Question

Number Theory Problem!

Answer 1

If we know that \(\Large \sqrt{7} - \frac{ m }{ n } > 0\) then prove \(\LARGE \sqrt{7} - \frac{ m }{ n } > \frac{ 1 }{ mn }\)

Answer 2

the idea i hava given so far is that we can use floor ...

Answer 3

and i forgot to add that \[m,n \in N\]

Answer 4

guys if you can help me with this problem please give me only the ideas...i would like to solve it my self :)

Answer 5

i got a solution here
http://mathhelpboards.com/challenge-questions-puzzles-28/prove-7-0-5-m-n-gt-1-mn-12170.html

im trying to see if it can be done in a short way

Answer 6

thanks....i would try out...

Answer 7

wait is sqrt 7-m/n an integer ?

Answer 8

\[\Large \sqrt{7} - \frac{ m }{ n } \gt 0 \implies 7n^2 - m^2 \gt 0\]

Answer 9

was struggling about if sqrt 7- m/n
plz say its integer xD

Answer 10

guys i've already got the ideas (and perhaps the solution :D) Thanks!

Answer 11

sqrt 7- m/n
how can it be an integer when sqrt(7) is irrational

Answer 12

m/n is irrational

Answer 13

I don't know you're the one that's rational do something about it!

Answer 14

Bad joke I know

Answer 15

m/n is rational since m,n are naturals
you should go back to sleep >.<

Answer 16

wait no

Answer 17

i mean

Answer 18

lol @iambatman i got your joke now :P

Answer 19

what if
m/n= t+(sqrt 7 -[sqrt 7])

Answer 20

yeah hehe what ever im making no sense xD

Answer 21

Okay the problem simplifies to solving the diophantine equation : \(\large 7n^2-m^2=0\)

Answer 22

**\(\large 7n^2-m^2\gt 0\)

Answer 23

just need to show it has solutions only when it satisfies the given constraint

Answer 24

anyways i stop at this point as the problem has a neat solution already :)