Glasnik Matematicki, Vol. 40, No.1 (2005), 13-20.

ON SHIFTED PRODUCTS WHICH ARE POWERS

Florian Luca

Abstract.   In this note, we improve upon results of Bugeaud, Gyarmati, Sarkozy and Stewart concerning the size of a subset A of {1,...,N} such that the product of any two distinct elements of A plus 1 is a perfect power. We also show that the cardinality of such a set is uniformly bounded assuming the ABC-conjecture, thus improving upon a result of Dietmann, Elsholtz, Gyarmati and Simonovits.

2000 Mathematics Subject Classification.   11B75, 11D99.

Key words and phrases.   Shifted products, perfect powers.

DOI: 10.3336/gm.40.1.02

References:

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

2. Y. Bugeaud and A. Dujella, On a problem of Diophantus for higher powers, Math. Proc. Cambridge Philos. Soc. 135 (2003), 1-10.
   CrossRef

3. Y. Bugeaud and K. Gyarmati, On generalizations of a problem of Diophantus, Illinois J. Math. 48 (2004), 1105-1115.

4. R. Dietmann, C. Elsholtz, K. Gyarmati and M. Simonovits, Shifted products that are coprime pure powers, J. Comb. Theor. A, to appear.

5. A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150.
   CrossRef

6. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.

7. R. L. Graham, B. L. Rothschild, J. H. Spencer, Ramsey Theory, John Wiley & Suns, 1980.

8. K. Gyarmati, On a problem of Diophantus, Acta Arith. 97 (2001), 53-65.

9. K. Gyarmati, A. Sárközy and C. L. Stewart, On shifted products which are powers, Mathematika 49 (2002), 227-230.

10. P. Kövari, V. Sós and P. Túran, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50-57.

11. P. Túran, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941), 436-452.