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


Florian Luca

Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58180, Morelia, Michoacan, Mexico

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.

Full text (PDF) (free access)

DOI: 10.3336/gm.40.1.02


  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.

  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.

  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.

Glasnik Matematicki Home Page