MTH 337 Homework #3 due 5pm Thursday, Feb 12

The whole thing to be submitted as a single "odt" or "sxw" document
(or pdf if you have to). Explicitly or implicitly include the
questions in your report, so that the report is self-contained.
 
1. Number searches
Write up the searches you did for
(i) a "Goldbach" ("dollar") number
(ii) a "Carmichael" ("prize") number 
Include descriptions of both the task, the process, and the results,
and include the Maple code used and its output, all formatted
in a clear and compact way.

2. Computer arithmetic
(i) Explain why it is important to have the use of (at least) two kinds
of computational number systems: one that is exact, using integers with
arbitrarily large numbers of digits, and another that is
approximate, using a fixed number of digits.

(ii) Supose a "floating-point" number system analogous to IEEE 64-bit
uses 32 bits: 1 for sign, 23 bits for the mantissa, and 8 bits for
the biased exponent, and the exponent bias is 127.
Also suppose the highest exponent code is reserved for non-numbers
such as "infinity" (e.g. resulting from doing 1/0) or "undefined" (e.g.
resulting from doing 0/0), and the two codes with all zeros in both mantissa
and exponent are re-purposed to represent "0",
what are, rounded to the nearest power of 2 and also expressed as
in decimal scientific notation,
(a) the largest positive number represented?
(b) the smallest positive number* represented?
(c) the relative spacing of the numbers (machine epsilon)?


*In case you start reading up on this, we will assume for
simplicity that there is no "denormalization".