Solutions of the Pell Equations x2 − (a2b2 + 2b)y2 = N when N ∈ {±1,±4}
... solution to the equation x − dy = 4 is 2a b + 2 + 2a d. Assume that b is odd. If a is odd, then d ≡ 3(mod4) and if a is even, then d ≡ 2(mod4). Thus, by Theorem 3.4 and Corollary 2.8, it follows that the fundamental solution to the equation x2 − dy 2 = −4 is (2(a2 b + 1), 2a). Then the proof follows ...
... solution to the equation x − dy = 4 is 2a b + 2 + 2a d. Assume that b is odd. If a is odd, then d ≡ 3(mod4) and if a is even, then d ≡ 2(mod4). Thus, by Theorem 3.4 and Corollary 2.8, it follows that the fundamental solution to the equation x2 − dy 2 = −4 is (2(a2 b + 1), 2a). Then the proof follows ...
Musings on Factoring of Polynomials Bob Rosenbaum
... To teach factoring as a method for solving quadratic equations may be considered an exercise in futility, because the factorable quadratic expressions form so insignificant a subset of all quadratic expressions. But, ―completing the square‖ to obtain the ―quadratic formula‖—now there’s something wor ...
... To teach factoring as a method for solving quadratic equations may be considered an exercise in futility, because the factorable quadratic expressions form so insignificant a subset of all quadratic expressions. But, ―completing the square‖ to obtain the ―quadratic formula‖—now there’s something wor ...
41(3)
... actually characterizes a-words (Theorem 4.4). The result used to prove this characterization is itself a characterization of a-words (Lemma 2.1) with other interesting consequences besides Theorem 4.4. In section 3, we obtain characterization of elements of the set P E R and standard Sturmian words ...
... actually characterizes a-words (Theorem 4.4). The result used to prove this characterization is itself a characterization of a-words (Lemma 2.1) with other interesting consequences besides Theorem 4.4. In section 3, we obtain characterization of elements of the set P E R and standard Sturmian words ...
