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63 is carréphylic - approach of √63=3√7 ~ 7.9372539332

Subsequent approximations of √63 - the position of a fraction indicates whether it is over or under the root-value.
 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 82-63*12 = 1 satisfies the diophantine equation.
 Diophantine equation: s2-63p2 = 1 d = distance to nearest square N2: -1 Smallest non-trivial s: (2*64-1)/1 rational: 127 actual: 127 (8) ⇒ F=254 (16) Smallest non-trivial p: 2*8/1 rational: 16 actual: 16 (1) ⇒ primus foldage=16 (1) v-value tq-blocks: 72-63*12: -14 Number of series: 9

Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
 s 1 8 127 2024 32257 ... p 0 1 16 255 4064 ...

 In the numerator: U(1,8)16 = 1/2*U(2,16)16 - half the secundus of 16. In the denominator: U(0,1)16 = - the primus of 16. as well as ... In the numerator: U(0,63)16 = 63*U(0,1)16 - the 63-fold primus of 16. In the denominator: U(1,8)16 = 1/2*U(2,16)16 - half the secundus of 16. and ... In the numerator: U(-7,7)16 = 7*U(-1,1)16 - the 7-fold quartus of 16. In the denominator: U(1,1)16 = - the tertius of 16.