Question

Prove that there exists no natural numbers , n and n such that m^2 =n^2 +2002

Answer 1

I would compare this to the pythagorean theorem

Answer 2

so 2002 would need to be a perfect square and it is not

Answer 3

for example we can have 5^2 = 3^2 + 4^@

Answer 4

\[3^2=2^2+5\]

Answer 5

then in order for m and n to be natural numbers 2002 would have to be a perfect square
and it clearly isn't

Answer 6

and 5 is not a perfect square

Answer 7

5^2 is a perfect square

Answer 8

My teacher gave a hint to make cases like , but after that i don't know what to do ,

Answer 9

@Miracrown what i said was
\[3^2=2^2+5\] and \(5\) is not a prefect square

Answer 10

think this has to do with remainders of perfect squares

Answer 11

they want us to compare what happens when m and n are odd and even

Answer 12

so the idea is if m is odd
then m^2 has to be odd also

Answer 13

(m+n)(m-n)=2002

It can be represented like this

Answer 14

ok do you need to go through the algebra for that step?

Answer 15

m^2 - n^2 = 2002
(m+n)(m-n) = factors of 2002

m+n = 2002
m-n = 1

m+n = 1001
m-n = 2

m+n = 286
m-n = 7

m+n = 182
m-n = 11

m+n = 154
m-n = 13

m+n = 143
m-n = 14

m+n = 91
m-n = 22

m+n = 77
m-n = 26

all case above solve by elimination or subtituion, obvious m is not a natural number

Answer 16

looks good to me!

Answer 17

yes neat thanks

Answer 18

using odd and even you can write a more elegant proof in just a few lines

Answer 19

How

Answer 20

(m+n) *(m-n)=2002
since 2002 is even number
if either m or n is odd then m+n and m-n are both odds and hence their product must be odd but 2002 is even
if both m and n are even then 2|m+n and 2|m-n as well
and hence
(m+n)*(m-n) must be divisible by 4 but 2002 is not divisible by 4

Answer 21

for even

assume n,m are even then m=2k and n=2l

then \[(2k)^2=(2l)^2+2002\]
\[4k^2=4l^2+2002\]
\[2k^2-2l^2=1001\]

the lhs is even the rhs is odd a contadiction

Answer 22

similarly if m and n are both odd
then 2|m+n and 2|m-n as well and hence (m+n)*(m-n) must be divisible by 4 but 2002 is not divisible by 4
so the result follows...