3 is carréphylic - approach of √3 ~ 1.7320508076
Subsequent approximations of √3 - the position of a fraction indicates whether it is over or under the root-value.
3 is one less than a square, so the exception mentioned in
on root approach applies: 7 and 4, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being
2 and
1 because
22-3*12 = 1 satisfies the diophantine equation.
| Diophantine equation: | s2-3p2 = 1 | | | |
| d = distance to nearest square N2: | -1 | | | |
| Smallest non-trivial s: | (2*4-1)/1 | rational: 7 | actual: 7 (2) | ⇒ F=14 (4) |
| Smallest non-trivial p: | 2*2/1 | rational: 4 | actual: 4 (1) | ⇒ primus foldage=4 (1) |
| v-value qt-blocks: | 12-3*12: | -2 | | |
| Number of series: | 3 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 2 | 7 | 26 | 97 | 362 | 1351 | 5042 | 18817 | 70226 | 262087 | 978122 | ... |
| p | 0 | 1 | 4 | 15 | 56 | 209 | 780 | 2911 | 10864 | 40545 | 151316 | 564719 | ... |
| In the numerator: | U(1,2)4 | = | 1/2*U(2,4)4 | - | half the secundus of 4. |
| In the denominator: | U(0,1)4 | = | | - | the primus of 4. |
| as well as ... |
| In the numerator: | U(0,3)4 | = | 3*U(0,1)4 | - | the 3-fold primus of 4. |
| In the denominator: | U(1,2)4 | = | 1/2*U(2,4)4 | - | half the secundus of 4. |
| and ... |
| In the numerator: | U(-1,1)4 | = | | - | the quartus of 4. |
| In the denominator: | U(1,1)4 | = | | - | the tertius of 4. |
