Class Note for MATH 294A at UA
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.
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Date Created: 02/06/15
Elementary Number Theory Cam McLeman October 26 2006 One of number theoryls claims to fame is the unusual ease with which one can pose exceedingly dif cult 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 theoremi Instead there are a select few elementary resultstopics that can cover a wide range of possible questions 0 Modular arithmetici o Eulerls theorem in particular Fermatls Little Theorem 0 Prime factorization gcdls and divisibilityi Example 1 Show that the sequence 11111111111111 contains no perfect squares Example 2 Prove that the fraction 21n4 14n3 is irreducible for every positive integer no Example 3 What are the last two digits of 3 32006 Example 4 Suppose that the number of prime divisors of a positive integer n is a prime number p which does not divide no Show that n is one more than a multiple of pi Problem 1 If 2nl and 3nl 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 7 Problem 3 Find all positive integers d such that d divides both n2 l and n 12 1 for some n Problem 4 For any prime p prove that every prime divisor of 2p 7 l is at least pi Problem 5 Prove that for any integers m and n7 the quantity god m7 n n n m is an integer Problem 6 Let pk denote the kth prime number Show that pk lt 22 Problem 7 Prove that for any integer k the number n 9k 2006 1 cannot be expressed in the form n 12 y2 22 for any integers z y and 2 Problem 8 Count the number of pairs of positive integers 17y such that 1 1 1 z 3 2006 Problem 9 Find the sum of the digits of the sum of the digits of the sum of the digits of 2006200 Problem 10 Show that for any prime p the number 21 31 is never a perfect power greater than 1 of an integer Problem 11 Show there are no nontrivial ie other than I y 2 0070 integer solutions to the equation 13 SyS 923 7 QIyz 0 Problem 12 For a given positive integer m7 nd all triples nzy of positive integers With m and n relatively prime7 Which satisfy the relation 12 y2m WW
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