# 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

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## Abstract

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In this note we prove that the equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1={q}^{n}+1$, $q\ge 2,n\ge 3$, has only finitely many positive integer solutions $\left(k,q,n\right)$. Moreover, all solutions $\left(k,q,n\right)$ satisfy $k{10}^{{10}^{182}}$, $q{10}^{{10}^{165}}$ and $n2·{10}^{17}$.

## How to cite

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Le, 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

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1. [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. [2] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62. Zbl0788.11026
3. [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. [4] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equation, Cambridge Tracts in Math. 87, Cambridge Univ. Press, Cambridge, 1986. Zbl0606.10011

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