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Explicit Solutions of Pyramidal Diophantine Equations

Published online by Cambridge University Press:  20 November 2018

Leon Bernstein
Leon Bernstein*
Affiliation:
Illinois Institute of Technology, Chicago, Illinois
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Let Pm, k denote the set of pyramidal numbers

1.1

The question has been asked whether there exist elements p, q, r in Pm, k such that p+q = r or, as the problem is usually posed,

1.2

The case k=2 has been studied by Sierpinski [6] and Khatri [3]; the case k=3 by Oppenheim [4] and Segal [5]; recently Fraenkel [2] has generalized (1.1) to the larger set

1.3

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 177 - 184
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Leon, Bernstein, A probability function for partitions, Amer. Math. Monthly (1968), 882-886.Google Scholar
Leon, Bernstein, New infinite classes of periodic Jacobi-Perron algorithms, Pacific J. Math., (3) 16 (1966), 439-469.Google Scholar
Leon, Bernstein, The generalized Pellian equation, Trans. Amer. Math. Soc. (1) 127 (1967), 76-89.Google Scholar
2. Fraenkel, Aviezri S., Diophantine equations involving generalized triangular and tetrahedral numbers (to appear).Google Scholar
3. Khatri, M. N., Triangular numbers andpythagorean triangles, Scripta Math. 21 (1955), 94.Google Scholar
4. Oppenheim, A., On the diophantine equation x3 + y3 + z 3 = x + y + z, Proc. Amer. Math. Soc. 16 (1965), 148-153.Google Scholar
Oppenheim, A., On the diophantine equation x 3 +y 3 + z 3 ?px +py?qz , Publ. Faculté D'electrotech. Univ. Belgrade, Ser. Math. Physique, No. 230-No. 241 (1968), 33-35.Google Scholar
5. Segal, S. L., A note on pyramidal numbers, Amer. Math. Monthly (1962), 637-638.Google Scholar
6. Sierpinski, W., Elementary theory of numbers, Translated from Polish by A. Hulamiki, Oxford Pergamon Press, 1964.Google Scholar