Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Prove that for any integerp>1,ifrole="math" localid="1663222073626" pisn’tpseudoprime, thenpfails the Fermat test for at least half of all numbers inΖp+

Short Answer

Expert verified

To solve the problem understanding the concepts of subgroup i.e., it is required to show that it is nonempty and closed under the inverses and multiplication.

Step by step solution

01

Understanding the given concept 

To show that the set is a subgroup, it is required to show that it is nonempty and closed under the inverses and multiplication.

02

To prove that elements set inΖp+ that pass the Fermat test forms multiplicative subgroup ofΖp+

Since the subgroup order divides the group order, if subgroup is a strict subgroup, it must contain at most half of elements of group.

First, the set is nonempty, since.1p-11modp

03

To prove if p is not pseudoprime then p fails Fermat Test for at least half of number 

Ifap-11(modp) and, then(ab)p-1ap-1bp-11modp , which shows closure under multiplication.

If ,ap-11(modp) then multiplying both sides of the equation by (a-1)p-1the shows that. 1(a-1)p-1modpThus, the set is closed under inverses.

Hence, on the other side if is not pseudo prime then fails Fermat Test for at least half of number.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

See all solutions

Recommended explanations on Computer Science Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free