By A. Barlotti, etc.
Curiosity in combinatorial concepts has been tremendously better through the purposes they could provide in reference to desktop expertise. The 38 papers during this quantity survey the state-of-the-art and document on fresh leads to Combinatorial Geometries and their applications.
Contributors: V. Abatangelo, L. Beneteau, W. Benz, A. Beutelspacher, A. Bichara, M. Biliotti, P. Biondi, F. Bonetti, R. Capodaglio di Cocco, P.V. Ceccherini, L. Cerlienco, N. Civolani, M. de Soete, M. Deza, F. Eugeni, G. Faina, P. Filip, S. Fiorini, J.C. Fisher, M. Gionfriddo, W. Heise, A. Herzer, M. Hille, J.W.P. Hirschfield, T. Ihringer, G. Korchmaros, F. Kramer, H. Kramer, P. Lancellotti, B. Larato, D. Lenzi, A. Lizzio, G. Lo Faro, N.A. Malara, M.C. Marino, N. Melone, G. Menichetti, okay. Metsch, S. Milici, G. Nicoletti, C. Pellegrino, G. Pica, F. Piras, T. Pisanski, G.-C. Rota, A. Sappa, D. Senato, G. Tallini, J.A. Thas, N. Venanzangeli, A.M. Venezia, A.C.S. Ventre, H. Wefelscheid, B.J. Wilson, N. Zagaglia Salvi, H. Zeitler.
Read or Download Combinatorics ’81, in honour of Beniamino Segre: proceedings of the International Conference on Combinatorial Geometrics and their Applications, Rome, June 7-12, 1981 PDF
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Extra info for Combinatorics ’81, in honour of Beniamino Segre: proceedings of the International Conference on Combinatorial Geometrics and their Applications, Rome, June 7-12, 1981
In fact, 1 is m-gonal for any m, 15 is 3-, 6- and 15-gonal, 36 is 3-, 4- and 13-gonal, 45 is 3-, 6- and 16-gonal, 325 is 3-, 6- and 9-gonal, 561 is 3-, 6- and 12-gonal, 1225 is 3-, 4- and 6-gonal, 1540 is 3-, 6- and 10-gonal, 3025 is 4-, 12- and 15-gonal, 4186 is 3-, 6- and 13-gonal, etc. , up to m = 16) less than 1012 , except 1 (see [Weis11]). 3. Consider now one more simple question: how to check whether a number N is an m-gonal number Sm (n) for some n? An arbitrary number N can be checked for m-gonality as follows.
2. A pentagonal square number is a number which is simultaneously pentagonal and square. Such numbers correspond to the positive integer solutions of the Diophantine equation 1 u(3u − 1) = v 2 . 2 Completing the square gives (6u − 1)2 − 24v 2 = 1. Substituting x = 6u − 1 and y = 2v gives the Pell’s equation x2 − 6y 2 = 1. It has positive integer solutions (x, y) = (5, 2), (49, 20), (495, 198), . . In 2401 terms of (u, v), these give (u, v) = (1, 1), ( 25 3 , 10), (81, 99), ( 3 , 980), (7921, 9701), .
22. The table below summarizes the set of m-gonal k-gonal numbers for small values of m and k. m 4 5 5 6 6 6 7 7 7 7 8 8 8 8 8 9 9 9 9 9 9 k Sequence 3 1, 36, 1225, 41616, 1413721, . . 3 1, 210, 40755, 7906276, 1533776805, . . 4 1, 9801, 94109401, 903638458801, 8676736387298001, . . 3 1, 6, 15, 28, 45, . . , all hexagonal numbers) 4 1, 1225, 1413721, 1631432881, 1882672131025, . . 5 1, 40755, 1533776805, 57722156241751, 2172315626468283465, . . 3 1, 55, 121771, 5720653, 12625478965 . . 4 1, 81, 5929, 2307361, 168662169, .
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