[Number Theory] How do you solve question 1? The second picture is for definitions. 1. Evaluate the following Legendre symbols:2. Use Gauss' Lemma,(i)to compute(ii)(iii)In (ii) p is an odd prime, p ‡ 7; in (iii) p is an odd prime p ‡ 3. In (ii) express the answerin terms of a congruence condition on p mod 28, and in (iii) express the answer in terms ofa congruence condition on p mod 24, analogous to= 1 if p = 1 mod 4 and -1 if p = 3 mod 4.463541= (-1)",4. For an odd prime p, suppose that3189331047293. For an odd prime p≥ 7, show that there are consecutive quadratic residues mod p, i.e.,that there exists an element a € Up such thatμ = |aPN,a(²) - ( ² + ¹) -=Hint: First show that at least one of the numbers 2, 5 and 10 is a quadratic residue mod p.x =a = 1 and consider the congruence= 1.=x² = a mod p.(i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence aretak+1or x =1(ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence arex = ±ak+¹,+2²k+1 k+1aExplain how to see which is the correct choice.(iii) Use the results of (i) and (ii) to find the solutions to the congruencesx² = 23 mod 751x² = 15 mod 797 Example 7.2If n = 7 then we can take =9 3 as a primitive root. The powers of g in U7are g = 3, g² = 2, g³ = 6, gª = 4, g5 = 5 and g6 = 1; of these, the quadraticresidues a = 1,2 and 4 correspond to the even powers of g. Thus Q7 is thecyclic group of order 3 generated by g² = 2.7.3 The Legendre symbolWe now consider the problem of determining whether or not a given elementa € Un is a quadratic residue. Unfortunately, Lemma 7.3 is not very effectivehere: Un is not always cyclic, and even when it is, it can be difficult to find aprimitive root g and then express a as a power of g (see Chapter 6). We thereforeneed more powerful techniques. Quadratic residues are easiest to determine inthe case of prime moduli; the case n = 2 is trivial, so we assume for the timebeing that n is an odd prime p. The following piece of notation greatly simplifiesthe problem of determining the elements of Qp:DefinitionFor an odd prime p, the Legendre symbol of any integer a isif pla,if a € Qp,if a € Up\Qp.0Clearly this depends only on the congruence class of a mod (p), so we canregard it as being defined either on Z or on Zp.09-{= 11-1Example 7.3Let p = 7. Then as in Example 7.2 we have-1if a = 0 mod (7),if a 1,2 or 4 mod (7),if a = 3,5 or 6 mod (7).

Question

[Number Theory] How do you solve question 1? The second picture is for definitions. 1. Evaluate the following Legendre symbols:2. Use Gauss' Lemma,(i)to compute(ii)(iii)In (ii) p is an odd prime, p ‡ 7; in (iii) p is an odd prime p ‡ 3. In (ii) express the answerin terms of a congruence condition on p mod 28, and in (iii) express the answer in terms ofa congruence condition on p mod 24, analogous to= 1 if p = 1 mod 4 and -1 if p = 3 mod 4.463541= (-1)&#34;,4. For an odd prime p, suppose that3189331047293. For an odd prime p≥ 7, show that there are consecutive quadratic residues mod p, i.e.,that there exists an element a € Up such thatμ = |aPN,a(²) - ( ² + ¹) -=Hint: First show that at least one of the numbers 2, 5 and 10 is a quadratic residue mod p.x =a = 1 and consider the congruence= 1.=x² = a mod p.(i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence aretak+1or x =1(ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence arex = ±ak+¹,+2²k+1 k+1aExplain how to see which is the correct choice.(iii) Use the results of (i) and (ii) to find the solutions to the congruencesx² = 23 mod 751x² = 15 mod 797 Example 7.2If n = 7 then we can take =9 3 as a primitive root. The powers of g in U7are g = 3, g² = 2, g³ = 6, gª = 4, g5 = 5 and g6 = 1; of these, the quadraticresidues a = 1,2 and 4 correspond to the even powers of g. Thus Q7 is thecyclic group of order 3 generated by g² = 2.7.3 The Legendre symbolWe now consider the problem of determining whether or not a given elementa € Un is a quadratic residue. Unfortunately, Lemma 7.3 is not very effectivehere: Un is not always cyclic, and even when it is, it can be difficult to find aprimitive root g and then express a as a power of g (see Chapter 6). We thereforeneed more powerful techniques. Quadratic residues are easiest to determine inthe case of prime moduli; the case n = 2 is trivial, so we assume for the timebeing that n is an odd prime p. The following piece of notation greatly simplifiesthe problem of determining the elements of Qp:DefinitionFor an odd prime p, the Legendre symbol of any integer a isif pla,if a € Qp,if a € Up\Qp.0Clearly this depends only on the congruence class of a mod (p), so we canregard it as being defined either on Z or on Zp.09-{= 11-1Example 7.3Let p = 7. Then as in Example 7.2 we have-1if a = 0 mod (7),if a 1,2 or 4 mod (7),if a = 3,5 or 6 mod (7).

Accepted Answer

Answered: Image /qna-images/answer/4c35e099-8c54-4f31-84fb-5afbd22c5eb0.jpg

1. Evaluate the following Legendre symbols: (i) (ii) р 463 541 (iii) 8933 104729 (-).