show that $$\frac{(m-1)(n-1)}{4} = \sum\limits_{i=1}^{(m-1)/2}\left\lfloor \frac{in}{m}\right\rfloor + \sum\limits_{j=1}^{(n-1)/2}\left\lfloor \frac{jm}{n}\right\rfloor$$

$m\ne n$ are odd primes

Question

show that $$\frac{(m-1)(n-1)}{4} = \sum\limits_{i=1}^{(m-1)/2}\left\lfloor \frac{in}{m}\right\rfloor + \sum\limits_{j=1}^{(n-1)/2}\left\lfloor \frac{jm}{n}\right\rfloor$$

$m\ne n$ are odd primes

anonymous · Answer

hehe can we use p and q instead xD

ganeshie8 · Answer

sure :)

anonymous · Answer

ok was jk :P

anonymous · Answer

see last line typo its 4 not 2 lol

anonymous · Answer

you wanna proof this lemma too ?

ganeshie8 · Answer

im looking for a proof without using any number theory fancy stuff

anonymous · Answer

ohh got you

anonymous · Answer

i like quadratic reciprocity proof itself , wanna me to assume we don't know it and we don't know Guass as well ?

ganeshie8 · Answer

yes assume i am a 10th grader

anonymous · Answer

ok lets see if i can convince you hmm i only know one proof xD

ganeshie8 · Answer

OK i am waiting :)

anonymous · Answer

assume we have this xy coordinate with this triangle
(0,0) (m/2,0) (0,n/2) (m/2,n/2)
|dw:1422998853621:dw|

ganeshie8 · Answer

you mean rectangle ?

anonymous · Answer

hehe yeah

anonymous · Answer

that was typo

ganeshie8 · Answer

ok assumed

anonymous · Answer

so since n and m are odd primes 
we can make grid of integer points like this :-
|dw:1422999042103:dw|

anonymous · Answer

on x axis integers would be numbers from 1 to m-1/2 and on y from 1 to n-1/2
ok ?

ganeshie8 · Answer

Okay im wid you so far

anonymous · Answer

so the number of integers coordinates points is
(m-1)/2 * (n-1)/2

ganeshie8 · Answer

that gives the left hand side of the equation, nice

anonymous · Answer

now lets consider the line from (0,0) to (m/2,n/2)
consider this graph not on scale 
|dw:1422999369001:dw|