Try the assignment f(1) = 2; f(2) = 1; f(3) = 4; f(4) = 3;.

Transcription

1 I. Precisely complete the following definitions: 1. A natural number n is composite whenever... See class notes for the precise definitions 2. Fix n in N. The number s(n) represents For A and B sets, we say f is a function from A to B when... II. Give an example of a function from N to N that is onto and assigns every natural number to a natural number different from itself. Explain why your function is onto in five [5] sentences or less. Try the assignment f(1) = 2; f(2) = 1; f(3) = 4; f(4) = 3;. More precisely, but not necessary for credit, f(m) = m + 1 if m is odd and f(m) = m 1 if m is even. This is a function; and given any natural n, we can find something that f assigns to n. If n is even f(n-1) = n; if n is odd f(n+1) = n.

2 III. Consider the diophantine equation 1694X Y = In five [5] sentences or less, outline the method developed in this course to determine a solution to this equation. Use the Euclidean Algorithm to find d = (1694,4060) and check that this divides 56. Then use this algorithm to express d as a linear combination of 1694 and Take this express and multiply through by an appropriate integer (the integer that when multiplied by d gives 56) to find a solution. 2. Find all the solutions to this equation. Check d = (1694,4060) = 14. The linear combination that emerges from the Euclidean Algorithm is (1694)(-139) + (4060)(58) = 14. Multiplying through by 4 gives X = (4)(-139) = -556 and Y = (4)(58) = 232. Now by Thm 2.2 all solutions are given by X = (-556) + (290)t and Y = 232 (121)t where t is an integer.

3 IV. Which number has more positive divisors: 1848 or 1850? Justify your choice in five [5] sentences or less. Note 1848 = (2 3 )(3)(7)(11) and 1850 = (2)(5 2 )(37). Computing nu values, 1848 has more divisors. NB. The number of distinct primes in the factorization of a number alone does not tell you how many divisors it can have. For example: 1848 has more distinct prime divisors than 2 40, but 2 40 has more divisors! V. Prove the following for all natural numbers n: (2n 1) = n 2. Standard proof using PMI.1. Be sure that your argument doesn t contain any statement that suggests the result you are attempting to prove!

4 VI. Let a, b, c be in Z with (a,b) = 1. Assume c = aq and c = br for some integers q and r. 1. Prove that there is an integer m for which r = am. Go with a direct proof: beginning with the observation that you can express 1 as a linear combination of a and b. Multiply this expression by r and manipulate the new expression using appropriate substitutions to see r as a multiple of a. There s your m! 2. Prove this m also gives q = bm. Take the expression for m found above and check bm gives q.

5 VII. Only one of the following will be graded. Please circle the number of the statement you want me to consider. 1. Let p be prime and a, b, m, n be naturals. We say p m exactly divides a when p m divides a, but p m+1 does not. Let p m exactly divide a and p n exactly divide b where m n. What power of p exactly divides a + b. Prove your claim. 2. Prove there are infinitely many primes of the form 6k + 5 with k > The power you want is the minimum of the numbers m and n. Prove this power of p divides a + b by a direct argument. Show no larger power of p divides a + b by contradiction. 2. See the proof by contradiction for the theorem proved in class: there are infinitely many primes of the form 4k + 3. There are only a couple modifications of this argument required to show that there are infinitely many primes of the form 6k + 5.