63 is carréphylic - approach of √63=3√7 ~ 7.9372539332
Subsequent approximations of √63 - the position of a fraction indicates whether it is over or under the root-value.
63 is one less than a square, so the exception mentioned in
on root approach applies: 127 and 16, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being
8 and
1 because
82-63*12 = 1 satisfies the diophantine equation.
| Diophantine equation: | s2-63p2 = 1 | | | |
| d = distance to nearest square N2: | -1 | | | |
| Smallest non-trivial s: | (2*64-1)/1 | rational: 127 | actual: 127 (8) | ⇒ F=254 (16) |
| Smallest non-trivial p: | 2*8/1 | rational: 16 | actual: 16 (1) | ⇒ primus foldage=16 (1) |
| v-value tq-blocks: | 72-63*12: | -14 | | |
| Number of series: | 9 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 8 | 127 | 2024 | 32257 | ... |
| p | 0 | 1 | 16 | 255 | 4064 | ... |
| In the numerator: | U(1,8)16 | = | 1/2*U(2,16)16 | - | half the secundus of 16. |
| In the denominator: | U(0,1)16 | = | | - | the primus of 16. |
| as well as ... |
| In the numerator: | U(0,63)16 | = | 63*U(0,1)16 | - | the 63-fold primus of 16. |
| In the denominator: | U(1,8)16 | = | 1/2*U(2,16)16 | - | half the secundus of 16. |
| and ... |
| In the numerator: | U(-7,7)16 | = | 7*U(-1,1)16 | - | the 7-fold quartus of 16. |
| In the denominator: | U(1,1)16 | = | | - | the tertius of 16. |
