69 is carréphobic - approach of √69 ~ 8.3066238629
Subsequent approximations of √69 - the position of a fraction indicates whether it is over or under the root-value.
| Diophantine equation: | s2-69p2 = 1 | | | |
| d = distance to nearest square N2: | +5 | | | |
| Smallest non-trivial s: | (2*64+5)/5 | rational: 133/5 | actual: 7775 | ⇒ F=15550 |
| Smallest non-trivial p: | 2*8/5 | rational: 16/5 | actual: 936 | ⇒ primus foldage=936 |
| v-value tq-blocks: | 1082-69*132: | +3 | | |
| v-value qt-blocks: | 2992-69*362: | -23 | | |
| Number of series: | 22 | | | |
Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
| s | 1 | 7775 | 120901249 | ... |
| p | 0 | 936 | 14554800 | ... |
| In the numerator: | U(1,7775)15550 | = | 1/2*U(2,15550)15550 | - | half the secundus of 15550. |
| In the denominator: | U(0,936)15550 | = | 936*U(0,1)15550 | - | the 936-fold primus of 15550. |
| as well as ... |
| In the numerator: | U(0,68584)15550 | = | 68584*U(0,1)15550 | - | the 69*936-fold primus of 15550. |
| In the denominator: | U(1,7775)15550 | = | 1/2*U(2,15550)15550 | - | half the secundus of 15550. |
| and ... |
| In the numerator: | U(108,108)15550 | = | 108*U(1,1)15550 | - | the 108-fold tertius of 15550. |
| In the denominator: | U(-13,13)15550 | = | 13*U(-1,1)15550 | - | the 13-fold quartus of 15550. |
| and ... |
| In the numerator: | U(-299,299)15550 | = | 299*U(-1,1)15550 | - | the 299-fold quartus of 15550. |
| In the denominator: | U(36,36)15550 | = | 36*U(1,1)15550 | - | the 36-fold tertius of 15550. |
