Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1171.11020
Kihel, Omar; Luca, Florian
Variants of the Brocard-Ramanujan equation. (English)
[J] J. Théor. Nombres Bordx. 20, No. 2, 353-363 (2008). ISSN 1246-7405

It is expected that the only solutions to the Diophantine equation $$n!+1=y^2\tag 1$$ are $(y,n)=(5,4),(11,5),(71,7)$ (the Brocard-Ramanujan problem). {\it M. Overholt} [Bull. Lond. Math. Soc. 25, No. 2, 104 (1993; Zbl 0805.11030)] showed that the weak form of Szpiro's conjecture (a special case of the ABC conjecture) implies that (1) has only finitely many solutions. Generalizing Overholt's result (and {\it A. Dabrowski}'s result [Nieuw Arch. Wiskd., IV. Ser. 14, No. 3, 321--324 (1996; Zbl 0876.11015)]), the second author showed that the full ABC conjecture implies that the Diophantine equation $n!=P(x)$ has only finitely many solutions [Glas. Mat. (3) 37, No. 2, 269--273 (2002; Zbl 1085.11023)]. The authors discuss some variants of the above Diophantine equations. Here are the main results of this paper (we write $k\times n$ to mean that $k$ does not divide $n)$. \par Theorem 1. The Diophantine equation $x^p\pm y^p=\prod^n_{k\times n,k=1}k$ admits only finitely many integer solutions $(x,y,p,n)$ with $p\ge 3$ a prime number and $\gcd(x,y)=1$. \par Theorem 2. Let $P\in{\Bbb Q}[x]$ be a polynomial of degree $\ge 2$. The ABC conjecture implies that the equation $P(x)=\prod^n_{k \times n,k=1}k$ has only finitely many solutions $(x,n)$, where $x$ is a rational number and $n$ is a positive integer. \par They also prove the following generalization of a result of {\it A. Dabrowski} [loc. cit.]. \par Theorem 3. Let $A$ be an integer which is not a perfect square. Then the Diophantine equation $x^2-A= \prod^n_{k \times n,k=1}k$ has only finitely many solutions.
[Andrzej Dabrowski (Szczecin)]

MSC 2000:: *11D85 Representation problems of integers
11D61 Exponential diophantine equations

Keywords: Brocard-Ramanujan problem; Diophantine equation involving factorials; ABC conjecture

Citations: Zbl 0876.11015; Zbl 1085.11023; Zbl 0805.11030

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]