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48 is carréphylic - approach of √48=4√3 ~ 6.9282032303

Subsequent approximations of √48 - the position of a fraction indicates whether it is over or under the root-value.
 1 0 1 2 3 4 5 6 7 48 55 62 69 76 83 90 97 672 769 866 963 1060 1157 1254 1351 9360 10711 12062 13413 14764 16115 17466 18817 130368 149185 168002 186819 205636 224453 243270 262087 1815792 2077879 2339966 2602053 2864140 3126227 3388314 3650401 25290720 ... 0 1 1 1 1 1 1 1 1 7 8 9 10 11 12 13 14 97 111 125 139 153 167 181 195 1351 1546 1741 1936 2131 2326 2521 2716 18817 21533 24249 26965 29681 32397 35113 37829 262087 299916 337745 375574 413403 451232 489061 526890 3650401 ...

48 is one less than a square, so the exception mentioned in on root approach applies: 97 and 14, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 7 and 1 because 72-48*12 = 1 satisfies the diophantine equation.
 Diophantine equation: s2-48p2 = 1 d = distance to nearest square N2: -1 Smallest non-trivial s: (2*49-1)/1 rational: 97 actual: 97 (7) ⇒ F=194 (14) Smallest non-trivial p: 2*7/1 rational: 14 actual: 14 (1) ⇒ primus foldage=14 (1) v-value qt-blocks: 62-48*12: -12 Number of series: 8

Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
 s 1 7 97 1351 18817 262087 3650401 ... p 0 1 14 195 2716 37829 526890 ...

 In the numerator: U(1,7)14 = 1/2*U(2,14)14 - half the secundus of 14. In the denominator: U(0,1)14 = - the primus of 14. as well as ... In the numerator: U(0,48)14 = 48*U(0,1)14 - the 48-fold primus of 14. In the denominator: U(1,7)14 = 1/2*U(2,14)14 - half the secundus of 14. and ... In the numerator: U(-6,6)14 = 6*U(-1,1)14 - the 6-fold quartus of 14. In the denominator: U(1,1)14 = - the tertius of 14.