# On some class of t -discriminants and t -Pell equations.

A special class of quadratic irrationalities is described in article. This class consists of irrationalities, which has next decomposition in periodic continues fraction

$$α = \frac{\sqrt{D} − b}{a} = [−q_0, \overline{q_1, q_2, ...q_2, q_1, tq_0}], t ≥ 2$$

Part of period is palindrome. This numbers we will call the negative $t$-discriminants.

In ﬁrst time positive $t$-discriminants considered in 2007.

Theorem of negative $t$-discriminants characterization is proved. The necessary and suﬃcient conditions of indicated decomposition are

$$2b = (t + 2)aq_0, q_0 = [α].$$

For example, a number

$$α = \frac{\sqrt{5689} − 91}{26} = [−1, \overline{2, 1, 1, 2, 5}]$$

is negative 5-discriminant.

Formulas for calculate of $a, b,$ and $D$ proved too:

$$a = 2P_{n−1}, b = (t + 2)q_0P_{n−1}, D = (tq_0P_{n−1} + 2Q_{n−1})^2 + 4(−1)^n,$$

where

$$\frac{P_{n−1}}{Q_{n−1}} = [q_1, q_2, ..., q_2, q_1].$$

One of negative $t$-discriminants application are solutions of Diophantine equations

$$(ax + by)^2 − Dy^2 = ±a^2.$$

This equations we will call $t$-Pell equation and minus-$t$-Pell equation respectively.

Descriptions of solutions structure are obtain in article. The structure of solutions is closely connected with some cyclic group in every case. Cyclicity of this groups proofs with help from parametrization respective Pell equations. Be introduces the binary operation on the sets of $t$-Pell equations solutions too.

Minus-$t$-Pell equation is solvable iﬀ period length of continues fraction decomposition is odd.

Examples are considered.