## 23609Re: Primes again [really Mordell's equation]

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• Nov 2, 2011
>
> I think I've cracked it:
> http://www.inf.unideb.hu/~pethoe/cikkek/72_S_INTGPZ.pdf
> shows that Mike found their record (with goading from Iago?):
> 54225: 48 in S_0 (no cheating)

Thanks for making all that so clear, David.

It's reassuring that I had found all their "top" 9 Elliptic Curves, with up to 24*2 integer points.

I have extended the investigation to abs(delta) <= 10^7, again for n <= 10^8.

There are exactly 3 delta's with > 24 solutions, namely:-
delta solutions
-3470400 25
-5472225 27
9754975 32 [outright winner!]

Here are the 32 solutions to n^3-m^2=9754975:-
214^3=9800344 213^2=45369
220^3=10648000 945^2=893025
236^3=13144256 1841^2=3389281
266^3=18821096 3011^2=9066121
350^3=42875000 5755^2=33120025
416^3=71991296 7889^2=62236321
475^3=107171875 9870^2=97416900
530^3=148877000 11795^2=139122025
680^3=314432000 17455^2=304677025
700^3=343000000 18255^2=333245025
799^3=510082399 22368^2=500327424
814^3=539353144 23013^2=529598169
1019^3=1058089859 32378^2=1048334884
1435^3=2954987875 54270^2=2945232900
1690^3=4826809000 69405^2=4817054025
1870^3=6539203000 80805^2=6529448025
1880^3=6644672000 81455^2=6634917025
2680^3=19248832000 138705^2=19239077025
3364^3=38068692544 195087^2=38058937569
5026^3=126960157576 356301^2=126950402601
6910^3=329939371000 574395^2=329929616025
8120^3=535387328000 731695^2=535377573025
14450^3=3017196125000 1737005^2=3017186370025
14800^3=3241792000000 1800495^2=3241782245025
19910^3=7892485271000 2809355^2=7892475516025
36815^3=49896997643375 7063780^2=49896987888400
68891^3=326954611071971 18081886^2=326954601316996
89710^3=721975682611000 26869605^2=721975672856025
129530^3=2173257047177000 46618205^2=2173257037422025
168896^3=4817903450587136 69411119^2=4817903440832161
232100^3=12503322161000000 111818255^2=12503322151245025
626866^3=246333878914829896 496320339^2=246333878905074921

I conjecture that, with >=64 integer points, the Mordell Elliptic Curve
y^2=x^3-9754975
will be found to have high rank, say 4 or 5.

Puzzle: compute its rank, torsion subgroup, etc.

Mike
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