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