An inquiry into the Pell equation, x² - Dy² = N
Bentsen, Doreen Joyce
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The Diophantine equation, x² - Dy² = N, where D and N are known integers, is called the Pell equation. If D is a positive integer not a perfect square, solutions of the Pell equation are closely connected with the convergents to the continued fraction expansion of D. If N = ± 1, methods of finding all integral solutions are known. It is the purpose of this study to take specific forms of √D and to determine those values of N for which the convergents pₙ/qₙ of the continued fraction expansion of √D provide solutions, x = pₙ, y = qₙ, to the Pell equation. The continued fraction expansion of the quadratic irrational √D is periodic. This investigation uses those forms of D for which the continued fraction expansion of √D has a repeating part of length r = 1, 2, or 3. If r = 1, √D has a continued fraction expansion given by √D = [aₒ,̅2̅aₒ] where aₒ denotes the greatest integer in √D and the bar over 2aₒ indicates the repeating part of the continued fraction expansion. It is proved that if r = 1, then D = aₒ² + 1 and the Pell equation, x² - Dy² = N, has solutions x = pₙ, y = qₙ for N = ± 1, where pₙ/qₙ denotes the n-th convergent to the continued fraction expansion of √D. Also, if r = 1, then x² - Dy² = ± 1 has all positive integral solutions given by the convergents Pₙ/qₙ to √D. With aₒ and Pₙ/qₙ as defined above and r = 2, √D = [aₒ,̅a₁,̅2̅aₒ], it is proved that D = aₒ² + 2aₒ/a₁ where a₁ is a positive integer which divides 2aₒ. Also, the Pell equation has solutions x = pₙ, y = qₙ for N = + 1 and N = aₒ² - D. In this case, the convergents pₙ/qₙ to √D provide all positive integral solutions of x² - Dy² = 1 and x² - Dy² = aₒ² - D. For r = 3, a special form of √D is considered, namely √D = [aₒ,̅2,̅2,̅2̅aₒ]. for this form of √D, it is proved that D = aₒ² + (4aₒ + 1)/5 where a ≡ 1 mod 5. Also, it is shown that x = pₙ, y = qₙ constitute all positive integral solutions of the Pell equation, x² - Dy² = N, for N = ± 1 or N = ±(D - aₒ²). Following the proofs of the preceding statements, a brief history of the Pell equation is given as well as some of its mathematical applications.