计算数论作业代写 Mth 440/540代写 数学作业代写 数学代写

Mth 440/540 Homework

计算数论作业代写 (1) Part I a.) Use Mathematica to determine the 100th prime, the 1000th prime, and the 10,000th prime. b.) Use Mathematica to determine the prime

(1) Part I

a.) Use Mathematica to determine the 100th prime, the 1000th prime, and the 10,000th prime.

b.) Use Mathematica to determine the prime factorization of 2022.

c.) Mathematica uses PrimePi[x] to denote the function π(x), giving the number of primes less than or equal to x. It uses LogIntegral[x] to denote the logarithmic integral li(x).

Using Mathematica, fifirst plot on the same graph π(x), li(x), and x/ log(x) for x ∈ [2, 100], then repeat with x ∈ [2, 10000].

(2) Part II

1. Suppose m ∈ N and a, b ∈ Z. Show that if a ≡ b (mod m) then gcd(a, m) = gcd(b, m).

2. Fix n ∈ N and suppose a, b ∈ Z such that gcd(a, b) = 1. Show that there exists some x ∈ Z such that gcd(ax+b, n) = 1.

(Hint: as a first step consider the prime divisors p_i of n, and thereafter use the Chinese Remainder Theorem.)

3. Prove that a natural number n is prime if and only if Ø(n) = n − 1.

(3) Part III 计算数论作业代写

For this, your web browser will need to invoke the OSU library proxy. Open this OSU library page and click on MathSciNet there. This should bring you to http://www.ams.org/mathscinet/search after perhaps requiring that you sign in with your OSU account.

1.) In the Title box, type “Chinese remainder theorem”, then click on Search. How many articles with these words in the title are reviewed?

2.) Go back to the search page. Find the review of an article entitled A centennial history of the prime number theorem.

What is the final sentence of the review?

3) Go back to the search page. Delete the title entry, and in-sert “shoup, v*” in the author field. Find the review of (the first edition) the textbook we are using. What is the first sentence of the fourth paragraph of the review?

4) Now choose your own topic and use the Search to find something of interest. Explain.

(4) Part IV. 计算数论作业代写

You must clearly comment your code — any fellow student should be able to understand the logic of your code. For functions that you write, clearly indicate what type of input is expected and the form of the output.

1. Find the first 20 primes found by the classical proof of the infinitude of the set of primes. That is: begin with P = {2};

then form, m, the sum of 1 with the product over all elements of P. Place the smallest prime factor of m into P and repeat.

(For this problem, you may wish to use FactorInteger.)

2. A Wilson prime is a prime p such that

(p − 1)! ≡ −1 mod p² .

Write a procedure which determines all Wilson primes less than 10⁴ .

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