19 is carréphobic - approach of √19 ~ 4.3588989435
Subsequent approximations of √19 - the position of a fraction indicates whether it is over or under the root-value.
| Diophantine equation: | s2-19p2 = 1 | | | |
| d = distance to nearest square N2: | +3 | | | |
| Smallest non-trivial s: | (2*16+3)/3 | rational: 35/3 | actual: 170 | ⇒ F=340 |
| Smallest non-trivial p: | 2*4/3 | rational: 8/3 | actual: 39 | ⇒ primus foldage=39 |
| v-value qt-blocks: | 132-19*32: | -2 | | |
| Number of series: | 12 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 170 | 57799 | 19651490 | ... |
| p | 0 | 39 | 13260 | 4508361 | ... |
| In the numerator: | U(1,170)340 | = | 1/2*U(2,340)340 | - | half the secundus of 340. |
| In the denominator: | U(0,39)340 | = | 39*U(0,1)340 | - | the 39-fold primus of 340. |
| as well as ... |
| In the numerator: | U(0,741)340 | = | 741*U(0,1)340 | - | the 19*39-fold primus of 340. |
| In the denominator: | U(1,170)340 | = | 1/2*U(2,340)340 | - | half the secundus of 340. |
| and ... |
| In the numerator: | U(-13,13)340 | = | 13*U(-1,1)340 | - | the 13-fold quartus of 340. |
| In the denominator: | U(3,3)340 | = | 3*U(1,1)340 | - | the 3-fold tertius of 340. |
