# A note on the diophantine equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1={q}^{n}+1$

Colloquium Mathematicae (1998)

- Volume: 76, Issue: 1, page 31-34
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topLe, Maohua. "A note on the diophantine equation ${k\atopwithdelims ()2}-1=q^n+1$." Colloquium Mathematicae 76.1 (1998): 31-34. <http://eudml.org/doc/210550>.

@article{Le1998,

abstract = {In this note we prove that the equation $\{k\atopwithdelims ()2\}-1=q^n+1$, $q\ge 2, n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k10^\{10^\{182\}\}$, $q10^\{10^\{165\}\}$ and $n2\cdot 10^\{17\}$.},

author = {Le, Maohua},

journal = {Colloquium Mathematicae},

keywords = {exponential diophantine equations; coding theory; upper bounds; estimates of linear forms in logarithms},

language = {eng},

number = {1},

pages = {31-34},

title = {A note on the diophantine equation $\{k\atopwithdelims ()2\}-1=q^n+1$},

url = {http://eudml.org/doc/210550},

volume = {76},

year = {1998},

}

TY - JOUR

AU - Le, Maohua

TI - A note on the diophantine equation ${k\atopwithdelims ()2}-1=q^n+1$

JO - Colloquium Mathematicae

PY - 1998

VL - 76

IS - 1

SP - 31

EP - 34

AB - In this note we prove that the equation ${k\atopwithdelims ()2}-1=q^n+1$, $q\ge 2, n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k10^{10^{182}}$, $q10^{10^{165}}$ and $n2\cdot 10^{17}$.

LA - eng

KW - exponential diophantine equations; coding theory; upper bounds; estimates of linear forms in logarithms

UR - http://eudml.org/doc/210550

ER -

## References

top- [1] R. Alter, On the non-existence of perfect double Hamming-error-correcting codes on q=8 and q=9 symbols, Inform. and Control 13 (1968), 619-627. Zbl0165.22101
- [2] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62. Zbl0788.11026
- [3] C. Hering, A remark on two diophantine equations of Peter Cameron, in: Groups, Combinatorics and Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press, Cambridge, 1992, 448-458.
- [4] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equation, Cambridge Tracts in Math. 87, Cambridge Univ. Press, Cambridge, 1986. Zbl0606.10011

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.