Math 502
Written Assignment No. 3

due Thursday, April 3, 2008

Directions: Use Maple for assistance in responding to the following problems. Please typeset your solutions. Explain what you have done. Maple session details are not necessary unless you think it important to include them. Accuracy is important.

Although you may refer to books and notes, you may not seek help from others on this written assignment.

Answer the following questions:

1. Determine polynomials $f_0\left(x\right),f_1\left(x\right),f_2\left(x\right),\dots$ such that $${\left(x{y}^3+1+x+x^{2}y\right)}^3=\sum _{k\ge 0}{f}_k\left(x\right)y^k\text{.}$$

2. How can the length 15 string represented by the vector of ASCII codes

     [97, 39, 98, 96, 99, 34, 100, 10, 101, 39, 102, 34, 103, 96, 104]

   — which contains a newline and various quoting characters — be entered as an input string in Maple without reference to the ASCII codes?

3. In this problem let $p$ denote the prime $40487$.

   1. Find all non-negative residues $k\left(modp-1\right)$ for which

      (i) $2^k\equiv 5\left(modp\right)$.

      (ii) $5^k\equiv 2\left(modp\right)$.

   2. Find the smallest positive primitive root modulo the prime $p$.

   3. Find the smallest positive primitive root modulo $p^2$.

   4. Find the least non-negative integer $k$ for which $${10}^k\equiv 5\left(mod{p}^2\right)\text{.}$$

4. The vector

     [1374123367812492240294819411451017240070232394572901767016863846,

      839316502484928463600482820810738376821725250432320253174346182,

      1239738327596678208340985633338387227858526317346603640791773335,

      1786140405373535737086895440853230782772901282581414488521399167,

      500665432788843543145295210921142588622600876314918005209142550,

      1134791724566968592421020750873161158581657388017051854867699847,

      2289042576082270612059353048240349198604028364564225150215042759]

   represents a sequence of ASCII codes (range $0$ to $127$) that has been squeezed into blocks of 30 and then encrypted by taking a certain power modulo $$m=2468256835981809063232453777824906406995466865716803865301401733\text{.}$$ It may be decrypted by using the exponent $5^{15}$.

   1. What string was encrypted?

   2. Can you find the encryption exponent that is paired with this decryption exponent?

5. Provide a listing of code for a Maple procedure that takes an expression $e$ in two variables $x,y$ as first argument and the variable names as second and third arguments and returns the anonymous function of two arguments $$\left(x,y\right)\mapsto x\frac{\partial e}{\partial x}+y\frac{\partial e}{\partial y}\text{.}$$ Make sure that the code you submit works correctly.