40 is carréphylic - approach of √40=2√10 ~ 6.3245553203
Subsequent approximations of √40 - the position of a fraction indicates whether it is over or under the root-value.
| Diophantine equation: | s2-40p2 = 1 | | | |
| d = distance to nearest square N2: | +4 | | | |
| Smallest non-trivial s: | (2*36+4)/4 | rational: 19 | actual: 19 | ⇒ F=38 |
| Smallest non-trivial p: | 2*6/4 | rational: 3 | actual: 3 | ⇒ primus foldage=3 |
| v-value qt-blocks: | 62-40*12: | -4 | | |
| Number of series: | 9 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 19 | 721 | 27379 | 1039681 | 39480499 | ... |
| p | 0 | 3 | 114 | 4329 | 164388 | 6242415 | ... |
| In the numerator: | U(1,19)38 | = | 1/2*U(2,38)38 | - | half the secundus of 38. |
| In the denominator: | U(0,3)38 | = | 3*U(0,1)38 | - | the 3-fold primus of 38. |
| as well as ... |
| In the numerator: | U(0,120)38 | = | 120*U(0,1)38 | - | the 4*3-fold primus of 38. |
| In the denominator: | U(1,19)38 | = | 1/2*U(2,38)38 | - | half the secundus of 38. |
| and ... |
| In the numerator: | U(-6,6)38 | = | 6*U(-1,1)38 | - | the 6-fold quartus of 38. |
| In the denominator: | U(1,1)38 | = | | - | the tertius of 38. |
