[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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[Number Theory] How do you solve question 2? The second picture is for definitions. Thank you! 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)
[Number Theory] How do you solve question 3? The second picture is for definitions. Thank you! 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)
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Reading Question 6.2.1. Let G be a cyclic group of order 20. Which of the following claims is true? Select all that apply. (a) G has 5 elements of order 4. (b) G has 4 elements of order 5. (c) G has 1 element of order 2. (d) G contains 10 generators. (e) If a is a generator of G, then the order of ak is equal to the gcd(20, k).
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