On the integer represented as the product of k prime numbers in arithmetic progression;

This paper presents the number n of representations of the nonzero integer z∈Z-2 as the difference of two squares of integers in Z-2,and the number n of representations of the nonzero integer z∈Z-2 as(z=(x~2+2y~2)),where x,y∈x~2+2y~2.

This paper utilizes the Reciprocal Law Two Times to prove that the rational number of a kind of form is not the integer,and proves that the number of another kind of prime is limitless.

The paper uses the tools about algebraic number theory to find a class of quartic algebraic integer ±p~(1/2)±q~(1/2),then,it is determined and proved that their minimal polynomial is [x2-(p+q)]2-4pq,and in their normal closure,there are four real inserts and no complex inserts.

Smyth proposed the following problem: Let r≥0 be a given integer, one tries to find all totally positive algebraic integers a which satisfya) Tr(α) - deg(α) = r;b)α_i >0, i = 1,…,d,whereα_i are the conjugates ofα(setα_1 = a), Tr(α) = a_1 +α_2+…+α_d is the trace ofα, and d = deg(α) is the degree of its minimal polynomial.

Let beαalgebraic integer of degree d, not 0 or a root of unity, all of whose conjugatesα_i are confined to a set S_θ= {α_i∈C : |arg(α_i)|≤θ}, 0 <θ< (?), i = 1,2,…, d.