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001-es BibID:	BIBFORM111672
Első szerző:	Győry Kálmán (matematikus)
Cím:	Power values of products of consecutive integers and binomial coefficients / K. Győry
Dátum:	1999
Megjegyzések:	In this article the author gives a description of the latest developments concerning power values of products of consecutive integers, binomial coefficients and related questions. After a brief historical introduction the remarkable result of Erdős and Selfridge is stated. This result says that for any integer k?2 a product of k consecutive integers is never a power; i.e., the equation (1) n(n+1)?(n+k?1)=xl in integers n?1, k,x,l?2, has no solution. A related equation treated by Erdős is (2) (n+k?1k)=xl in integers k?2, n?k+1, x,l?2. In 1951 Erdős showed that for k?4 equation (2) has no solution. The method of Erdős does not work for the equation (2) if k<4. The author shows how the case k=2 follows from a result of H. Darmon and L. Merel [J. Reine Angew. Math. 490 (1997), 81?100; MR1468926] and also settles the case k=3. The equations (1) and (2) can be combined as (3) n(n+1)?(n+k?1)=bxl in integers n,b, x?1, k,l?2, with P(b)?k, where b is an lth power, and P(b) denotes the greatest prime factor of b. By a result of Saradha for k?4, and Györy for k?3, we have that equation (3) has only the solution (n,k,b,x,l)=(48,3,6,140,2) with P(x)>k apart from the case k=b=l=2. Another generalisation of equation (3) is (4) n(n+d)?(n+(k?1)d)=bxl in integers n,b, x?1, k?3, l?2, with gcd(n,d)=1, P(b)?k. A conjecture of Erdős says that equation (4) implies that k is bounded by an absolute constant. In fact, this constant is believed to be 3. This equation has been studied by several authors. The results are partial. Saradha has shown that equation (4) with l?3 and d?6 has no solution. By the works of Saradha, Hajdu and Filakovszky, it is known that equation (4) with l=2, b=1 and d?30 has as its only solution (n,d,k,x)=(18,7,3,120) and (1,24,3,35). The author shows in this article that equation (4) with b=1, k=3 and l?3 has no solution by using the results of Darmon, Merel and Ribet.
ISBN:	0-7923-5952-6
Tárgyszavak:	Természettudományok Matematika- és számítástudományok előadáskivonat könyvrészlet
Megjelenés:	Number theory and its applications / Shigeru Kanemitsu; Kálmán Győry (eds.). - p. 145-156. -
Internet cím:	Intézményi repozitóriumban (DEA) tárolt változat Szerző által megadott URL
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