**Carréphylic ancestors**

Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that x = a

^{2}y ⇔ √x = a√y.

Since y is carréphylic, the approach of √x should pose no

*practical*problems.

Here's how to find the 's' and 'p' of the rational approach of x:

s

^{2}-xp

^{2}= 1 ⇔ s

^{2}-y(ap)

^{2}= 1.

The fraction s/ap is the a

^{th}fraction of the set of sp-blocks of the rational approach of y, starting from the trivial block.

It is therefore the first fraction of the base-a series accelleration of the rational approach of y.

The corresponding factor '2s' is F

_{a}or the factor of y 'base-a'.

Since √x = a√y the sp-fraction of x is a(s/ap) = s/p.

__Example carréphylic ancestry__

The position of a fraction indicates whether it is over or under the root-value.

__The approach of √7 - carréphylic__

1 | 0 | 1 | 2 | 3 | 5 | 8 | 21 | 29 | 37 | 45 | 82 | 127 | 336 | 463 | 590 | 717 | 1307 | 2024 | 5355 | 7379 | 9403 | 11427 | 20830 | 32257 | 85344 | 117601 | 149858 | 182115 | 331973 | 514088 | 1360149 | 1874237 | 2388325 | 2902413 | 5290738 | 8193151 | 21677040 | 29870191 | 38063342 | 46256493 | 84319835 | 130576328 | 345472491 | ... |

0 | 1 | 1 | 1 | 1 | 2 | 3 | 8 | 11 | 14 | 17 | 31 | 48 | 127 | 175 | 223 | 271 | 494 | 765 | 2024 | 2789 | 3554 | 4319 | 7873 | 12192 | 32257 | 44449 | 56641 | 68833 | 125474 | 194307 | 514088 | 708395 | 902702 | 1097009 | 1999711 | 3096720 | 8193151 | 11289871 | 14386591 | 17483311 | 31869902 | 49353213 | 130576328 | ... |

Accellerations | numerator | denominator | factor |

base-2 | U(1,127) | U(0,48) | 254 |

base-3 | U(1,2024) | U(0,765) | 4048 |

base-4 | U(1,32257) | U(0,12192) | 64514 |

__The approach of √28=2√7 - carréphobic__

1 | 0 | 1 | 2 | 3 | 4 | 5 | 11 | 16 | 21 | 37 | 90 | 127 | 672 | 799 | 926 | 1053 | 1180 | 1307 | 2741 | 4048 | 5355 | 9403 | 22854 | 32257 | 170688 | 202945 | 235202 | 267459 | 299716 | 331973 | 696203 | 1028176 | 1360149 | 2388325 | 5804826 | 8193151 | 43354080 | ... |

0 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 7 | 17 | 24 | 127 | 151 | 175 | 199 | 223 | 247 | 518 | 765 | 1012 | 1777 | 4319 | 6096 | 32257 | 38353 | 44449 | 50545 | 56641 | 62737 | 131570 | 194307 | 257044 | 451351 | 1097009 | 1548360 | 8193151 | ... |

Using the above data we find that the first non trivial sp-fraction of 28 is twice the second non-trivial fraction of 7, or 127/24.

The corresponding factor 2*127 is the base-2 accelleration of the factor of 7: 16

^{2}-2.

__The approach of √63=3√7 - carréphylic__

1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 63 | 71 | 79 | 87 | 95 | 103 | 111 | 119 | 127 | 1008 | 1135 | 1262 | 1389 | 1516 | 1643 | 1770 | 1897 | 2024 | 16065 | 18089 | 20113 | 22137 | 24161 | 26185 | 28209 | 30233 | 32257 | 256032 | ... |

0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 127 | 143 | 159 | 175 | 191 | 207 | 223 | 239 | 255 | 2024 | 2279 | 2534 | 2789 | 3044 | 3299 | 3554 | 3809 | 4064 | 32257 | ... |

63 is one less than a square, so the exception mentioned in on root approach applies: 127 and 16, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 8 and 1 because 8

^{2}-63*1

^{2}= 1 satisfies the diophantine equation.

Nevertheless, trice the third non-trivial fraction of 7 is 2024/255 and the resulting series is the base-3 accelleration of the sp-series of √63.

The corresponding factor 2*2024 is the base-3 accelleration of the factor of 7: 16

^{3}-3*16.

__The approach of √112=4√7 - carréphobic__

As it happens 112 falls outside the first 100 non-squares, so I didn't do a search, but using the above data the first non trivial sp-fraction of 112 is four times the fourth non-trivial fraction of 7, or 32257/3048.

The corresponding factor 2*32257 is the base-4 accelleration of the factor of 7: 16

^{4}-4*16

^{2}+2.

**Carréphylic descendants**

Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that a

^{2}x = y ⇔ a√x = √y.

Here's how to find the 's' and 'p' of the rational approach of x:

For y the first non-trivial 's' and 'p' satisfy: s

^{2}-yp

^{2}= 1 ⇔ s

^{2}-x(ap)

^{2}= 1 ⇔ 's' and 'ap' are the first non-trivial element of the set of sp-blocks of the rational approach of x. The corresponding factor is the same.

__Example carréphylic descendency__

__The approach of √13 - carréphobic__

The fractional approach of root 13, the first carréphobic number, as provided by Ed's straightforward program, looks like this:

1 | 0 | 1 | 2 | 3 | 4 | 7 | 11 | 18 | 83 | 101 | 119 | 137 | 256 | 393 | 649 | 2340 | 2989 | 3638 | 4287 | 4936 | 9223 | 14159 | 23382 | 107687 | 131069 | 154451 | 177833 | 332284 | 510117 | 842401 | 3037320 | 3879721 | 4722122 | 5564523 | 6406924 | 11971447 | 18378371 | 30349818 | 139777643 | 170127461 | 200477279 | 230827097 | 431304376 | 662131473 | 1093435849 | 3942439020 | ... |

0 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 5 | 23 | 28 | 33 | 38 | 71 | 109 | 180 | 649 | 829 | 1009 | 1189 | 1369 | 2558 | 3927 | 6485 | 29867 | 36352 | 42837 | 49322 | 92159 | 141481 | 233640 | 842401 | 1076041 | 1309681 | 1543321 | 1776961 | 3320282 | 5097243 | 8417525 | 38767343 | 47184868 | 55602393 | 64019918 | 119622311 | 183642229 | 303264540 | 1093435849 | ... |

As it happens, 5

^{2}*13 = 325 = 18

^{2}-1 and thus carréphylic. The formula renders 649/36 as the first sp-block of the approach of 5√13, making 649/180 the first sp-block of √13 with the same factor 1298.

Easy as fruitpie.