MindSports

As far as the practical side of the approach of the square root of any natural number 'n' is concerned, just find the smallest non-trivial numbers 's' and 'p' satisfying the equation:

• s²-np² = 1

Apart from the general and trivial solution s=1 and p=0, an infinite number of solutions are always possible for any 'n'.
If the smallest non-trivial 's' and 'p' are found, then the approach of the root of 'n' follows the term by term quotient of two series:

• U(1,s)2s or half the secundus of the factor 2s or: 1, s, 2s²-1 4s³-3s ...
• U(0,p)2s or the p-fold primus of the factor 2s or: 0, p, 2sp, 4s²p-p ...

Using series acceleration there's no limit to the speed and precision of the approach.

Carréphylic & carréphobic numbers
Every square N² has a 'domain' of 2N numbers (including the square itself), ranging from (N²-N+1) up to and including (N²+N). These domains cover the series of natural numbers without overlapping, so any non-square number 'n' is in the domain of exactly one square. It's distance to that square |N²-n| is called 'd'.

• Definition:
  If a number's distance 'd' to the nearest square N² divides 2N, it is called 'carréphylic', otherwise 'carréphobic'

The 'rational s and p'
For carréphylic numbers the above solves all practical problems concerning root approach:

These values are called the 'rational s, p and F' - for carréphylic numbers they are integers.


For carréphobic numbers the 'rational s and p' come out as fractions. Here chaos rules as emphatic as order does in the carréphylic realm: there appears to be no pattern in the sp-blocks and factors found.
Some carréphobic numbers may be cracked using a carréphylic ancestor or descendent but apart from that the as yet only way to find integers 's' and 'p' satisfying s²-np² = 1 is a search program. Ed van Zon provided one using a standard root approach to measure against, and thus implicitly prone to inaccuracy beyond a certain horizon.

An example
Here is the rational approach of root 13, the first carréphobic number, as provided by Ed's straightforward program.
The position of a fraction indicates whether it is over or under the root-value.

Top- and middle row give the numerators and denominators of the successively better rational approaches of root 13. The bottom row gives the index.

The program's output began at index4: 3/1, 4/1, 7/2, 11/3, 18/5 ... up to the 10 digits realm. The order that it obeys - the very subject of this investigation - is superimposed on it, and the series involved have been extended sufficiently to highlight the fabfour. In this case it shows that the approach is made up of 15 different series of fractions, divided in sections.

The trivial sp-block
The 'primary section' starts on indices 0-1 with the trivial sp-block that embodies the general solution 1²-n*0² = 1 for every 'n'.
Its left part, 1/0, disqualifies as a fraction - the price for universality.
This block is at the center of the approach of every natural number! It's very existence is at the core of the universal solvability of the equation s²-np² = 1 for every natural non-square 'n'.

The qt-block
Next follow a number of a-symmetric series, interrupted by the qt-block. This block involves the other two fabfour series that are always part of the set of series of a fractional approach, the quartus and the tertius - in this case U(-18,18)1298 and U(5,5)1298, the 18-fold quartus and 5-fold tertius of the factor 1298 respectively.
QT-blocks come in three kinds:

1. A multiple of the tertius in the numerator, a multiple of the quartus in the denominator.
   In this case v = t²-nq² > 1 and the value of the fraction is above the root.
2. A multiple of the quartus in the numerator, a multiple of the tertius in the denominator.
   In this case v = q²-nt² < 0 and the value of the fraction is below the root.
3. Both of the above in successive fractions. In this case the t/q fraction precedes the q/t fraction and v₁*v₂ = -n.

The value of 'v' is the same for the qt-block in all sections of a series. In the example:
18²-13*5² = 23382²-13* 6485 ² = 30349818²-13*8417525² = -1.

The first non-trivial sp-block
In the first non-trivial sp-block is characterized by an index 'i' where the numerator equals the denominator of the next index.
In the example it is index 15 where we find that s/p = 649/180 and Np/s = 2340/649.
Cross multiplication renders 649²-13*180² = 1
The factor 1298 follows from F = 2s.
The approach of root-13 therefore is the term by term quotient of the series:

• U(1,649)1298 or half the secundus of 1298 or: 1, 649, 842401, 1093435849 ... and
• U(0,180)1298 or the 180-fold primus of 1298 or: 0, 180, 233640, 303264540 ...

Carréphylic numbers follow predictable patterns and can be divided in classes whose members show similar profiles.
No such convenient signposts here: there appears to be no way to link the 'rational s and p' of 13 (29/3 and 8/3 respectively) to 649/180, or to link any carréphobic number to its 's' and 'p' for that matter. The best I can give is the improved search procedure, that recognizes the qt-block (18/5) and on hitting it would have delivered s=649 and p=180, without the need to develop the series up to that point. See QT-blocks.

General structure
Every fractional approach, whether the number is carréphylic or carréphobic, consists of a limited set of series with the same factor, that can be divided in sections. A section consists of the corresponding terms of every series, which always include the fabfour and a number of a-symmetric series (the latter with the exception of root-2 and root-3, which are too crammed to allow them in).
The sp-blocks are unique in that they have the same core for all 'n' and can thus be predetermined for carréphylic numbers. They are related in yet another way. The fraction Np/s is, in local terms, only the slightest improvement on s/p, and since their approaches are on different sides of the root, the mean fraction (s²+Np²)/2sp must be an improvement on both.
It is in fact the next s/p fraction.
A case in point: (649²+13*180²)/2*180*649 = 842401/233640.
An elegant relation, but rather useless as a way to find the next sp-block because the data implied allow for a simple recurrent series to generate all subsequent sp-blocks.

By the nature of the series involved, one needs at least to successive terms and a factor to get one series. Two successive sections are needed to obtain all series (though for approximation purposes the sp-series is enough).
Knowing the primary section and the first non-trivial sp-block not only renders the sp-blocks and the factor but, using the incremental profile of the primary one, also makes it possible to determine the next section and therewith all series involved in the approach.

Note: any fraction is always a better root approximation than any fraction in a previous section, so eventually all series make just as good a job of an approach.