da
i
dijele
.
Svakim takvim "popravljanjem" (tj dodavanjem novog prostog broja kojeg moramo imati) prelazimo iz
u
, pa korijen prelazi iz
u ![\sqrt{pn}](/media/m/9/5/6/956e93e8678ff87f996a74a88de8f123.png)
a između
i
sigurno postoji prost broj pa moramo opet "popravljati".
Dakle ne postoji
veći od 50.
Za
i
provjerom dobivamo da vrijedi. Za neparni
sigurno ne vrijedi, a za
mora biti djeljiv s
. Provjerom do 9 dobivamo da vrijedi za
,
i
.
Provjerom brojeva
,
i
dobivamo da vrijedi za
i
, ali ne i za
(jer
nije kongruentno
).
Ako je
mora biti djeljiv s
, a jedini broj manj od
i veći od
je
, za koji ne vrijedi jer
.
Dakle jedina rješenja su
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![p<\sqrt{n}](/media/m/4/0/c/40c5a104140533e0d83d8ff74d801659.png)
![p \equiv p \pmod n](/media/m/a/8/8/a883f5271ab1d5d15f30f7024b5cd698.png)
![p^2 < n](/media/m/7/c/2/7c24bb2fc0090cba52578c6efbd28844.png)
![p^2](/media/m/f/0/f/f0fbfedc204c557f55c06eceeb024b6c.png)
![kn+1](/media/m/4/4/d/44df5557f81e5a7dc778b8aa0bb92a73.png)
Pretpostavimo da postoji
![n \geqslant 50](/media/m/d/5/b/d5b8676f822c3bdc34535c437bfcb636.png)
![n \geqslant 210](/media/m/a/9/8/a98f4e808b2394b28a88ed635a0adc0a.png)
Kako je
![\sqrt{210} > 14](/media/m/8/0/b/80b0e91f5e715f993b6db5f8b1c96b0b.png)
![11](/media/m/0/d/2/0d2d0ab9a023da1d30a2ddc91cbc38db.png)
![13](/media/m/a/f/0/af007727d79ff468853d32d8393cc8de.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
Svakim takvim "popravljanjem" (tj dodavanjem novog prostog broja kojeg moramo imati) prelazimo iz
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![pn](/media/m/0/9/7/0972c36e8a5729133e71cd5a783a3523.png)
![\sqrt{n}](/media/m/2/c/3/2c3e61b80b7a18a1dc5cc33b8f55fa2e.png)
![\sqrt{pn}](/media/m/9/5/6/956e93e8678ff87f996a74a88de8f123.png)
![p>4 \Rightarrow 2\sqrt{n}< \sqrt{pn}](/media/m/9/1/d/91decad64357a2df3df20e1f22b379ee.png)
![\sqrt{n}](/media/m/2/c/3/2c3e61b80b7a18a1dc5cc33b8f55fa2e.png)
![2\sqrt{n}](/media/m/0/6/9/069365d94abf5134ef41cca3e07e09b1.png)
Dakle ne postoji
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
Za
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![n>4](/media/m/3/1/2/312f54b65a7dbf1d3d1e6e17a9de2cc4.png)
![n>9](/media/m/c/9/5/c954a13175d2da886b1a945f3b959ff0.png)
![6](/media/m/e/e/e/eeec330d59a70f8ed1d6882474cb02a3.png)
![4](/media/m/d/a/6/da6087359ae47e86dcb2e49565050046.png)
![6](/media/m/e/e/e/eeec330d59a70f8ed1d6882474cb02a3.png)
![8](/media/m/3/d/2/3d2c45264dbff498f9bcb16af5f83881.png)
Provjerom brojeva
![12](/media/m/e/f/6/ef6c8e9eecc5ee3d49031ee4f0e20f98.png)
![18](/media/m/b/2/d/b2d2f451c0676c0584107ade6c87edfa.png)
![24](/media/m/0/0/d/00d47baa5fbe61055e1bb5135b2c8fd4.png)
![12](/media/m/e/f/6/ef6c8e9eecc5ee3d49031ee4f0e20f98.png)
![24](/media/m/0/0/d/00d47baa5fbe61055e1bb5135b2c8fd4.png)
![18](/media/m/b/2/d/b2d2f451c0676c0584107ade6c87edfa.png)
![25](/media/m/6/b/3/6b3be22b73368e0c46171dca7fbbc637.png)
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
Ako je
![n>25](/media/m/e/b/7/eb708b26d8e66861d6b7136896afd5ae.png)
![30](/media/m/f/7/2/f7237ad0b4286650c3902269b3d01bdb.png)
![50](/media/m/2/f/c/2fceaac52c4bae9bf67827f9dc8c6130.png)
![25](/media/m/6/b/3/6b3be22b73368e0c46171dca7fbbc637.png)
![30](/media/m/f/7/2/f7237ad0b4286650c3902269b3d01bdb.png)
![49 \equiv 19 \pmod {30}](/media/m/b/d/d/bdd4a69a11d44c0d8a4b57c9ff8e6d17.png)
Dakle jedina rješenja su
![n \in \{2,3,4,6,8,12,24 \}](/media/m/2/5/8/2585a07558ccb2f7e8eaf8fbc87e73d7.png)