Question 45
(α) Write the expression $A = k^4 + 4$, where $k$ is a positive integer, as a product of two factors each of them being a sum of two squares of integers.
(β) Simplify the expression
$$
K = \frac{\left(2^4 + \frac{1}{4}\right)\left(4^4 + \frac{1}{4}\right)\left(6^4 + \frac{1}{4}\right) \cdots \left((2n)^4 + \frac{1}{4}\right)}{\left(1^4 + \frac{1}{4}\right)\left(3^4 + \frac{1}{4}\right)\left(5^4 + \frac{1}{4}\right) \cdots \left((2n-1)^4 + \frac{1}{4}\right)}
$$
and write it as a sum of the squares of two successive integers.
(β) Simplify the expression
$$
K = \frac{\left(2^4 + \frac{1}{4}\right)\left(4^4 + \frac{1}{4}\right)\left(6^4 + \frac{1}{4}\right) \cdots \left((2n)^4 + \frac{1}{4}\right)}{\left(1^4 + \frac{1}{4}\right)\left(3^4 + \frac{1}{4}\right)\left(5^4 + \frac{1}{4}\right) \cdots \left((2n-1)^4 + \frac{1}{4}\right)}
$$
and write it as a sum of the squares of two successive integers.