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.
 1 0 1 2 3 4 5 6 7 8 9 19 47 66 151 217 283 500 5217 5717 6217 6717 7217 7717 8217 8717 9217 18934 47085 66019 150972 216991 283010 500001 4717000 5217001 5717002 6217003 6717004 7217005 7717006 8217007 8717008 9217009 18934019 47085047 66019066 150972151 216991217 283010283 500001500 5217006217 5717007717 6217009217 6717010717 7217012217 7717013717 8217015217 8717016717 9217018217 ? ? ? ? ? ? 500002000001 4717009434000 ... 0 1 1 1 1 1 1 1 1 1 1 2 5 7 16 23 30 53 553 606 659 712 765 818 871 924 977 2007 4991 6998 16003 23001 29999 53000 500001 553001 606001 659001 712001 765001 818001 871001 924001 977001 2007002 4991005 6998007 16003016 23001023 29999030 53000053 553001553 606001606 659001659 712001712 765001765 818001818 871001871 924001924 977001977 ? ? ? ? ? ? 53000106000 500002000001 ...

 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.