math 420  FINAL EXAM  Wednesday December 15, 1993

Do any seven problems.

(15) 1. Prove that if a ≠ 1 then a^{3} + 1 is not prime.

(15) 2. Prove that if (a, b) = 1 then b ∤ a − b.

(20) 3. Prove that if 2^{m} − 1 is prime then 2^{m−1}(2^{m} − 1) is a perfect number.

(20) 4. Find the continued fraction expansions for the following real numbers: a) 87/38

b) \sqrt{28}

(30) 5. Prove that if n is a composite number then there exists a prime divisor p of n such that p ≤ \sqrt{n}.

(30) 6. Use Fermat’s Little Theorem to prove that if p is a prime number larger than 17 then p^{16} has the form 51k + 1.

(30) 7. a) State the Well-ordering Principle for the natural numbers.

b) Use the Well-ordering Principle and the Division Algorithm to prove that if d = (a, b) then there exist integers x and y such that ax + by = d.

(30) 8. Use the Well-ordering Principle to prove that the simple continued fraction expansion for any rational number is finite. (hint: Consider the denominator.)

(35) 9. Let q be prime number of the form 4k + 1.

a) Explain why there exist primes p such that p does not divide x^{2} − q for any x.

b) Determine those primes p which divide x^{2} − 5 for some x.

(40) 10. a) Let p be an odd prime and let a ≢ 0 (mod p). Prove that the congruence x^{2} ≡ a (mod p) has either no solutions or exactly two solutions c such that 0 < c < p.

b) Prove that if p is an odd prime then exactly half the numbers between 1 and p − 1 are quadratic residues modulo p.