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 √106.
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 21 | 31 | 72 | 103 | 175 | 278 | 453 | 731 | 1184 | 2821 | 4005 | 45239 | 49244 | 53249 | 57254 | 61259 | 65264 | 69269 | 73274 | 77279 | 81284 | 166573 | 247857 | 576998 | 824855 | 1401853 | 2226708 | 3628561 | 5855269 | 9483830 | 22596221 | 32080051 | 330284340 | 362364391 | 494444442 | 426524493 | 458604544 | 490684595 | 522764646 | 554844697 | 586924748 | 619004799 | 651084850 | 1334249751 | 1985334601 | 4621754052 | 6607088653 | 11228842705 | ? | ? | ? | ? | ? | 256961212515 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 2058259344325201 | 21191076943402680 | ... |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 7 | 10 | 17 | 27 | 44 | 71 | 115 | 274 | 389 | 4394 | 4783 | 5172 | 5561 | 5950 | 6339 | 6728 | 7117 | 7506 | 7895 | 16179 | 24074 | 56043 | 80117 | 136160 | 216277 | 352437 | 568714 | 921151 | 2194739 | 3115890 | 32080051 | 35195941 | 38311831 | 41427721 | 44543611 | 47659501 | 50775391 | 53891281 | 57007171 | 60123061 | 63238951 | 129593792 | 192832743 | 448904437 | 641737180 | 1090641617 | ? | ? | ? | ? | ? | 24958279289 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 199915820220780 | 2058259344325201 | ... |
| Diophantine equation: | s2-106p2 = 1 | |||
| d = distance to nearest square N2: | +6 | |||
| Smallest non-trivial s: | (2*100+6)/6 | rational: 206/6 | actual: 32080051 | ⇒ F=64160102 |
| Smallest non-trivial p: | 2*10/6 | rational: 20/6 | actual: 3115890 | ⇒ primus foldage=3115890 |
| v-value qt-blocks: | 40052-106*3892: | -1 | ||
| Number of series: | 43 |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 32080051 | 2058259344325201 | ... |
| p | 0 | 3115890 | 199915820220780 | ... |
| In the numerator: | U(1,32080051)64160102 | = | 1/2*U(2,64160102)864160102 | - | half the secundus of 64160102. |
| In the denominator: | U(0,3115890)64160102 | = | 3115890*U(0,1)64160102 | - | the 3115890-fold primus of 64160102. |
| as well as ... | |||||
| In the numerator: | U(0,330284340)64160102 | = | 330284340*U(0,1)64160102 | - | the 106*3115890-fold primus of 64160102. |
| In the denominator: | U(1,32080051)64160102 | = | 1/2*U(2,64160102)64160102 | - | half the secundus of 64160102. |
| and ... | |||||
| In the numerator: | U(-4005,14005)64160102 | = | 4005*U(-1,1)64160102 | - | the 4005-fold quartus of 64160102. |
| In the denominator: | U(389,389)64160102 | = | 389*U(1,1)64160102 | - | the 389-fold tertius of 64160102. |