Recall Fermat's Little Theorem: For any prime p and integer a, ap−1≡mod p It happens that the converse to FLT is often but not always true. That is, if n is composite and a is an integer, then more often than not a^(n-1)≢1 mod n We can use this as the basis of a simple primality test, called the Fermat Test. For a∈Zn we make the following definitions. 1) We call a a Fermat Liar for n if a^(n−1)≡1 mod n , where a∉(0,1,n−1) 2) We call a a Fermat Witness for n if a^(n−1)≢1modn where a∉(0,1,n−1) If a number is composite, then 2 is very often a Fermat Witness. What is the smallest composite integer n greater than 72697269 for which 2 is not a Fermat Witness?
Recall Fermat's Little Theorem: For any prime p and integer a, ap−1≡mod p It happens that the converse to FLT is often but not always true. That is, if n is composite and a is an integer, then more often than not a^(n-1)≢1 mod n We can use this as the basis of a simple primality test, called the Fermat Test. For a∈Zn we make the following definitions. 1) We call a a Fermat Liar for n if a^(n−1)≡1 mod n , where a∉(0,1,n−1) 2) We call a a Fermat Witness for n if a^(n−1)≢1modn where a∉(0,1,n−1) If a number is composite, then 2 is very often a Fermat Witness. What is the smallest composite integer n greater than 72697269 for which 2 is not a Fermat Witness?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
Related questions
Question
Recall Fermat's Little Theorem:
For any prime p and integer a, ap−1≡mod p
It happens that the converse to FLT is often but not always true.
That is, if n is composite and a is an integer, then more often than not a^(n-1)≢1 mod n
We can use this as the basis of a simple primality test, called the Fermat Test.
For a∈Zn we make the following definitions.
1) We call a a Fermat Liar for n if a^(n−1)≡1 mod n , where a∉(0,1,n−1)
2) We call a a Fermat Witness for n if a^(n−1)≢1modn where a∉(0,1,n−1)
If a number is composite, then 2 is very often a Fermat Witness.
What is the smallest composite integer n greater than 72697269 for which 2 is not a Fermat Witness?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 19 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,