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.

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TopNewestc) \(x^2+4xy+y^2=(x+2y)^2-3y^2\) 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.

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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 \(n\).

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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.

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