<s>[INST] No Integer Solutions and Congruences Using congruences, I seek to prove two things:
1) $x^2 - 4y^2 = 3$ has no solutions in integers $x,y,z$.
I think this can be done using modulo 4? How so?
2) $3x^3 - 7y^3 + 21 z^3 = 2$ has no solutions in integers $x,y,z$.
Not sure how to attack this one...
 [/INST] (1) If $x$ is even, is $x^2-4y^2$ odd or even? If $x$ is odd, can $x^2-3$ be divisible by $4$ if $x$ is an integer?
(2) $3x^3-7y^3+21z^3=7(3z^3-y^3)+3x^3$; can $3x^3-2$ be divisible by $7$ if $x$ is an integer?
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