Question

Prove or disprove: There exist odd integers a and b such that 4 | (3m^2 + 7n^2)

Answer 1

what a and b?

Answer 2

wonder if they meant odd integers m and n

Answer 3

the second part should be 4 | (3a^2 + 7b^2)

Answer 4

k

Answer 5

\[4i=3a^2+7b^2 , i \in \Integer \\ \text{ so we are given } a=2L+1 \text{ and } b=2K+1 \\ 3(2L+1)^2+7(2K+1)^2 \\ =3(4L^2+4L+1)+7(4K^2+4K+1) \\ =4(3L^2+7K^2)+4(3L+7K)+(3+7) \\ \]
\[=4(3L^2+7K^2)+4(3L+7K)+2(5) \\ =2[2(3L^2+7K^2)+2(3L+7K)+5]\]
hmmm... I wonder if there is some integer L and K we can chose so that thing in the [ ] is a multiple of 2

Answer 6

actually try I think you can write the thing in brackets as a odd number
and odd numbers aren't divisible by 2

Answer 7

Thanks for the help!