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.
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. |
