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 √97.
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 39 | 49 | 59 | 69 | 128 | 197 | 325 | 522 | 847 | 3063 | 3910 | 4757 | 5604 | 60797 | 66401 | 72005 | 77609 | 83213 | 88817 | 94421 | 100025 | 105629 | 111233 | 439328 | 550561 | 661794 | 773027 | 1434821 | 2207848 | 3642669 | 5850517 | 9493186 | 34330075 | 43823261 | 53316447 | 62809633 | 618603144 | 681412777 | 744222410 | 807032043 | 869841676 | 932651309 | 995460942 | 1058270575 | 1121080208 | 1183889841 | 1246699474 | 4923988263 | 6170687737 | 7417387211 | 8664086685 | ? | ? | ? | ? | ? | ? | ? | ? | 703970372268 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 7890099995189377 | 77708472894572304 | ... |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 5 | 6 | 7 | 13 | 20 | 33 | 53 | 86 | 311 | 397 | 483 | 569 | 6173 | 6742 | 7311 | 7880 | 8449 | 9018 | 9587 | 10156 | 10725 | 11294 | 44607 | 55901 | 67195 | 78489 | 145684 | 224173 | 369857 | 594030 | 963887 | 3485691 | 4449578 | 5413465 | 6377352 | 62809633 | 69186985 | 75564337 | 81941689 | 88319041 | 94696393 | 101073745 | 107451097 | 113828449 | 120205801 | 126583153 | 499955260 | 626538413 | 753121566 | 879704719 | ? | ? | ? | ? | ? | ? | ? | ? | 71477361785 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 801118277263632 | 7890099995189377 | ... |
| Diophantine equation: | s2-97p2 = 1 | |||
| d = distance to nearest square N2: | -3 | |||
| Smallest non-trivial s: | (2*100-3)/3 | rational: 197/3 | actual: 62809633 | ⇒ F=125619266 |
| Smallest non-trivial p: | 2*10/3 | rational: 20/3 | actual: 6377352 | ⇒ primus foldage=6377352 |
| v-value qt-blocks: | 56042-97*5692: | -3 | ||
| Number of series: | 47 |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 62809633 | 7890099995189377 | ... |
| p | 0 | 6377352 | 801118277263632 | ... |
| In the numerator: | U(1,62809633)125619266 | = | 1/2*U(2,125619266)125619266 | - | half the secundus of 125619266. |
| In the denominator: | U(0,6377352)125619266 | = | 6377352*U(0,1)125619266 | - | the 6377352-fold primus of 125619266. |
| as well as ... | |||||
| In the numerator: | U(0,618603144)125619266 | = | 618603144*U(0,1)125619266 | - | the 97*6377352-fold primus of 125619266. |
| In the denominator: | U(1,62809633)125619266 | = | 1/2*U(2,125619266)125619266 | - | half the secundus of 125619266. |
| and ... | |||||
| In the numerator: | U(-5604,5604)125619266 | = | 5604*U(-1,1)125619266 | - | the 5604-fold quartus of 125619266. |
| In the denominator: | U(569,569)125619266 | = | 569*U(1,1)125619266 | - | the 569-fold tertius of 125619266. |