Solving Equations - Assignment Example

In this project, Bluman (2004, p. 331) challenged students to examine an equation, x2 – x + 41, which purportedly yields prime numbers. Students choose numbers to substitute to x in the formula and observe if prime numbers occur. An added challenge was to venture into finding a number, which when substituted in the given equation, will result in a composite number. It is presupposed that a prime number is one whose factors are itself and 1, is common knowledge. On the other hand, a composite number may not be that much of a common knowledge.

A definition is thus, provided: a composite number is “any whole number greater than one that is not a prime number … [it] always has at least one divisor other than itself and 1” (p. 175). Based on the instructor’s specifications, numbers to be substituted should include zero (0), any two even numbers and any two odd numbers. The numbers chosen for substitution aside from 0 were: 10 and 16; and 9 and 11. Table 1 presents the computations involved in this project in five columns: the given equation, the number substituted, the computation, the result and an indicator of whether the result is a prime or a composite number.

As shown in Table 1, all the five numbers substituted to the equation yielded all prime numbers. However, the greater challenge in this project was to find if the equation could also yield composite numbers. For this purpose, an attempt was made to find results which are composite number using the first 201 integers from 0 -200, which should serve as the delimitation for the scope of the substitutions. This mini-experiment revealed that the equation x2 – x + 41 does not result in all prime numbers.

Substituting 0 – 40 yielded all prime numbers. From 41 onwards, many composite number outcomes were found. Table 2 shows the numbers substituted which came out with non-prime or composite results, together with the factors of the composite number other than one and itself. The substituted