Diophantine equations, Tzanakis articles

[Hugo Pfoertner]

SeqFans,

in a current thread in the NG sci.math.research "Finding all integral
solutions of some Diophantine equations" Joe Silverman mentions the
following article

"Tzanakis, N.(GR-CRET); de Weger, B. M. M.(NL-TWEN-A) On the practical 
solution of the Thue equation. J. Number Theory 31 (1989), no. 2, 
99--132."

and writes
<<
The review of this last article says: 
The algorithm is based on a combination of Baker's method with the 
so-called basis reduction algorithm due to Lenstra, Lenstra and Lovász 
and several skilful lemmas. As a remarkable application of the main 
result all the 22 integral points of the elliptic curve y^2=x^3-4x+1 
are determined.
>>

With a little program I computed the first few solutions:

   x     y
   2     1
   3     4
   4     7
  10    31
  12    41
  20    89
 114  1217
1274 45473

both not found in the OEIS. Can someone here look into the article and
provide the remaining 14 solutions (or compute it)?

Googling for "tzanakis thue equation" I found a related thread in the NG 
sci.math "when is a perfect square?", dated July 27 2001, in which 
Erick Bryce Wong
http://groups.google.com/group/sci.math/msg/56fc1206d948f0fe
wrote
<<
So the question is, when is (3^m-1)/2 a perfect square? 

I suspect the only solutions are m=0,1,2,5, but I don't have a reference 
for this.  There are certainly only finitely many solutions, and if I
recall 
correctly there exists an effective bound on m, even for the far more
general 
case of (k^m-1)/2 being a perfect square.
>>

In the same thread Dave Rusin mentioned: (in
http://groups.google.com/group/sci.math/msg/38ca2ed4ff177f52
".... Diophantine equaitons of the form  y^2 = (quartic in x), which is
a task which can be solved effectively, at least for many quartics,
using the work of Tzanakis and others. (There is a reduction to Thue
equations which can be effectively solved.)"

and gave a list of several papers by Nicholas Tzanakis and John
Wolfskill in the J. Number Theory between 1982 and 1987.

I don't have immediate access to this Journal, but the titles suggest
that several interesting sequences might be found in these papers. Can
someone have a look into the articles?

Assuming that Dave Rusin's answer to
"when is (3^m-1)/2 a perfect square?" is correct, we get a funny short
new sequence

Perfect squares that can be written in the form (3^m-1)/2
0,1,4,121

The corresponding values of m: 0,1,2,5 are so unspecific that a new
sequence will not be of much use.

For the more general question "When is (k^m-1)/2 a perfect square?" I
found a few small non-trivial solutions (k>1,m>1):

    k   m   (k^m-1)/2  sqrt((k^m-1)/2)
    3   2          4       2
    3   5        121      11
   17   2        144      12
   99   2       4900      70
  577   2     166464     408
 3363   2    5654884    2378
19601   2  192099600   13860

For 4,121,144,4900,166464 Lookup produces:

http://www.research.att.com/projects/OEIS?Anum=A075114
4,121,144,4900,166464,5654884
Germain perfect powers: perfect powers n such that 2*n+1 is also a
perfect power.
by Zak Seidov, extended by RGWV.

Is 192099600 the next term and can more terms be found?

Numbers whose squares can be written in the form (k^m-1)/2, k>1,m>1
2,11,12,70,408,2378,13860,
is not in the OEIS.

Hugo Pfoertner