Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla
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- by R. A. Mollin PDF
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Abstract:
Based on the fundamental unit of $Q\left ( {\sqrt n } \right )$, an arbitrary real quadratic field, we provide a necessary condition for the class number $h\left ( n \right )$ to be 1. For $n = 4{m^2} + 1$ we prove the equivalence of three necessary and sufficient conditions for $h\left ( n \right )$ to be 1. One of these conditions is that $- {x^2} + x + {m^2}$ is prime for all integers $x$ such that $1 < x < m$. This is the exact analogue of the complex quadratic field case. We discuss the connection with a conjecture of S. Chowla as well as with other related topics.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 17-21
- MSC: Primary 11R29,; Secondary 11R11
- DOI: https://doi.org/10.1090/S0002-9939-1988-0915707-1
- MathSciNet review: 915707