Chapter 1, Problem 6NT

NOW TRY THIS

A prime number is a natural number with exactly two distinct positive numbers, $1$ and the number itself, that divide it with $0$ remainder; for example, $2,3,5,7,11$, and $13$ are primes. One day, Amy makes a conjecture that the formula $y=x^2+x+11$ will produce only prime numbers if she substitutes the natural numbers $1,2,3,4,5,\mathrm{...}$ for $x$. She shows her work in Table $5$ for $x=1,2,3,4$.

Table 5
x	$1$	$2$	$3$	$4$
y	$13$	$17$	$23$	$31$

a. What type of reasoning is Amy using?

b. Try the next several natural numbers and see whether they seem to work.

c. Show that Amy’s conjecture is false for $x=11$.