Question

fun question :)

Answer 1

Find all 4 pairs (p,q,r,s) of natural numbers such that \[p \le q \le r\] and \[p!+q!+r!=3^s\]

Answer 2

dude don't give away hints 2 minutes after posting the question

Answer 3

lol ok cx

Answer 4

this isn't a very fun question

Answer 5

this is indeed a fun question,
the hints only act as spoilers for those who genuinely want to try

Answer 6

i didnt saw any hint ?

Answer 7

good :)

Answer 8

It seems (1,1,1,1) is the only solution...
I kinda have a proof too but I'll let others try :)

Answer 9

there are 3 solutions :) and this is one of them

Answer 10

right,
(1,2,3,2) is the second one

Answer 11

yes :D

Answer 12

(1,2,4,3) is the last one

Answer 13

yes ᗡ:

Answer 14

should i post the solution? or some hints? :)

Answer 15

no wait :)

Answer 16

ok (:

Answer 17

\(p!+q!+r!=3^s\)

1) Clearly \(p=1\) (why ?)
2) so we look at \(q!+r!=3^s-1\)
3) since the right hand side is not divisible by \(3\), it must be the case that \(q=2\) for all \(r\gt 1\).
4) now we look at \(r! = 3^s-3\)
5) Clearly this equation has no solutions for \(r\ge 9\) and \(s\ge 2\) because the right hand side wont be divisible by \(9\).

Answer 18

yes correct :) our solutions are the same

Answer 19

please do post the solution for the completeness sake..

Answer 20

ok wait a min :)

Answer 21

Note that if p > 1 then the left-hand side is even, and therefore p = 1. If q > 2
then 3 divides q!+r! and hence 3 does not divide the left-hand side. Therefore q = 1 or q = 2.
If q = 1 then r! + 2 = 3^s, so r < 2 and hence s = 1. If q = 2 then r! = 3^s -3. Note that
s = 1 does not give any solution. If s> 1 then 9 does not divide r!, so r < 6. By checking
the values for r = 2, 3, 4, 5 we see that r = 3 and r = 4 are the only two solutions. Thus
(p, q, r, s) = (1, 1, 1, 1), (1, 2, 3, 2) or (1, 2, 4, 3).