Solving Putnam Problems with Number Theory Techniques, Study notes of Mathematics

Download Solving Putnam Problems with Number Theory Techniques and more Study notes Mathematics in PDF only on Docsity! Elementary Number Theory Cam McLeman October 26, 2006 One of number theory’s claims to fame is the unusual ease with which one can pose exceedingly difficult problems, and the enormity of the full toolkit needed to tackle all of these problems. Fortunately, the good news is that solving a Putnam problem is rarely about having memorized the applicable theorem. Instead, there are a select few elementary results/topics that can cover a wide range of possible questions: • Modular arithmetic. • Euler’s theorem (in particular Fermat’s Little Theorem). • Prime factorization, gcd’s, and divisibility. Example 1. Show that the sequence 11, 111, 1111, 11111, · · · contains no perfect squares. Example 2. Prove that the fraction 21n + 4 14n + 3 is irreducible for every positive integer n. Example 3. What are the last two digits of 33 2006 ? Example 4. Suppose that the number of prime divisors of a positive integer n is a prime number p which does not divide n. Show that n is one more than a multiple of p. Problem 1. If 2n+1 and 3n+1 are both perfect squares, show that n is divisible by 40. Problem 2. How many trailing zeros are at the end of the decimal expansion of( 150 72 ) ? Problem 3. Find all positive integers d such that d divides both n2 + 1 and (n + 1)2 + 1 for some n. Problem 4. For any prime p, prove that every prime divisor of 2p − 1 is at least p. 1