The problem of the existence of infinitely many prime values of a number-theoretic function f (x) has been one of the most important topics in Number Theory. Note that if f (x) represents infinitely many primes, then we can get this necessary condition: for any positive integer h , there exists a positive integer k such that ( f (k), h) =1and f (k) >1. Naturally, we are interested in the number-theoretic functions f (x) that satisfy the aforementioned necessary condition. Thus, there must exist the least positive integer n such that (f(n),h)-1andf(k)>1. Denote this least positive integer n byF_f(h). In this paper, we mainly focus on three famous number-theoretic functions:&f(x)=2^{2^x}+1,m(x)=2^x-1 and l(x)=x^2+1& proving they satisfy the aforementioned necessary condition respectively. Furthermore, we approximately estimate the upper bound of F_f(x)(h),Fm(x)(h)，Fl(x)(h) respectively, and obtain some interesting results.

The traditional kaleidoscope uses the prism as a basic unit. Through the reflections of the mirrors on the inner surfaces of a prism, we can see many symmetric axles and patterns. However, traditional kaleidoscope only presents two-dimensional images. The three-dimensional kaleidoscope we designed uses the pyramid as a basic unit. We can see three-dimensional images via mirror faces on the inner surfaces of the pyramid. Our examination shows that not all pyramids are able to show three-dimensional images but only pyramids with specific angles do. The kaleidoscope we designed differs from the traditional kaleidoscope in the following ways: ①Our three-dimensional kaleidoscope not only shows three-dimensional space, but also shows it through two-dimensional planes. ②Our three-dimensional kaleidoscope is based on three kinds of pyramid: three-sided pyramid (tetrahedron), four-sided pyramid (square pyramid) and five-sided pyramid (pentagonal pyramid). Each kind of pyramid has its own charm in showing its reflected images. ③Each kind of pyramid is separated from its entire geometry. However, it can present the whole view of the geometry after mirrored. Our exploration shows that two dimensional and three dimensional worlds can change into one another under specific conditions. Perhaps before long we can discover more dimensions and explain the relationships between each kind of dimension. This treatise mainly involves the principles of Euclidean Geometry and Projective Geometry.

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A circle family is a group of separate or tangent circles in the plane. In this paper, we study how many parts at most a plane can be divided by several circle families if the circles in a same family must be separate (resp. if the circles can be tangent). We also study the necessary conditions for the intersection of two circle families. Then we primarily discuss the similar problems in higher dimensional space and in the end,raise some conjectures.

Lights Out is a video game, which was released by Tiger Toys in 1995. Generally, there are 3×3, 4×4, 5×5, 10×10 grids of games. Under the normal circumstances, it’s easy to deal with some special configurations manually. However, the possibility of solving the randomly-generated configuration by hand is small, for it’s a bit difficult to find the law. Out of interest and enthusiasm about the game, our team reads some reference and gathered four mathematic methods to solve the game. What’s more, we extend the methods to m×n games, the cycling game, the lit-only game and N-color game. We independently propose a method to classify all configurations in the game for 5×5, which can also be extended to general cases. We also put forward the issues about N-order hexagon games, and provide some feasible solutions.