From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Uhm...
Date: 5 Mar 1999 22:56:11 GMT
Newsgroups: sci.math
Keywords: Known cases of ABC conjecture; adjacent numbers with few divisors
Zdislav V. Kovarik wrote:
>Question: I have seen, but did not save, a proof that no more "Pythagorean
>twins" (adjacent integers, each of them a product of non-negative
>integer powers of 2, 3, and 5) with larger entries than (81,80) exist.
>Does anyone know a reference to this statement?
>
>And: What if we throw in 7 as a prime factor, is the number of "shepherd
>twins" also finite, and if so, is the list known?
The ABC conjecture would imply that if the prime factors of A, B, C are
prescribed in advance, then there is only a finite number of solutions
to the equation A + B = C (indeed it would bound C to be no more than
"roughly" the product of those primes). So in particular there ought to be
only finitely many pairs of adjacent integers whose prime factors are
limited to {2, 3, 5}, or to {2, 3, 5, 7}, or whatever.
The ABC conjecture is open, but some partial results are known; these
may be among them. Certainly it's trivial to check for pairs of such integers
up to 10^10, say; we find the lesser numbers in the pair to be
{1, 2, 3, 4, 5, 8, 9, 15, 24, 80}
for the primes {2,3,5} and
{1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125,
224, 2400, 4374}
for the primes {2,3,5,7}. These data certainly suggest the lists are complete.
For related information see
index/11PXX.html
dave
==============================================================================
From: nikl+sm000775@mathematik.tu-muenchen.de (Gerhard Niklasch)
Subject: S-unit equation (Was: Re: Uhm...)
Date: 6 Mar 1999 19:32:48 GMT
Newsgroups: sci.math
In article <7bpneb$ovt$1@gannett.math.niu.edu>,
rusin@vesuvius.math.niu.edu (Dave Rusin) writes:
|> Zdislav V. Kovarik wrote:
|> >Question: I have seen, but did not save, a proof that no more "Pythagorean
|> >twins" (adjacent integers, each of them a product of non-negative
|> >integer powers of 2, 3, and 5) with larger entries than (81,80) exist.
|> >Does anyone know a reference to this statement?
|> >
|> >And: What if we throw in 7 as a prime factor, is the number of "shepherd
|> >twins" also finite, and if so, is the list known?
|>
|> The ABC conjecture would imply that if the prime factors of A, B, C are
|> prescribed in advance, then there is only a finite number of solutions
|> to the equation A + B = C (indeed it would bound C to be no more than
|> "roughly" the product of those primes). So in particular there ought to be
|> only finitely many pairs of adjacent integers whose prime factors are
|> limited to {2, 3, 5}, or to {2, 3, 5, 7}, or whatever.
But when the set of primes is fixed in advance as it is here, far
stronger results are known. The complete set of solutions of A + B = C
in integers each composed of the prime factors {2,3,5,7} and no others
can be and has been effectively determined. The canonical approach is
to use estimates for linear forms in logarithms and p-adic logarithms
to obtain a priori upper bounds on the exponents attached to the prime
factors, and diophantine approximation (or lattice reduction) techniques
to reduce those bounds to a manageable size. For a complete description,
covering the larger set of primes {2,3,5,7,11,13}, see Benne de Weger's
CWI Tract 65, `Algorithms for Diophantine Equations', Amsterdam 1989.
However, elementary congruence considerations often allow one to handle
the problem directly if the number of admitted primes is small. There
are a largeish number of papers by Leo J. Alex and Lorraine L. Foster
(jointly and separately and sometimes with other co-authors) about
this. Doing {2,3} is a short and amusing exercise.
|> Certainly it's trivial to check for pairs of such integers
|> up to 10^10, say; we find the lesser numbers in the pair to be
|> {1, 2, 3, 4, 5, 8, 9, 15, 24, 80}
|> for the primes {2,3,5} and
|> {1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125,
|> 224, 2400, 4374}
|> for the primes {2,3,5,7}. These data certainly suggest the lists are
|> complete.
All my reference material is 4 miles from where I'm typing this,
but if you did search up to 10^10 then those lists should indeed be
complete; the exponent bound after reduction would be small enough.
(I thought I had an ASCII file somewhere with all the solutions, but
I can't seem to find it right now.)
(And of course one should search among exponent combinations, not
among all integers up to a given size...)
Enjoy, Gerhard
--
* Gerhard Niklasch * spam totally unwelcome
* http://hasse.mathematik.tu-muenchen.de/~nikl/ ******* all browsers welcome
* This .signature now fits into 3 lines and 77 columns * newsreaders welcome
==============================================================================
[Some related results using four summands. --djr]
96i:11035 11D61
Alex, L. J.(1-SUNY); Foster, L. L.(1-CASN)
On the Diophantine equation $w+x+y=z$, with $wxyz=2\sp r3\sp s5\sp t$.
(English. English summary)
Rev. Mat. Univ. Complut. Madrid 8 (1995), no. 1, 13--48.
This paper is one in a series by the authors and J. L. Brenner, in
which they use only elementary tools to completely determine the sets
of solutions of certain four-term exponential Diophantine equations.
See the earlier papers in this series [e.g., J. L. Brenner and L. L.
Foster, Pacific J. Math. 101 (1982), no. 2, 263--301; MR 83k:10035; L.
