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3^(2p) + 3^(3q) + 3^(5r) = 3^(7s) Find the minimum value of p+q+r+s where p,q,r,s are all positive integer.

Mathematics
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9^p+27^q+3^5r=3^7s
ok

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Other answers:

What was that?
we just square and cube the value
The question is to find minimum value of p+q+r+s
then common 3 as a base
and then add powers
then you solve it
I have never seen that rule.
ok so go through wolframalpha
http://www.wolframalpha.com/input/?i=3%5E%282p%29+%2B+3%5E%283q%29+%2B+3%5E%285r%29+%3D+3%5E%287s%29+Find+the+minimum+value+of+p%2Bq%2Br%2Bs
I dont think that will also help here.
go through
There is nothing.
I am seeing it.
So, what is the answer?
Give a min, but at tell u the answer can't be find the above way.
What I figured out till now is s must be an odd number....
@experimentX PLZ see this
this doesn't look nice ... saying that find the minimum value of p+q+r+s <--- this should be close to first instances.
Cant we find that......
exponential diophantine equations are awful!! you should probably tag mukushla
This question is not of mine........ Someone posted this today and said a 8th grade challenging question.
But he is not online.
you don't do diophantine equations on 8th grade.
Even I dont know that.
This surely cant be a 8th grade question.
@mathslover i think he do this
very dhinchach question
Now, what I am trying to PROVE is 3^(2p) + 3^(3q) + 3^(5r) = 3^(7s) is never true for any value of p,q,r,s
anyway this was the output I got from mathematica \[ \left\{\left\{p\to \frac{8}{5}+2 i,q\to \frac{223}{10}+\frac{53 i}{5},r\to -\frac{71}{10}-\frac{49 i}{10},\\ s\to \frac{180 i \pi +\text{Log}\left[3^{-\frac{71}{2}-\frac{49 i}{2}}+3^{\frac{16}{5}+4 i}+3^{\frac{669}{10}+\frac{159 i}{5}}\right]}{7 \text{Log}[3]}\right\}\right\} \] I trolled enough ... not time to go strolling!!
I dont know it even has a solution.
*now .. seeya later!!
ok...... BYE thanx for trying.
call it hardly a try .. hehehe
how about this : i dont know if this'll work : ( 3^(2p) + 3^(3q) + 3^(5r) )/3 >= 3^( (2p + 3q + 5r)/3 ) or 3^(7s-1) >= 3^( (2p + 3q + 5r)/3 ) ??
@ujjwal p,q,r,s are only positive integers..
@shubhamsrg ( 3^(2p) + 3^(3q) + 3^(5r) )/3 >= 3^( (2p + 3q + 5r)/3 ) HOW???
from AM>=GM
Oh >= got it.
Yeah nice one.
and continuing that, we have 21s >= 2p + 3q + 5r + 3 but still no idea if that'll help..
And I got: |dw:1349955495301:dw|
I guess NOW, we can prove it has no SOLUTION
hmm ??
*
THAT GIVES: |dw:1349955687180:dw|
Now, |dw:1349955794330:dw|
Now, can I say No solution...... Or I lack something.
well you have made an assumption that 5r<3q .. only that makes LHS an integer..otherwise you cant say..
am sorry,,you assumed 5r>3q ..
but conclusion to my saying is that you made an assumtion..
oh...... no assumtion.
L.H.S or R.H.S can be both negative
Just looking for integer....... L.H.S will always be integer R.H.S will always be in decimal.
LHS is not necessary an integer.. we only know that p,q,r,s are integers..
and agreed that RHS will always be non-integer..
Oh ya....... got it @shubhamsrg
So,from there we conclude that 5r<3q
guess not,,how can you say that ?
Because if 5r>=3q then R.H.S will always be integer.
I mean L.H.S
but there is also 5r<=3q possibility ,, the ques doesnt say that 5r - 3q > 0 /// even if that's true..the proof you mention is invalid..
oh wait........ I guess R.H.S can be integer too.
also, we may always conclude that 7s > 2p.3q,5r ..but nothing can be said for inequalities in between 2p,3q and 5r..
Ok further simplification of it gives: |dw:1349956922192:dw|
divide both sides by \(3^{2p}\)\[1+3^{3q-2p}+3^{5r-2p}=3^{7s-2p}\]right hand side is a power of 3 so does the left hand side so we must have\[2p=3q=5r\]\[7s-2p=1\]
GREAT.
Now, we need to find its minimum value.
what is that fraction?
p+q+r+s = (207p+15)/105
right?
ahh yeah :)
Now, we need to find its minimum integer value.
Oh wait...... But that can make other numbers decimal.
u better start with \[7s-2p=1\]ans\[p=\frac{7s-1}{2}\]its immidiate that s is odd let s=2m+1\[p=7m+3\]now find the smallest value of m for which p is a multiple of 3 and 5
if u wanna check the answer i found is 106 for \[m=6\]\[p=45\]\[q=30\]\[r=18\]\[s=13\]
Thanx @mukushla u r really amazing.

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