Difference between neighbor square and cube

[Max A.]

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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