# Pillow-Problems: Problem #1

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Subjects Classified | Problem #2
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

## Problem:

1.

Find a general formula for two squares whose sum = 2.

[24/3/84

## Solution:

1.

Let u, v be the Nos.

Then u² + v² = 2.

Evidently ‘(1 + k), (1 − k)’ is a form for the squares.

Also, if we write ‘2m²’ (which will not interfere with the problem, as we can divide by m², and get
 (m²)/ (m²)
+
 (m²)/ (m²)
= 2), the above form becomes ‘(m² + k), (m² − k)’.

Now, as these are squares, their resemblance to

‘(a² + b² + 2ab), (2² + b² − 2ab)’
at once suggests itself; so that the problem depends on the known one of finding a, b, such that (a² + b²) is a square; and we can then take 2ab as k.

A general form for this is

a = x² − y²,
b = 2xy;
a² + b² = (x² + y²)²;
∴ the formula ‘u² + v² = 2m²’ becomes
(x² − y² + 2xy)² + (x² − y² − 2xy)² = 2(x² + y²)²;
i.e. (
 (x² − y² + 2xy)/ (x² + y²)
)² +
 (x² − y² − 2xy)/ (x² + y²)
)² = 2.

Q.E.F.

Subjects Classified | Problem #2
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).