1. If g is a primitive root of p, show that two consecutive powers of g have consecutive least residues. That is, show that there exists k such that g^(k+1)=g^k+1(mod p) (Fibonacci primitive root)2. Show that if p=12k+1 for somek , then (3/p)=13. Show that if a is aquadratic residue (mod p) and ab=1(mod p) then b is a quadratic residue (mod p)4. Suppose that p=1+4a, where p and q are odd primes. show that (a/p)=(a/q).Im attaching the proofs to all the in .docx and .pdf formats.Please see the attached file for the complete solution response.1. If g is a primitive root of p, show that two consecutive powers of g have consecutive least residues; that is, show that there exists a k such that (please see the attached file).Proof: Note that the property does not hold for (please see the attached file) since the only primitive root of 2 is 1 and (please see the attached file).
Now suppose p is an odd prime and g is a primitive root of p. Then g is a least residue modulo p and the order of g modulo p is (please see the attached file) that is, (please see the attached file) and the powers of g, (please see the attached file) are a permutation of least residues 1, 2, …, modulo p. In other words, if r is a nonzero least residue modulo p, then (please see the attached file) for some positive integer k.Now consider the linear congruence (please see the attached file). Since (please see the attached file) it follows that (please see the attached file). So (please see the attached file). Thus there exist integers a and b such that (please see the attached file)So there exists some nonzero least residue r such that (please see the attached file). But then (please see the attached file) for some positive integer k. Thus we have:
(please see the attached file).Therefore, for every odd prime p, there exists a k such that (please see the attached file).Note: If g is a primitive root of p and satisfies the congruence (please see the attached file) then g is called a Fibonacci primitive root of p since the roots of the equation (please see the attached file) are related to the Fibonacci sequence.2. Show that if (please see the attached file) for some k, then (please see the attached file). Proof: Let p be a prime such that (please see the attached file) for some k. Lets use the quadratic reciprocity theorem.
Theorem. The Quadratic Reciprocity Theorem.
If p and q are odd primes and (please see the attached file) then (please see the attached file). Otherwise,
(please see the attached file)Since (please see the attached file) it follows by the theorem, that (please see the attached file).
Now (please see the attached file) and so (please see the attached file). So p is a quadratic residue modulo 3. Therefore,
(please see the attached file).3. Show that if a is a quadratic residue (mod p) and (please see the attached file) then b is a quadratic residue (mod p).Proof: Let a is a quadratic residue (mod p) and suppose (please see the attached file). Then there exists a least residue c modulo p such that (please see the attached file). Since (please see the attached file) it follows that (please see the attached file). Then since (please see the attached file) the congruence (please see the attached file) has a solution. Thus there exist some least residue d modulo p such that (please see the attached file). Then we have
(please see the attached file)
So
(please see the attached file)Therefore, b is quadratic residue (mod p).
…(please see the attached file)…




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