(a) Let a > 1 be an integer. Prove that any composite divisor of a −...

(a) Let a > 1 be an integer. Prove that any composite divisor of a − 1 is a pseudoprime of base a.

(b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod m). Prove that if n is composite, then n is a pseudoprime of base a.

(c) Use (b) to give two examples pseudoprimes of base a with a = 2 and a = 3 (hint: take m = 2k to be an even number).

Expert Solution

A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime to describe all probable primes, both composite numbers and actual primes.

Colby Messinger answered 2 years ago

Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

Let a and b be positive integers, and let d be their greatest common divisor. Prove...

Let a and b be positive integers, and let d be their greatest common divisor. Prove that there are infinitely many integers x and y such that ax+by = d. Next, given one particular solution x0 and y0 of this equation, show how to find all the solutions.

a. Let A be a square matrix with integer entries. Prove that if lambda is a...

a. Let A be a square matrix with integer entries. Prove that if lambda is a rational eigenvalue of A then in fact lambda is an integer. b. Prove that the characteristic polynomial of the companion matrix of a monic polynomial f(t) equals f(t).

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...

Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.

For any nonnegative integer n, let Y1 < Y2 < · · · < Y2n+1 be...

For any nonnegative integer n, let Y1 < Y2 < · · · < Y2n+1 be the ordered statistics of 2n + 1 independent observations from the uniform distribution on [−2, 2]. (i) Find the p.d.f. for Y1 and the Y2n+1. (ii) Calculate E(Yn+1). Use your intuition to justify your answer.

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F . Show further that the smallest positive integer with this property is a prime number.

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime...

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime number.

8.Let a and b be integers and d a positive integer. (a) Prove that if d...

8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

show that an integer n > 4, is prime iff it is not a divisor of...

show that an integer n > 4, is prime iff it is not a divisor of (n-1)!

Question