Fundamental Number Theory with Applications

Arithmetic progressions on Pell equations

“…Do not confuse this notation with the complex conjugation! A theorem due to Nagell states (see [11] or [10]). Theorem 6 (Nagell).…”

The Diophantine Equation8x2-y2+8x(1+t)+<

“…which is a Pell equation (see [11][12][13][14]). Before considering its integer solutions, we need some notations.…”

Continued fractions expansion of √D and Pell equation x2 - Dy2 = 1

“…The first non-trivial solution (x 1 , y 1 ) of this equation, from which all others are easily computed, can be found using, e.g., the cyclic method [1], known in India in the 12th century, or using the slightly less efficient but more regular English method [1] (17th century). There are other methods to compute this so-called fundamental solution, some of which are based on a continued fraction expansion of the square root of D (For further details on Pell equation see [1,2,3,4,5,6,7,8,9] Recall that a simple continued fraction of order n is an expression of the form (3) a 0 + 1…”