The Pell sequence and cyclotomic matrices involving squares over finite fields

By using some arithmetic properties of the Pell sequence and some p-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let 1=s1,s2,…,s(q−1)/2 be all the nonzero squares over Fq, where q=pf is an odd prime power with q≥7. We prove that the matrix Bq((q−3)/2)=[(si+sj)(q−3)/2]2≤i,j≤(q−1)/2 is singular whenever f≥2. Also, for q=p, we show that detBp((p−3)/2)=0⟺Qp≡2(modp2Z), where Qp is the pth term of the companion Pell sequence {Qi}∞i=0 defined by Q0=Q1=2 and Qi+1=2Qi+Qi−1.