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# Algebra

Let $$A$$ be the set of all integers $$n$$ of the form $$n= a^2 + 4ab+ b^2$$ where $$a$$ and $$b$$ are integers.
a) Show that if $$x$$ and $$y$$ are in $$A$$, then $$xy$$ is in $$A$$.
b) Show that 11 is not in $$A$$.
c) Show that the equation $$x^2 + 4xy + y^2 = 1$$ has infinitely many integer solutions.

Note by Bulbuul Dev
6 months, 2 weeks ago

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c) $$x^2+4xy+y^2=(x+2y)^2-3y^2$$ which is of the form $$a^2-3b^2$$. Since $$3$$ is not a perfect square by Pell's equation there are infinitely many solutions.

- 6 months, 2 weeks ago

a)

$$n=a^2+4ab+b^2=(a+2b)^2-3b^2$$.So $$x$$ is of the form $$m^2-3n^2$$. Similarly $$y$$ is of the form $$p^2-3q^2$$.

By Brahmgupta's identity, $$xy=(m^2-3n^2)(p^2-3q^2)=(mp+3pq)^2-3(mq+np)^2$$ which is also of the form of $$n$$.

- 6 months, 2 weeks ago

b) All perfect squares are of the form $$0,1 \pmod 4$$. So $$x^2+4xy+y^2 \equiv 0,1,2 \pmod 4$$ but $$11 \equiv 3\pmod 4$$.

So no solution.

- 6 months, 2 weeks ago