Difference between neighbor square and cube

[zak seidov]

Max,
thanks a lot for your (excellent as usual) reply!!

Using it I found:
A054504(*
Numbers n such that Mordell's equation y^2 = x^3 + n
has no integral solutions. *)=
{6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, 45,
46, 47, 51, 53, 58, 59, 60, 61, 62, 66, 67, 69, 70,
74, 75, 77, 78, 83, 84, 85, 86, 87, 88, 90, 93, 95,
96, 102, 103, 104, 109, 110, 111, 114, 115, 116, 118,
123, 124, 130, 133, 135, 137, 139, 140, 146, 147, 149,
153, 155};

A081121=(*
Numbers n such that Mordell's equation y^2 = x^3 - n
has no integral solutions. *){
3, 5, 6, 9, 10, 12, 14, 16, 17, 21, 22, 24, 29, 30,
31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 46, 50, 51,
52, 57, 58, 59, 62, 65, 66, 68, 69, 70, 73, 75, 77,
78, 80, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96,
97, 98, 99};

Intersection[A081121,A054504]
ND={6,14,21,29,32,34,42,46,51,58,59,62,66,69,70,75,77,78,84,85,86,88,90,93,96}

And ND list gives 
(Absolute values of) Non-difference between square and
cube (not only neighbor as in %C A125643).

Hence any number in ID should be in %C A125643.
But 21 is in ND and absent in %C A125643!

Indeed in A125643,
we have:
..,64,81,100,121,125,...,
and 121-100=11^2-10^2=21!!

21 is not difference between neighbor square and cube,
but it is difference between neighbor terms in
A125643.

As a result: my subject 'Difference between neighbor
square and cube' is not correct,
and in %C A125643 we have 
'Difference between neighbor square/cube and
square/cube'.

Sorry and thanks to Max & all seqfans,
Zak  


--- "Max A." <maxale at gmail.com> wrote:

> Zak,
> 
> Your conjecture is related to Mordell curves of the
> form y^2 = x^3 + n, see
> http://mathworld.wolfram.com/MordellCurve.html
> http://tnt.math.metro-u.ac.jp/simath/MORDELL/
> 
> The latter link can help to resolve the most
> difficult case of whether
> a given number is the difference of a square and a
> cube. In
> particular, for n=5 and n=-5 the corresponding
> equation y^2 = x^3 + n
> has no positive solutions, meaning that 5 is not a
> difference of
> positive square and cube (not necessary neighbors,
> so this is even
> stronger than needed by conjecture).
> It is also easy to see that 5 is not the difference
> of two cubes. And
> 5 can be represented in an unique way as the
> difference of two
> squares: 5 = 3^2 - 2^2 but 2^2 and 3^2 are not
> neighbors in A125643
> since they are separated by 2^3.
> Therefore, 5 indeed never appears as the difference
> between neighbor
> terms of A125643.
> 
> Max
> 
> 
> On 11/29/06, zak seidov <zakseidov at yahoo.com> wrote:
> > Just submitted (with no autoreply).
> > My Q is:
> > Is the Conjectured list (see %C)..
> > old hat or what?
> > Thanks, Zak
> >
> > %I A125643
> > %S A125643
> >
>
0,0,1,1,4,8,9,16,25,27,36,49,64,64,81,100,121,125,144,169,196,216,225,256,289,324,343,361,400,441,484,512,529,576,625,676,729,729,784,841,900,961,1000,1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681,1728,1764,1849,1936,2025,2116,2197
> > %N A125643 Squares and cubes (with repetitions)
> > %C A125643 Cf. A002760 Squares and cubes (without
> > repetitions).
> > Conjectured list of numbers not appeared as
> difference
> > between neighbor terms:
> >
>
5,6,10,14,16,22,23,29,32,34,42,44,46,50,52,54,58,62,64,66,70,72,78,82,84,86,88,90,92,93,94,96,98,102,105,110,112,114,117,118,120,122,124,126,130,132,134,136,140,141,142,144,153,156,158,160,162,164,165,166,172,176,178,179,182,188,190,192,194
> > %Y A125643 A002760
> > %O A125643 0
> > %K A125643 ,nonn,
> > %A A125643 Zak Seidov  (zakseidov at gmail.com), Nov
> 29 2006
> >
> >
> >
> >
>
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> 



 
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