*always*part of the set of series of a fractional approach, the quartus and the tertius.

We've defined a fraction's value 'v' = (numerator)

^{2}-n*(denominator)

^{2}. With regard to this, qt-blocks come in three kinds:

- A multiple of the tertius in the numerator, a multiple of the quartus in the denominator.

In this case v = t^{2}-nq^{2}> 1 and the value of the fraction is above the root. - A multiple of the quartus in the numerator, a multiple of the tertius in the denominator.

In this case v = q^{2}-nt^{2}< 0 and the value of the fraction is below the root. - Both of the above in successive fractions.

In this case the t/q fraction*precedes*the q/t fraction and v_{1}*v_{2}= -n.

The value of 'v' is the same for all qt-blocks in a particular series.

For carréphobic numbers, all fractions involved are indeed

Carréphylic numbers on the other hand always have a single tertius or quartus in the denominator.

For carréphobic numbers, all fractions involved are indeed

*multiples*of the tertius and quartus.Carréphylic numbers on the other hand always have a single tertius or quartus in the denominator.

__From the first qt-block ⇒ the first non trivial sp-block__

The following formulas hold for both carréphylic and carréphobic numbers. For the former they are inconsequential because the diophantine equation s

^{2}-np

^{2}= 1 can be solved. For the latter they allow the calculation of 's' and 'p' in cases where a direct search fails.

As far as this investigation is concerned, I used them to determine the sp-blocks of √61 (1766319049 / 226153980) and √109 (158070671986249 / 15140424455100), both of which were out of range of Ed's fractional approximations program.

**Conjecture**

- A fraction is part of a (semi-)symmetric series if and only if v | 2*(numerator).

If v = 1 we have the first fraction of a sp-block, otherwise it is a qt-block.

^{2}-nq

^{2}= v, the distance of nq

^{2}to the nearest square (t

^{2}) is v, which divides 2t. Hence nq

^{2}is

*carréphylic*, so for the sp-series linked to the approach of its root: s = (2t

^{2}-v)/v and p = 2t/v.

Since √nq

^{2}= Q√n, s/p is an approximation of Q√n, so for √n itself:

**s = (2t**and^{2}-v)/v**p = 2qt/v**

**s = -(2q**and^{2}-v)/v**p = -2tq/v**