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.
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. |
