The prime factors of Wendt’s binomial circulant determinant

Fee, Greg; Granville, Andrew

doi:10.1090/S0025-5718-1991-1094948-8

The prime factors of Wendt’s binomial circulant determinant
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by Greg Fee and Andrew Granville PDF

Math. Comp. 57 (1991), 839-848 Request permission

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 $\left ( {\begin {array}{*{20}{c}} m \\ {|i - j|} \\ \end {array} } \right )$ 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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