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The Pell numbers are defined by the recurrence relation: = {=; =; + In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number, plus the Pell number before that.
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. This equation was first studied extensively in India starting with Brahmagupta , [ 1 ] who found an integer solution to 92 x 2 + 1 = y 2 {\displaystyle 92x^{2}+1=y^{2}} in his Brāhmasphuṭasiddhānta circa 628. [ 2 ]
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.. An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients.
The solutions of the Pell's equation X 2 − DY 2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4). For D = 5, ω = 1+ √ 5 / 2 is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet. Its units have the form ±ω n, where n is an arbitrary integer
This is a list of limits for common functions such as elementary functions. In this article, the terms a , b and c are constants with respect to x . Limits for general functions
A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken Yamamura and is available on pages 915–919 of his article on the subject. [15] Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1. [16]