80 is carréphylic - approach of √80=4√5 ~ 8.9442719100
Subsequent approximations of √80 - the position of a fraction indicates whether it is over or under the root-value.
80 is one less than a square, so the exception mentioned in
on root approach applies: 161 and 18, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being
9 and
1 because
92-80*12 = 1 satisfies the diophantine equation.
| Diophantine equation: | s2-80p2 = 1 | | | |
| d = distance to nearest square N2: | -1 | | | |
| Smallest non-trivial s: | (2*81-1)/1 | rational: 161 | actual: 161 (9) | ⇒ F=322 (18) |
| Smallest non-trivial p: | 2*8/1 | rational: 18 | actual: 18 (1) | ⇒ primus foldage=18 (1) |
| v-value tq-blocks: | 82-80*12: | -16 | | |
| Number of series: | 10 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 9 | 161 | 2889 | ... |
| p | 0 | 1 | 18 | 323 | ... |
| In the numerator: | U(1,9)18 | = | 1/2*U(2,18)18 | - | half the secundus of 18. |
| In the denominator: | U(0,1)18 | = | | - | the primus of 18. |
| as well as ... |
| In the numerator: | U(0,80)18 | = | 80*U(0,1)18 | - | the 80-fold primus of 18. |
| In the denominator: | U(1,9)18 | = | 1/2*U(2,18)18 | - | half the secundus of 18. |
| and ... |
| In the numerator: | U(-8,8)18 | = | 8*U(-1,1)18 | - | the 8-fold quartus of 18. |
| In the denominator: | U(1,1)18 | = | | - | the tertius of 18. |
