Math 350 Practice Midterm, Exams of Number Theory

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MATH 350 (NUMBER THEORY) PRACTICE MIDTERM

Note: this exam consists of 5 questions. Do them all, and provide justifications for your asser- tions unless it is specified that this is not necessary.

Name:

Say whether the following statements are true or false (a proof is not required).

True False

(a) Let a, b, c be integers such that gcd(a, b) = 1. If c^2 = ab then a and b are themselves squares.

(b) Let p, q be distinct prime numbers. Then φ(pq) = φ(p)φ(q).

(c) There are infinitely many prime numbers p such that p ≡ 2 (mod 4).

(d) If p is a prime number, then 2 p^ − 1 is a prime number.

(e) The product of two sums of two squares (u^2 + v^2 )(s^2 + t^2 ) can be written as a sum of two squares.

Explain in your own words how the Euclidean algorithm can be used to compute the greatest common divisor gcd(a, b) of two integers a and b, and to express gcd(a, b) as a linear combi- nation ax + by for some integers x, y.

Are there any Carmichael numbers that have only two prime factors? Either find an example or prove that none exists.