OEIS A000047: Integers of Form x^2 − 2y^2

We explore the integers that can be written as x^2 − 2y^2. A practical test: an integer n is representable iff in its prime factorization no prime congruent to 3 or 5 mod 8 appears with an odd exponent. Through examples like 6 and 18 we see the rule in action, and we connect the modular-prime condition to the underlying algebra of the form, its relation to Pell-type equations in Z[√2], and why this makes quick membership tests possible. We also situate A000047 in the wider landscape of quadratic forms in number theory.

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