J. Alex and L. L. Foster, Rocky Mountain J. Math. 13 (1983), no. 2,
321--331; MR 84j:10015; L. J. Alex and L. L. Foster, Rocky Mountain J.
Math. 15 (1985), no. 3, 739--761; MR 87d:11023]. See also a further
paper by the authors [Forum Math. 7 (1995), no. 6, 645--663; see the
following review].
This paper is about the equation $w+x+y=z$, in which $w,x,y$ and $z$
are integers $>1$ built up from the primes 2, 3 and 5 only. A complete
solution is presented; the largest value for $z$ that occurs is
$2\,125\,764$. The case of $\min\{w,x,y\}=1$ was treated in another
paper in the same series [L. J. Alex and L. L. Foster, Rocky Mountain
J. Math. 22 (1992), no. 1, 11--62; MR 93e:11040].
The proofs of this paper, as in all papers of this series, are
elementary in the sense that no deeper mathematics than the theory of
congruences is needed. So other methods often used to solve
Diophantine equations, such as techniques from transcendence theory,
Diophantine approximation theory or algebraic number theory, are not
used at all. This does not mean that the proofs are easy or trivial:
sometimes they are very clever and ingenious. For example, the moduli
used can be as large as eleven-digit numbers. It is not obvious that
the method will always work, but in practice it seems to work very
often. The main problem seems to be to find the right moduli. It is
impressive indeed to see that no deep mathematics is needed to prove
such results. It is not yet clear at the moment where the practical
limits of these elementary methods lie.
Reviewed by B. M. M. de Weger
_________________________________________________________________
93e:11040 11D61
Alex, Leo J.(1-SUNY); Foster, Lorraine L.(1-CASN)
On the Diophantine equation $1+x+y=z$.
Rocky Mountain J. Math. 22 (1992), no. 1, 11--62.
The aim of the paper under review is to prove that the Diophantine
equation $1+x+y=z$ has 251 explicitly given solutions in positive
integers $x$, $y$ and $z$ such that the largest prime factor of $xyz$
is less than 7. The proof is elementary: congruences of the form
$1+x+y\equiv z\bmod p$ are examined for various well-chosen primes
$p$.
Reviewer's remark: The authors state in the introduction that it is
unknown whether the equation $(*)$ $1+2\sp a3\sp b=5\sp c+2\sp d3\sp
e5\sp f$ has only a finite number of nontrivial solutions. This is not
correct because by the main theorem of $S$-unit equations of
86c:11045J.-H. Evertse [Compositio Math. 53 (1984), no. 2, 225--244;
MR 86c:11045] equation $(*)$ has only finitely many nontrivial
solutions.
Reviewed by A. Petho
Cited in: 96i:11035
_________________________________________________________________
87d:11023 11D61
Alex, Leo J.(1-SUNY); Foster, Lorraine L.(1-CASN)
On the Diophantine equation $1+p\sp a=2\sp b+2\sp cp\sp d$.
Rocky Mountain J. Math. 15 (1985), no. 3, 739--761.
The authors consider the Diophantine equation $1+p\sp a=2\sp b+2\sp
cp\sp d$ where $p$ is an odd prime. Such equations arise in the
character theory of finite groups. The reviewer and G. Rhin [C. R.
Acad. Sci. Paris Ser. A-B 282 (1976), no. 21, A1211--A1214; MR 55
#2793] and H. P. Schlickewei [Acta Arith. 33 (1977), no. 2, 183--185;
MR 55 #12633] proved that the Diophantine equation $x\sb
1+\cdots+x\sb n=0$, where the prime factors of each $x\sb i$ lie in a
finite set $S\sb i$ with $S\sb i\cap S\sb j=\varnothing$ for $i\neq
j$, has only finitely many solutions. But their methods do not apply
when $S\sb i\cap S\sb j\not=\varnothing$. The authors give the
solutions (infinitely many) for some $p$ such that $p=2\sp q-1$,
$q\geq 2$, or $p=2�9\sp k+1$, $k\geq 1$, and for any $p$ with $p<500$.
Reviewed by Eugene Dubois
Cited in: 96i:11035
_________________________________________________________________
84j:10015 10B25
Alex, Leo J.; Foster, Lorraine L.(1-CASN)
On Diophantine equations of the form
$1+2\sp{a}=p\sp{b}q\sp{c}+2\sp{d}p\sp{e}q\sp{f}$.
Rocky Mountain J. Math. 13 (1983), no. 2, 321--331.
Certain relations among group characters lead to exponential
Diophantine equations of the type in the title. The authors' summary
reads: "In this paper several Diophantine equations of the form
$1+2\sp a=p\sp bq\sp c+2\sp dp\sp eq\sp f$, where $p$ and $q$ are
distinct odd primes and the exponents are nonnegative integers, are
solved. In particular this equation is solved for $(p,q)=(23,47)$,
$(7,23)$ and $(73,223)$. The related equations $1+73\sp a=2\sp b223\sp
c+2\sp d73\sp e223\sp f$ and $1+223\sp a=2\sp b73\sp c+2\sp d73\sp
e223\sp f$ are also solved. This work extends recent work of the
authors and J. L. Brenner."
Reviewed by J. L. Brenner
Cited in: 96i:11035 86j:11031
(c) 2001, American Mathematical Society