Theory of Numbers Problems Antonios-Alexandros Robotis Robotis October 2018 1 First Set 1. Find values of x and y so that 71x 50y = 1. 2. Prove that if n is odd, then n 2 1 is divisible by 8. 3. Define a valuation on Q \ {0} which extends the p adic valuation defined in class. 4. Let Q[x] denote the set of polynomials in a variable x with coefficients in Q Define what it means for one polynomial p(x) Q[x] to divide another polynomial q(x) Q[x]. Define what it means for a polynomial to be prime as an element of Q[x]. 5. Prove that 2 n = n =0 ( n ). 6. Prove that an integer is divisible by 3 if and only if the sum of its digits is divisible by 3. 7. If x and y are odd integers, prove that x 2 + y 2 cannot be a perfect square. 2 Second Set 1. Find all integers x with 1 x 100 so that x 7 mod 17. 2. Prove that any fourth power must have one of 0, 1, 5, 6 as its units digit. 3. Solve 3x 5 mod 11. 4. Show that 2, 4, 6,..., 2m is a complete residue system if m is odd. 5. What is the last digit in the decimal representation of 3 400. 6. Prove: If p is a prime number, then (p 1)! 1 mod p. (This result is sometimes nown as Wilson s Theorem.) 1
7. For any prime p, if a p b p mod p, show that a p b p mod p 2. 8. Prove that (a + b) p = a p + b p mod p. 9. For m odd prove that the sum of the elements of any complete residue system modulo m is congruent to zero modulo m. 10. Show that if f(x) is a polynomial with integral coefficients, and if f(a) mod m, then f(a + tm) mod m, for every integer t. 3 Third Set 1. Provide a closed form formula for a function ϕ : N N which to each n N assigns the number ϕ(n) of natural numbers m coprime to n with m n. This function is called Euler s phi function, or Euler s Totient function. 2. Prove that the number of multiplicative residues modulo n is ϕ(n). 3. Notice that for p a prime ϕ(p) = p 1. So, for p a prime, we can rewrite Fermat s Little Theorem in the form a ϕ(p) 1 (mod p). Prove the generalization of Fermat s Little Theorem for general moduli: a ϕ(m) 1 (mod m). 4. Compute ϕ(3600). 5. What are the last two digits of 2 1000 and 3 1000? 6. Solve the congruence x 3 9x 2 + 23x 15 0 (mod 503). 7. Characterize the set of positive integers satisfying ϕ(2n) = ϕ(n). 4 Fourth Set 1. Prove that if n has distinct odd prime factors, then 2 ϕ(n). 2. If φ(m) = φ(mn) prove that n = 2 and m is an odd number. 3. Find the smallest positive integer x giving remainders 1, 2, 3, 4 and 5 when divided by 3, 5, 7, 9, and 11, respectively. 4. Prove that n 7 n is divisible by 42 for any choice of n N. 2
5. A group is a set G equipped with a multiplication map G G G, written (g, h) g h such that (Associativity) For all g, h, G, (g h) = g (h ). (Identity) There exists an element e G with the property that e g = g e = g for all g G. e is called the identity element of G. (Inverses) For each element g G, there exists an element g 1 with the property that g 1 g = g 1 g = e. Chec that the integers Z form a group with multiplication + : Z Z Z defined by (x, y) x + y. 6. Give three more examples of groups. 7. Prove that identity element of a group G is unique. 8. Prove that for any g G, its inverse g 1 is unique. 9. For which m N is the set of nonzero residue classes mod m a group under multiplication? 5 Fifth Set 1. Find the smallest x Z such that τ(x) = 6. 2. What is the number of irreducible positive fractions 1 with denominator 1 n, for N. 3. If f(n) and g(n) are multiplicative functions and g(n) 0 for every n, show that F (n) = f(n)g(n) and G(n) = f(n)/g(n) are also multiplicative. 4. Prove that an integer q is prime if and only if σ(q) = q + 1. 5. Show that if σ(q) = q + and q and < q then = 1. 6. We say that m is a perfect number if σ(m) = 2m, that is: if m is the sum of all its positive divisors besides itself. Suppose 2 n 1 1 = p is a prime. Show that 2 n 1 p is a perfect number. 7. If 100! is written out in decimal notation without the factorial sign (for instance 4! = 24, 5! = 120, etc), how many zeroes would there be in a row at the right end? 8. Suppose f is a multiplicative function. Show that F (n) = d n f(d) is a multiplicative function. 9. Find a positive integer n such that µ(n) + µ(n + 1) + µ(n + 2) = 2. 10. Calculate j=1 µ(j!) or show that the sum diverges. 3
6 Sixth Set 1. Recall that the order of a modulo m is the least N such that a 1 (mod m). This is denoted by ord m (a). Calculate the orders of 1, 2, 3, 4, 5, 6 modulo 7. 2. Prove that a function f : N N is multiplicative if and only if for all products of distinct prime powers p α1 1 pα, we have i=1 f(p α1 1 pα ) = f(p αi i ). 3. Prove that if a has order h modulo m, then { : a 1 (mod m) has solutions} = { : h }. 4. Prove that if gcd(a, m) = 1, then ord m (a) ϕ(m). 5. If ord m (a) = h, show that a has order h/ gcd(h, ). 6. Prove that if ord m (a) = h, ord m (b) =, and gcd(h, ) = 1, then ord m (ab) = h. 7. Prove that if ord m (a) = h, then 1, a, a 2,..., a h 1 are distinct modulo m. 8. Provide a one line proof that: if a (p 1)/2 1 (mod p) and a 1 for < p 1 2 then ord p (a) = p 1. 9. Show that 3 is a primitive root modulo 17. 10. Show that if ord p (a) = h, then ord p (a 1 ) = h. 7 Seventh Set 1. Calculate the number of primitive roots modulo 35035 = 5 7 2 11 13. 2. What is the remainder when the prime number 1234567891 is divided by 11? What is the remainder when 11 1234567890 is divided by 1234567891? 3. Find an inverse to the matrix ( ) 1 2 3 9 modulo 29. 4. Suppose a and b are positive integers and a 4 b 3. Prove that a b. 4
5. Suppose n is a positive integer. Calculate lim n +1 σ(n ). 6. Show that n 4 + n 2 + 1 is composite for all n 2. 7. Find n N such that n 3 is a perfect cube, n 4 is a perfect fourth power, and is a perfect fifth power. n 5 8. Let n Z. Prove that n is a difference of squares if and only if n is odd or divisble by 4. 5