Math 706, Theory of Numbers Final Exam Review

The Final Exam consists of two parts as indicated below. You have one hour and 50 minutes. No notes, books or calculators are permitted. You may take the final exam any time during the semester after you have completed the midterm and at least six homework assignments. It must be taken before the last day of final exam week. Part I: Thirteen short answer problems similar to the type below and the type on the midterm review. (52 points) Part II: Eight computational/proof problems, from which you are to select four problems to turn in. Additional problems count as extra credit. (48 points)

1 Short answer and computational problems

Chapters 1 to 5, on the Midterm Review, are part of the Final Exam Review. Additional problems are listed below.

Chapter 5 a  1. Give the definition of the Jacobi symbol b , and state for what values of a, b it is defined.

2. State the generalized version of that applies to the Jacobi symbol.

3. Determine whether the following numbers can be expressed as a sum of two squares and explain why or why not? a) 233457. b) 1010 − 1.

1 Chapter 6

4. If 2n + 1 is a prime then what can be said about n? If 2n − 1 is a prime, what can be said about n?

5. How can Fermat’s Little Theorem be used to obtain a very simple way of showing that a number m is composite?

6. A prime p is called a Mersenne/Fermat prime if it has what form?

7. A positive m is called a pseudoprime to the base b if what condi- tions are satisfied?

Chapter 7

8. Let φ(n) be the Euler phi-function and τ(n) the number of of n, and σ(n) the sum of the divisors of n. Evaluate φ(1000), τ(1000), σ(1000).

9. State the definition of the M¨obiusfunction µ(n).

10. State the M¨obius inversion formula.

11. Define what it means for an arithmetic function f to be multiplicative.

12. Let f be a multiplicative function with f(p) = 2 for all primes p and F P e e be defined by F (n) = d|n µ(d)f(d). Find F (p ) for any p .

2 13. Determine the multiplicity of 7 dividing 4900! (4900 factorial).

14. If n is an even perfect number then n has what form?

15. If f and g are multiplicative functions, which of the following are also P P multiplicative? fg, f + g, F (n) := d|n f(d), G(n) := d|n f(n/d)g(d), P H(n) := d|n dg(d).

Chapter 8

I won’t put any problem from this chapter on Part I. Part II will include an optional problem such as the following.

16. Derive the formula for the n-th Fibonnaci number. (F1 = F2 = 1, Fn = Fn−1 + Fn−2 for n ≥ 3.)

0 1 F F  17. Let A = . Prove that for any n, An = n−1 n . 1 1 Fn Fn+1

18. Prove that if d|n then Fd|Fn.

2 2 19. Prove that for any positive n, Fn + Fn+1 = F2n+1.

20. Find the value of the n-th Fibonnaci number Fn (mod 19), and determine the Pisano period π(19). (Given: 92 ≡ 5 (mod 19), 9−1 ≡ −2 (mod 19), ord19(15) = 18.)

3 Chapter 9

I won’t put any problem from this chapter on Part I. Part II will include an optional problem such as the following. 21. Determine for what a, b the following system is solvable, and find the solution set. 4x + 6y − 2z = a 2x − 8y + 10z = b

22. Determine all rational points on the hyperbola x2 − 7y2 = 1.

23. Find an infinite family of primitive solutions of the equation x2 + y2 = z4.

Chapter 10

I won’t put any problem from this chapter on Part I. Part II will include an optional problem such as the following. 24. i) Explain why the equation y2 = x3 + 4 determines an elliptic curve. ii) Let G denote the group of real points on the elliptic curve y2 = x3 + 4, and P be the point P = (0, 2). Find the of P in G.

25. Let G be the group of rational points on the elliptic curve y2 = x3 − x + 1 with zero element P∞. Let P = (1, 1) and Q = (3, 5). Find P + Q and 2P .

