New Seq: Differences between squares are cubes

[franktaw at netscape.net]

Take a look at
http://www.research.att.com/~njas/sequences/id:A080761|id:A084388|id:A084389.

(These sequences could use some editing, IMO.  One has to (a) realize
that there are three sequences involved -- two of them are adjacent, 
but
the first is not, and there are no cross-refs -- and (b) stare at it 
for a while, in
order to figure out what is being presented.  Furthermore, looking at 
the
PARI program, an arbitrary limit has been imposed on the search, so 
these
are really only conjectured values -- that is, the triples in these 
sequences
certainly belong, but there may be others intermingled that are not
shown (this is noted in A080761, but not in the other two).  There are
other problems, too: the crossref lines say to see certain sequences; 
but
only a description is given, not the A numbers.  Finally, there is an 
error in
A084388: the value 17322 should be two values: 173, 22.)

Franklin T. Adams-Watters

-----Original Message-----
From: zak seidov <zakseidov at yahoo.com>

Dear seqfans,

My Qs, plz:

Is a(42)= 7821697850 the last term
(if corresponding q exists it is > ~ 10^10).
As i know (actually i don't)
there is no theory about a^2-b^2=c^3, right?
Anyway, more terms?

thanks, zak

PS i hope to get some responses before submitting
these two seq's:

%S A1
1,3,6,10,15,17,561,564,423564,1268439,2535189,4223814,6334314,8866689,118
20939,15197064,18995064,23214939,27856689,32920314,38405814,44313189,5064
2439,54153730,57782670,61529259,65393497,69375384,73474920,77692105,82026
939,86479422,91049554,95737335,100542765,105465844,110506572,115664949,12
0940975,126334650,126385350,126385350,7821697850
%N A1
a(1)=1, then a(n) is the smallest integer > a(n-1)
such that a(n)^2-a(n-1)^2 is cube q^3.

%C A1
Values of q's are (future A2):
2,3,4,5,4,68,15,5640,11265,16890,22515,28140,33765,39390,45015,50640,5626
5,
61890,67515,73140,78765,84390,71659,74060,76461,78862,81263,83664,86065,8
8466,
90867,93268,95669,98070,100471,102872,105273,107674,110075,23400,3940000

Note that sequence of q's is not monotonic.
The sequence presumably terminates with
a(42)=7821697850(?)

%e A1
3^2-1^2=2^3,
6^2-3^2=3^3,
10^2-6^2=4^3,
15^2-10^2=5^3,
17^2-15^2=4^3,
561^2-17^2=68^3.




zs> From seqfan-owner at ext.jussieu.fr  Thu Apr 10 09:56:42 2008
zs> Date: Thu, 10 Apr 2008 00:53:59 -0700 (PDT)
zs> From: zak seidov <zakseidov at yahoo.com>
zs> Subject: New Seq: Differences between squares are cubes
zs> To: seqfan at ext.jussieu.fr
zs> 
zs> Dear seqfans,
zs> 
zs> My Qs, plz:
zs> 
zs> Is a(42)= 7821697850 the last term
zs> (if corresponding q exists it is > ~ 10^10).
zs> As i know (actually i don't)
zs> there is no theory about a^2-b^2=c^3, right?

Ove Hemer, "Notes on the Diophantine equation y^2-k=x^3", Arkiv f"or Matematik,
Band 3 nr 3 (called issue 1 now ?), 1954, p67-77,
http://dx.doi.org/10.1007/BF02589282
This covers some aspects and provides tables for small positive and
negative k.

Richard