8 is carréphylic - approach of √8=2√2 ~ 2.8284271247
Subsequent approximations of √8 - the position of a fraction indicates whether it is over or under the root-value.
8 is one less than a square, so the exception mentioned in
on root approach applies: 17 and 6, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being
3 and
1 because
32-8*12 = 1 satisfies the diophantine equation.
| Diophantine equation: | s2-8p2 = 1 | | | |
| d = distance to nearest square N2: | -1 | | | |
| Smallest non-trivial s: | (2*9-1)/1 | rational: 17 | actual: 17 (3) | ⇒ F=34 (6) |
| Smallest non-trivial p: | 2*3/1 | rational: 6 | actual: 6 (1) | ⇒ primus foldage=6 (1) |
| v-value qt-blocks: | 22-8*12: | -4 | | |
| Number of series: | 4 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 3 | 17 | 99 | 577 | 3363 | 19601 | 114243 | 665857 | 3880899 | 22619537 | ... |
| p | 0 | 1 | 6 | 35 | 204 | 1189 | 6930 | 40391 | 235416 | 1372105 | 7997214 | ... |
| In the numerator: | U(1,3)6 | = | 1/2*U(2,6)6 | - | half the secundus of 6. |
| In the denominator: | U(0,1)6 | = | | - | the primus of 6. |
| as well as ... |
| In the numerator: | U(0,8)6 | = | 8*U(0,1)6 | - | the 8-fold primus of 6. |
| In the denominator: | U(1,3)6 | = | 1/2*U(2,6)6 | - | half the secundus of 6. |
| and ... |
| In the numerator: | U(-2,2)6 | = | 2*U(-1,1)6 | - | the 2-fold quartus of 6. |
| In the denominator: | U(1,1)6 | = | | - | the tertius of 6. |
