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.

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Comments

$n=a^2+4ab+b^2=(a+2b)^2-3b^2$ .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 $n$ .

A Former Brilliant Member - 4 years, 2 months ago

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.

A Former Brilliant Member - 4 years, 2 months 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.

A Former Brilliant Member - 4 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$