Here Ed's program failed to provide the last 24 fractions before the sp-block. The sp-block itself was derived from the qt-block, after it was identified as such. The positioning of the missing fractions is derived from a property of roots with the qt-block in the middle of the section: corresponding fractions in the first and second half are on opposite sides of the root-value.
Subsequent approximations of √109.
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 21 | 52 | 73 | 94 | 167 | 261 | 877 | 1138 | 1399 | 5335 | 6734 | 8133 | 9532 | 39527 | 49059 | 58591 | 68123 | 194837 | 262960 | 331083 | 730289 | 1061372 | 2853033 | 3914405 | 4975777 | 8890182 | 101706407 | 110596589 | 119486771 | 128376953 | 137267135 | 146157317 | 155047499 | 163937681 | 172827863 | 181718045 | 372326272 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 158070671986249 | 1650306265605900 | ... |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 7 | 9 | 16 | 25 | 84 | 109 | 134 | 511 | 645 | 779 | 913 | 3786 | 4699 | 5612 | 6525 | 18662 | 25187 | 31712 | 69949 | 101661 | 273271 | 374932 | 476593 | 851525 | 9741707 | 10593232 | 11444757 | 12296282 | 13147807 | 13999332 | 14850857 | 15702382 | 16553907 | 17405432 | 35662389 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 15140424455100 | 158070671986249 | ... |
| Diophantine equation: | s2-109p2 = 1 | |||
| d = distance to nearest square N2: | +9 | |||
| Smallest non-trivial s: | (2*100+9)/9 | rational: 209/9 | actual: 158070671986249 | ⇒ F=316141343972498 |
| Smallest non-trivial p: | 2*10/9 | rational: 20/9 | actual: 15140424455100 | ⇒ primus foldage=15140424455100 |
| v-value qt-blocks: | 88901822-109*8515252: | -1 | ||
| Number of series: | 73 |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 158070671986249 | ... |
| p | 0 | 15140424455100 | ... |
| In the numerator: | U(1,158070671986249)316141343972498 | = | 1/2*U(2,316141343972498)316141343972498 | - | half the secundus of 316141343972498. |
| In the denominator: | U(0,15140424455100)316141343972498 | = | 15140424455100*U(0,1)316141343972498 | - | the 15140424455100-fold primus of 316141343972498. |
| as well as ... | |||||
| In the numerator: | U(0,1650306265605900)316141343972498 | = | 1650306265605900*U(0,1)316141343972498 | - | the 109*15140424455100-fold primus of 316141343972498. |
| In the denominator: | U(1,158070671986249)316141343972498 | = | 1/2*U(2,316141343972498)316141343972498 | - | half the secundus of 316141343972498. |
| and ... | |||||
| In the numerator: | U(-8890182,8890182)316141343972498 | = | 8890182*U(-1,1)316141343972498 | - | the 8890182-fold quartus of 316141343972498. |
| In the denominator: | U(851525,851525)316141343972498 | = | 851525*U(1,1)316141343972498 | - | the 851525-fold tertius of 316141343972498. |