57 is carréphobic - approach of √57 ~ 7.5498344353
Subsequent approximations of √57 - the position of a fraction indicates whether it is over or under the root-value.
| Diophantine equation: | s2-57p2 = 1 | | | |
| d = distance to nearest square N2: | -7 | | | |
| Smallest non-trivial s: | (2*64-7)/7 | rational: 121/7 | actual: 151 | ⇒ F=302 |
| Smallest non-trivial p: | 2*8/7 | rational: 16/7 | actual: 20 | ⇒ primus foldage=20 |
| v-value qt-blocks: | 152-57*22: | -3 | | |
| Number of series: | 14 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 151 | 45601 | 13771351 | ... |
| p | 0 | 20 | 6040 | 1824060 | ... |
| In the numerator: | U(1,151)302 | = | 1/2*U(2,302)302 | - | half the secundus of 302. |
| In the denominator: | U(0,20)302 | = | 20*U(0,1)302 | - | the 20-fold primus of 302. |
| as well as ... |
| In the numerator: | U(0,1140)302 | = | 1140*U(0,1)302 | - | the 57*20-fold primus of 302. |
| In the denominator: | U(1,151)302 | = | 1/2*U(2,302)302 | - | half the secundus of 302. |
| and ... |
| In the numerator: | U(-15,15)302 | = | 15*U(-1,1)302 | - | the 15-fold quartus of 302. |
| In the denominator: | U(2,2)302 | = | 2*U(1,1)302 | - | the 2-fold tertius of 302. |
