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15 is carréphylic - approach of √15 ~ 3.8729833462

Subsequent approximations of √15 - the position of a fraction indicates whether it is over or under the root-value.
 1 0 1 2 3 4 15 19 23 27 31 120 151 182 213 244 945 1189 1433 1677 1921 7440 9361 11282 13203 15124 58575 73699 88823 103947 119071 461160 580231 699302 818373 937444 3630705 4568149 5505593 6443037 7380481 28584480 ... 0 1 1 1 1 1 4 5 6 7 8 31 39 47 55 63 244 307 370 433 496 1921 2417 2913 3409 3905 15124 19029 22934 26839 30744 119071 149815 180559 211303 242047 937444 1179491 1421538 1663585 1905632 7380481 ...

15 is one less than a square, so the exception mentioned in on root approach applies: 31 and 8, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 4 and 1 because 42-15*12 = 1 satisfies the diophantine equation.
 Diophantine equation: s2-15p2 = 1 d = distance to nearest square N2: -1 Smallest non-trivial s: (2*16-1)/1 rational: 31 actual: 31 (4) ⇒ F=62 (8) Smallest non-trivial p: 2*4/1 rational: 8 actual: 8 (1) ⇒ primus foldage=8 (1) v-value qt-blocks: 32-15*12: -6 Number of series: 5

Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
 s 1 4 31 244 1921 15124 119071 937444 7380481 ... p 0 1 8 63 496 3905 30744 242047 1905632 ...

 In the numerator: U(1,4)8 = 1/2*U(2,8)8 - half the secundus of 8. In the denominator: U(0,1)8 = - the primus of 8. as well as ... In the numerator: U(0,15)8 = 15*U(0,1)8 - the 15-fold primus of 8. In the denominator: U(1,4)8 = 1/2*U(2,8)8 - half the secundus of 8. and ... In the numerator: U(-3,3)8 = 3*U(-1,1)8 - the 3-fold quartus of 8. In the denominator: U(1,1)8 = - the tertius of 8.