[Number Theory] How do you solve question 3? The second picture is for definitions. thanksAdditional information:We say that a group G is abelian if its elements commute, that is, gh = hg for all g, h in GUn is an abelian group under multiplication mod (n) 1. (i) Show by hand that 2 is a primitive root mod 19.(ii) Find² a complete set of primitive roots mod 19.2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not aprimitive root mod p.3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and onlyif ord, (2)| (q-1) and ord, (2)| (p-1). Here, if gcd(a, p) = 1, we use the notation ord, (a)for the order of the element [a], in the group Up.4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given byLemma 6.4 and a calculator. Conclude that 2 is a primitive root mod 11e for any e.(ii) How many elements of order 5 are there in the group U1331? Note that 1331 = 11³.(iii) Find all elements of order 5 in U1331.¹1.e., do not use a calculatorHere you may use a calculator to compute residues mod 19. DefinitionIf Un is cyclic then any generator g for Un is called a primitive root mod (n).This means that g has order equal to the order (n) of Un, so that the powersof g yield all the elements of Un. For instance, 2 and 3 are primitive rootsmod (5), but there are no primitive roots mod (8) since Us is not cyclic.Finding primitive roots in Un (if they exist) is a non-trivial problem, andthere is no simple solution. One obvious but tedious method is to try each ofthe (n) units a E Un in turn, each time computing powers a' mod (n) to findthe order of a in Un; if we find an element a of order (n) then we know thatthis must be a primitive root. The following result is a rather more efficienttest for primitive roots:Lemma 6.4An element a E Un is a primitive root if and only if a(n)/9 ‡ 1 in Un for eachprime q dividing (n).

We shall be using the fact that a^p-1 =1 mod p

3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only if ord, (2)| (q-1) and ord, (2) | (p-1). Here, if gcd(a, p) = 1, we use the notation ord,(a) for the order of the element [a], in the group Up.

3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only if ord, (2)| (q-1) and ord, (2) | (p-1). Here, if gcd(a, p) = 1, we use the notation ord,(a) for the order of the element [a], in the group Up.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

[Number Theory] How do you solve question 3? The second picture is for definitions. thanks

Additional information:

We say that a group G is abelian if its elements commute, that is, gh = hg for all g, h in G
Un is an abelian group under multiplication mod (n)

1. (i) Show by hand that 2 is a primitive root mod 19.
(ii) Find² a complete set of primitive roots mod 19.
2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not a
primitive root mod p.
3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only
if ord, (2)| (q-1) and ord, (2)| (p-1). Here, if gcd(a, p) = 1, we use the notation ord, (a)
for the order of the element [a], in the group Up.
4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given by
Lemma 6.4 and a calculator. Conclude that 2 is a primitive root mod 11e for any e.
(ii) How many elements of order 5 are there in the group U1331? Note that 1331 = 11³.
(iii) Find all elements of order 5 in U1331.
¹1.e., do not use a calculator
Here you may use a calculator to compute residues mod 19.

Definition
If Un is cyclic then any generator g for Un is called a primitive root mod (n).
This means that g has order equal to the order (n) of Un, so that the powers
of g yield all the elements of Un. For instance, 2 and 3 are primitive roots
mod (5), but there are no primitive roots mod (8) since Us is not cyclic.
Finding primitive roots in Un (if they exist) is a non-trivial problem, and
there is no simple solution. One obvious but tedious method is to try each of
the (n) units a E Un in turn, each time computing powers a' mod (n) to find
the order of a in Un; if we find an element a of order (n) then we know that
this must be a primitive root. The following result is a rather more efficient
test for primitive roots:
Lemma 6.4
An element a E Un is a primitive root if and only if a(n)/9 ‡ 1 in Un for each
prime q dividing (n).

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