初等数论练习

作业次数：学号姓名作业成绩

第0章序言及预备知识

第一节序言（1）

1、数论人物、资料查询：（每人物写60字左右的简介）

（1）华罗庚

2、理论计算与证明：

（1

（2）Show that there are infinitely many Ulam numbers

3、用Mathematica数学软件实现

A Ulam number is a member of an integer sequence which was devised by Stanislaw Ulam

and published in SIAM Review in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with U1=1 and U2=2 being the first two Ulam numbers. Then for n > 2, U n is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way 。

By the definition, 3=1+2 is an Ulam number; and 4=1+3 is an Ulam number (The sum 4=2+2 doesn't count because the previous terms must be distinct.) The integer 5 is not an Ulam number because 5=1+4=2+3. The first few terms are

1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99

（1）Find the first 200 Ulam numbers

（2）What conjectures can you make about the number of Ulam numbers less than an integer n？

Do your computations support these conjetures？

作业次数： 学号 姓名 作业成绩

第2节 序言（2）

1、数论人物、资料查询：（每人物写60字左右的简介）

（2）陈景润

2、理论计算与证明：

（1）用数学归纳法证明：!n n n ≤

（2）用数学归纳法证明：2!(4)n n n ≤≥

3、用Mathematica 数学软件实现

The 3x + 1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's

algorithm , and Ulam's problem , concerns the behavior of the iterates of the function which takes odd integers n

to 3n+1 and even integers n to 2

n . The 3x+1 Conjecture asserts that, starting from any positive integer n , repeated iteration of this function eventually produces the value 1.

参考文献：Jeffrey C. Lagarias, "The 3x+1 problem and its generalizations".

作业次数： 学号 姓名 作业成绩

第3节 预备知识

1、数论人物、资料查询：（每人物写60字左右的简介）

（1）王小云（山东大学）

（2）

The tower of Hanoi

2、理论计算与证明： （1）设n f 是第n 个Fabonacci 数，11F 10⎛⎫= ⎪⎝⎭，则1n 1F n n n n f f f f +-⎛⎫= ⎪⎝⎭

（2）求证：212232122...n n n f f f f f f f -+++=

3、用Mathematica 数学软件实现

（The tower of Hanoi puzzle ）

The Tower of Hanoi or Towers of Hanoi is a mathematical game or puzzle. It consists of three

rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with

the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making

a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following

rules:

∙

Only one disk may be moved at a time.

∙ Each move consists of taking the upper disk from one of the rods and sliding it onto

another rod, on top of the other disks that may already be present on that rod.

∙ No disk may be placed on top of a smaller disk 参考文献：[1]、http://wipos.p.lodz.pl/zylla/games/hanoi5e.html

[2]、/wiki/Tower_of_Hanoi