26. Explain the Pollard (p − 1)-method for factoring a positive integer m.

2 Theorems you should be able to prove

Theorem 2.1. GCDLC Theorem. The greatest common of two a, b can be expressed as a linear combination of a, b with integer

4 coefficients. Lemma 2.1. Euclid’s Lemma. Suppose that d|ab and that (d, a) = 1. Then d|b.

Theorem 2.2. The Fundamental Theorem of Arithmetic (FTA). Z is a Unique Factorization Domain. Theorem 2.3. A few classical results in . (i) Fermat’s Little Theorem: For any prime p and integer a with (a, p) = 1, ap−1 ≡ 1 (mod p). Equivalently, ap ≡ a (mod p) for any integer a. (ii) Euler’s Theorem: For any m ∈ N and integer a with (a, m) = 1, aφ(m) ≡ 1 (mod m). (iii) Wilson’s Theorem: For any prime p, (p − 1)! ≡ −1 (mod p). Lemma 2.2. If d|(p − 1) then the polynomial xd − 1 has d distinct zeros in Fp. Theorem 2.4. Chinese Remainder Theorem. Algebraic Version. Let m1, m2, . . . , mk be pairwise relatively prime integers (that is (mi, mj) = 1 for all i 6= j), and let m = m1m2 ··· mk. Then we have the ring isomor- phism, Z/(m) ' Z/(m1) × · · · × Z/(mk). Theorem 2.5. Structure Theorem for G(m). Let m be a positive integer e1 ek with prime factorization m = p1 . . . pk . Then

e1 e2 ek G(m) ' G(p1 ) × G(p2 ) × · · · × G(pk ), as multiplicative groups. Corollary 2.1. Properties of the Euler Phi Function. (i) φ is multiplicative, that is, if a, b are integers with (a, b) = 1 then φ(ab) = φ(a)φ(b).   ei ei ei−1 1 (ii) If m = Πip then φ(m) = Πi(p − p ) = mΠi 1 − . i i i pi Theorem 2.6. For any prime p, G(p) is a cyclic group. Theorem 2.7. Primitive element theorem. If p is an odd prime, and a is a primitive element (mod p), with

ap−1 = 1 + kp, and p - k, then a is a primitive element (mod pn) for all n ∈ N. (On a test, I would just ask you to prove that a is a primitive root (mod p2).) Theorem 2.8. Euler’s Criterion. If p is an odd prime and (a, p) = 1 p−1 then a is a (mod p) if and only if a 2 ≡ 1 (mod p).

5 Theorem 2.9. Lucas’ Primality Test Let m > 1 . Suppose that there exists an integer a > 1 such that

(i) am−1 ≡ 1 (mod m),

(ii) ad 6≡ 1 (mod m), for every proper divisor d of m − 1. Then m is a prime. Lemma 2.3. Gauss’ Lemma. Let p be an odd prime and a be a posi- p−1 tive integer with p - a. Consider the numbers a, 2a, 3a, . . . , 2 a reduced p−1 p−1 (mod p) to values between − 2 and 2 . Let ν denote the number of  a  ν negative values. Then p = (−1) . Theorem 2.10. For any odd prime p, ( 2 1, if p ≡ ±1 (mod 8); = p −1, if p ≡ ±3 (mod 8). P Theorem 2.11. If f is multiplicative and F is defined by F (n) = d|n f(d), then F is also multiplicative. Theorem 2.12. An even number n is perfect if n is of the form n = 2e(2e+1 − 1), for some positive integer e such that 2e+1 − 1 is a prime (a Mersenne prime). (The converse is true for even numbers, but I won’t ask you to prove it on an exam.) Theorem 2.13. (a) For positive integers n, ( X 1 if n = 1 µ(d) = 0 if n > 1. d|n

(b) For any positive integer n,

X µ(d) φ(n) = . d n d|n P Theorem 2.14. For any natural number n, d|n φ(d) = n. Theorem 2.15. The M¨obiusinversion formula. If f is any arithmetic P function and F is defined by F (n) = d|n f(d) for n ∈ N , then for any n ∈ N, X f(n) = F (d)µ(n/d). d|n

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