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Problem

The Indian mathematician and astronomer Brahmagupta (597 - 668 AD) used the identity for solving quadratic indeterminate equations[1] of the form , where are integers. Verify that this identity is valid.

[1] In Europe, these equations are called Pell's equations, named after English mathematician John Pell (1611 – 1685 AD). John Pell was not the first European to study these equations, and was in fact first proposed by Pierre de Fermat (1607 - 1665 AD) in 1657. It was William Brouncker (1620 – 1684 AD) who was the first European to solve Pell's equation. The name of Pell's equation arose from Leonhard Euler (1707 – 1783 AD) mistakenly attributing Brouncker's solution of the equation to John Pell.

Solution

Expand the left hand side.

Expand the right hand side.

The expanded results on the left hand side matches the right hand side indeed.