Theory of Numbers Problems


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1 Theory of Numbers Problems AntoniosAlexandros Robotis Robotis October First Set 1. Find values of x and y so that 71x 50y = Prove that if n is odd, then n 2 1 is divisible by 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 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 Prove that any fourth power must have one of 0, 1, 5, 6 as its units digit. 3. Solve 3x 5 mod 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 Prove: If p is a prime number, then (p 1)! 1 mod p. (This result is sometimes nown as Wilson s Theorem.) 1
2 7. For any prime p, if a p b p mod p, show that a p b p mod p 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 and ? 6. Solve the congruence x 3 9x x 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
3 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) = 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 Show that if σ(q) = q + and q and < q then = 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) = Calculate j=1 µ(j!) or show that the sum diverges. 3
4 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 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 Show that 3 is a primitive root modulo Show that if ord p (a) = h, then ord p (a 1 ) = h. 7 Seventh Set 1. Calculate the number of primitive roots modulo = What is the remainder when the prime number is divided by 11? What is the remainder when is divided by ? 3. Find an inverse to the matrix ( ) modulo Suppose a and b are positive integers and a 4 b 3. Prove that a b. 4
5 5. Suppose n is a positive integer. Calculate lim n +1 σ(n ). 6. Show that n 4 + n is composite for all n 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
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