History of Good examples of Hall's conjecture

   points, Danilov, Elkies Definition: For a positive integer $x$, let $y$ be the nearest positive integer to-  $x^{3/2}$, $k=y^2-x^3$, and $r=\sqrt{x}/|k|$. This table holds the cases of Hall's-  conjecture CITE{WikiHall} with $k\ne0$ and $r>1$, recording the four parts $x$,-  $y$, $k$ and $r$ for each $x$.+  $x^{3/2}$, $k=y^2-x^3$, and $r=\sqrt{x}/|k|$. This table holds the positive integers+  $x$ with $k\ne0$ and $r>1$, called good examples of Hall's conjecture in CITE{WikiHall},+  recording the four parts $x$, $y$, $k$ and $r$ for each $x$. Parameters:   x:

     and $y$, so non-primitive examples from CITE{WikiHall} are kept. Formulas:-  hall-ratio:-    formula: $r=\sqrt{x}/|y^2-x^3|$-    description: The Hall ratio measures how small $|y^2-x^3|$ is compared with $\sqrt{x}$.+  hall-ratio: $r=\sqrt{x}/|y^2-x^3|$ Similar tables: - table: HREF{First_Mordell_curves_of_given_rank}[First known Mordell curves of given

     url: https://en.wikipedia.org/wiki/Hall%27s_conjecture   ElkiesHall:-    title: 'Noam D. Elkies: List of integers x,y with x<10^18 and 0<|x^3-y^2|<sqrt(x)'+    title: 'Noam D. Elkies: Hall''s conjecture examples'     url: https://people.math.harvard.edu/~elkies/hall.html Data properties:   type: R   complete: 'no'-  complete-note: it holds every example with $x\leq10^{29}$ in CITE{WikiHall}, together-    with the ten listed examples beyond that bound+  complete-note: it holds every $x\leq10^{29}$ with $r>1$, a range in which CITE{AKR}+    showed that no further examples occur, together with the ten larger known examples+    recorded in CITE{WikiHall}   sources:   - CITE{WikiHall}   - CITE{ElkiesHall}+  - CITE{AKR}   rigour: proven   rigour details: The integers $x$, $y$ and $k$ are checked by exact integer arithmetic.

     r: '1.65720364673277316306883384869'   '5853886516781223':-    x:-      number: '5853886516781223'-      comment: This row has the largest Hall ratio in the source list.+    x: '5853886516781223'     y: '447884928428402042307918'     k: '-1641843'-    r: '46.6004943471754121442137157318'+    r:+      number: '46.6004943471754121442137157318'+      comment: This is the largest Hall ratio among the entries here.   '12813608766102806':     x: '12813608766102806'

     x:       number: '23415546067124892'-      comment: This non-primitive example is obtained from the Elkies row $x=5853886516781223$+      comment: This non-primitive example comes from the example $x=5853886516781223$         by multiplying $x$, $y$ and $k$ by $4$, $8$ and $64$.     y: '3583079427427216338463344'

     k: '75512937817147150'     r: '1.03248327527037349400800325099'+References:+  AKR:+    bib: Stål Aanderaa, Lars Kristiansen and Hans Kristian Ruud, Search for good examples+      of Hall's conjecture, Mathematics of Computation 87 (2018), no. 314, 2903-2914.+    doi: 10.1090/mcom/3298