3[ Identical Transformations $]$

Authors: Proizvolov V.V., Senderov V.A.

Find some four pairwise distinct natural numbers $a, b, c, d$, for which the numbers $a^{2}+2 c d+b^{2}$ and $c^{2}+2 a b+d^{2}$ are perfect squares.

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It is sufficient to find such numbers that $a b=c d$. Then $a^{2}+2 c d+b^{2}=a^{2}+2 a b+b^{2}=(a+b)^{2}, c^{2}+2 a b+d^{2}=(c+$ $d)^{2}$. For this, we need to find a number $n$ that can be factored into the product of two factors in different ways.

For example, $6=1 \cdot 6=2 \cdot 3$.

## Answer

For example, $a=1, b=6, c=2, d=3$.