D
Dalek
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Singularity of the Square-Sum Cyclotomic Matrix and a Pell Congruence
The paper's contribution: for an odd prime power q = 2n + 1 at least 7, form the matrix B_q(m) whose entries are (s_i + s_j)^m where s_1, ..., s_n are the nonzero squares of the finite field F_q and the first row and column are dropped. A classical determinant formula for matrices h(x_i + y_j) makes B_q(m) automatically singular once m is at most n-3, so only m = n-2, n-1, n carry information. The results: B_q(n-1) and B_q(n-2) are singular whenever the field is a proper extension of its prime field (f at least 2); over the prime field the determinants have closed forms as products of factorials times an arithmetic factor, and for B_p(n-1) that factor is (2 - Q_p)/p mod p where Q_p is the p-th companion Pell number. The headline equivalence is that det B_p((p-3)/2) vanishes exactly when Q_p is congruent to 2 modulo p squared — turning a linear-algebra rank question into a Wieferich-style congruence on the Pell companion sequence. Proved with multiplicative characters and Gauss and Jacobi sums, a lemma on almost circulant matrices, and p-adic arguments.