Some hints on HW #1

• Problem 34/3: use the hint in the book at write out at least the first two terms and last two terms of the binominal expansion and note (a+b) divides all the rest.
• Problem 34/9a: show if (m,n) = 1 then (m^2,n)=1. [Here ^ = to the exponent so m^2 = m*m]. if d = (a,b) = (a,c) look at (a/d,b/d) = (a/d, c/d) = 1.
• Problem 34/12: look every two divisions. what is an upper bound on the number of steps needed if 2^N >= b?
• Problem 43/2: (a/m) - (b/n) = (an-bm)/mn. what is the smallest possible positive numerator?

Some hints on HW #2

• Problem 23/5: when is p^e a square?
• Problem 23/9: show any number is a difference of two squares. show that (x^2-y^2)(z^2-w^2) is a difference of two squares by factoring and multiplying wisely.
• Problem 125/2: if f is multiplicative show f^n is. also do not forget theorem 6.3.
• Problem 125/3: if p is an odd prime when is \sigma(p^e) odd? what if p =2? also look at things mod 2.

Some hints on HW #3

• Problem 24/5: show if (a,b) = 1 and (ab,c) =d then d=d_1d_2 with (d_1,d_2)=1 and d_1=(a,c), d_2=(b,c). also show (xy+z,x)=(z,x).
• Problem 50/1: look in (Z/rZ)[t].
• Problem 50/4: if a-b = k(m/d) divide k by m in the division algorithm.
• Problem 7a: look at (r-1)(s-1)/2
• Problem 56/4: let 1 <= a < mn with (a,mn)=1. write a = mx+ny (why can you do this) and use the division algorithm to write x=nq+x_1 with 0 <= x_1 < n and y=mg+y_1 with 0 <= y_1 < m so a congruent mx_1+ny_1 mod mn. show x_1, y_1 are independent of choice of x,y.

You can turn in 56/4 with Homework Four.

Some hints on HW #4

• Problem 11/4: you must show P(n) goes to infinity as n goes to infinity. this means: given any M > 0 you must find an N such that if n > N then P(n) > M. how many prime powers are less than N?
• Problem 56/23: use fermat for divisors of 561. look at the factorization of 560 compared do the divisors of 561. the second part is harder. (HINT EXTENDED) note if p^2 | m that either a^{p-1} or (a+p)^{p-1} is not congruent to 1 mod p^2 for (a,m) = 1. what is the ord_{p^2} of this element? what happens if you raise this element to the power m-1 and evaluate mod m, mod p^2?

FIRST MIDTERM MONDAY 5 MAY 08

The first midterm will cover material through Homework 4: gcds, primes, Fundamental Theorem of Arithmetic, Arithemetic functions (including the tau and sigma functions), Congruences, Euler phi-function and applications for sums of two squares.

The test will have four problems. One will be to solve a linear diophantine equation (probably with in positive integers), one homework problem (but not a (*) problem), one theorem proved in class, and one additional problem.