Journal of Integer Sequences, Vol. 22 (2019), Article 19.7.6

Primes and Perfect Powers in the Catalan Triangle


Nathaniel Benjamin
Department of Mathematics
Iowa State University
Carver Hall, 411 Morrill Road
Ames, IA 50011
USA

Grant Fickes
Department of Mathematics
Kutztown University of Pennsylvania
15200 Kutztown Road
Kutztown, PA 19530
USA

Eugene Fiorini
Department of Mathematics
Muhlenberg College
2400 Chew Street
Allentown, PA 18104
USA

Edgar Jaramillo Rodriguez
Department of Mathematics
University of California, Davis
1 Shields Avenue
Davis, CA 95616
USA

Eric Jovinelly
Department of Mathematics
Notre Dame University
255 Hurley
Notre Dame, IN 46556 USA

Tony W. H. Wong
Department of Mathematics
Kutztown University of Pennsylvania
15200 Kutztown Road
Kutztown, PA 19530
USA

Abstract:

The Catalan triangle is an infinite lower-triangular matrix that generalizes the Catalan numbers. The entries of the Catalan triangle, denoted by cn,k, count the number of shortest lattice paths from (0,0) to (n,k) that do not go above the main diagonal. This paper studies the occurrence of primes and perfect powers in the Catalan triangle. We prove that no prime powers except 2, 5, 9, and 27 appear in the Catalan triangle when k ≥ 2. We further prove that cn,k are not perfect semiprime powers when k ≥ 3. Finally, by assuming the abc conjecture, we prove that aside from perfect squares when k = 2, there are at most finitely many perfect powers among cn,k when k ≥ 2.

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(Concerned with sequences A275481 A275586 A317027.)

Received January 30 2019; revised versions received October 18 2019; October 31 2019. Published in Journal of Integer Sequences, November 8 2019.

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