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The prime factors of Wendt's binomial circulant determinant

Authors: Greg Fee and Andrew Granville
Journal: Math. Comp. 57 (1991), 839-848
MSC: Primary 11Y50; Secondary 11C20, 11D41, 11R18, 11Y40
MathSciNet review: 1094948
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Abstract: Wendt's binomial circulant determinant, $ {W_m}$, is the determinant of an m by m circulant matrix of integers, with (i, j)th entry $ \left( {\begin{array}{*{20}{c}} m \\ {\vert i - j\vert} \\ \end{array} } \right)$ whenever 2 divides m but 3 does not. We explain how we found the prime factors of $ {W_m}$ for each even $ m \leq 200$ by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and $ q = mp + 1$ are odd primes, 3 does not divide m, and $ m \leq 200$, then the first case of Fermat's Last Theorem is true for exponent p.

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Article copyright: © Copyright 1991 American Mathematical Society

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