A note on the diophantine equation ${k\choose 2}-1=q^n+1$

• Autorzy: Maohua Le
• Języki publikacji: EN

Abstrakty

EN

In this note we prove that the equation ${k\choose 2}-1=q^n+1$, $q\ge 2, n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k\lt10^{10^{182}}$, $q\lt10^{10^{165}}$ and $n\lt 2\cdot 10^{17}$.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1998
• Tom: 76
• Numer: 1
• Strony: 31-34

Daty

wydano

1998

otrzymano

1997-02-06

Twórcy

autor

Maohua Le

• Department of Mathematics Zhanjiang Teachers College 524048 Zhanjiang, Guangdong P.R. China

Bibliografia

• [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.
• [2] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62.
• [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.