89 is carréphobic - approach of √89 ~ 9.4339811321
Using the
incremental profile, the questionmarks can be filled in - the position of a fraction indicates whether it is over or under the root-value. The positioning of the missing fractions corresponds to the first section.
Subsequent approximations of √89.
| Diophantine equation: | s2-89p2 = 1 | | | |
| d = distance to nearest square N2: | +8 | | | |
| Smallest non-trivial s: | (2*81+8)/8 | rational: 170/8 | actual: 500001 | ⇒ F=1000002 |
| Smallest non-trivial p: | 2*9/8 | rational: 18/8 | actual: 53000 | ⇒ primus foldage=53000 |
| v-value qt-blocks: | 5002-89*532: | -1 | | |
| Number of series: | 33 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 500001 | 500002000001 | ... |
| p | 0 | 53000 | 53000106000 | ... |
| In the numerator: | U(1,500001)1000002 | = | 1/2*U(2,1000002)1000002 | - | half the secundus of 1000002. |
| In the denominator: | U(0,53000)1000002 | = | 53000*U(0,1)1000002 | - | the 53000-fold primus of 1000002. |
| as well as ... |
| In the numerator: | U(0,4717000)1000002 | = | 4717000*U(0,1)1000002 | - | the 89*53000-fold primus of 1000002. |
| In the denominator: | U(1,500001)1000002 | = | 1/2*U(2,1000002)1000002 | - | half the secundus of 1000002. |
| and ... |
| In the numerator: | U(-500,500)1000002 | = | 500*U(-1,1)1000002 | - | the 500-fold quartus of 1000002. |
| In the denominator: | U(53,53)1000002 | = | 53*U(1,1)1000002 | - | the 53-fold tertius of 1000002. |
