This article provides a simple criterion for the simultaneous solvability of the Diophantine equations X2 −DY 2 = c and x2 −Dy2 = −c when c ∈ Z, and D ∈ N is not a perfect square.

The U-statistic elegantly and usefully generalizes the notion of a sample mean. Typical examples include (i) sample mean: h(x1, x2) = 12 (x1 + x2); (ii) sample variance: h(x1, x2) = 12 (x1 − x2); (iii) Gini’s mean difference: h(x1, x2) = |x1 − x2|; (iv) one-sample Wilcoxon’s statistic: h(x1, x2) = 1(x1 + x2 ≤ 0). The non-degenerate U-statistic shares many limiting properties with the sample mea...

We adopt the following rules: n is a natural number and x1, x2, x3, x4, x5, y1, y2, y3 are sets. Let x1, x2, x3, x4, x5 be sets. We say that x1, x2, x3, x4, x5 are mutually different if and only if: (Def. 1) x1 6= x2 and x1 6= x3 and x1 6= x4 and x1 6= x5 and x2 6= x3 and x2 6= x4 and x2 6= x5 and x3 6= x4 and x3 6= x5 and x4 6= x5. Next we state several propositions: (1) If x1, x2, x3, x4, x5 ...

(i) ∥x1, x2, . . . , xn∥ = 0 if any only if x1, x2, . . . , xn are linearly dependent, (ii) ∥x1, x2, . . . , xn∥ is invariant under any permutation, (iii) ∥x1, x2, . . . , axn∥ = |a| ∥x1, x2, . . . , xn∥, for any a ∈ R (real), (iv) ∥x1, x2, . . . , xn−1, y + z∥ = ∥x1, x2, . . . , xn−1, y∥ + ∥x1, x2, . . . , xn−1, z∥ is called an n-norm on X and the pair (X, ∥•, . . . , •∥) is called n-normed li...