An idea to solve a particular problem in mathematics may play a crucial role in solving more general problems in mathematics. The present article clearly illustrate this point. In this article, we will see how an idea to solve a particular problem related to a particular arithmetic sequence may be used to solve a general problem related to general arithmetic sequence.
Consider the following polynomial: \[q(x) = x^2 + x + 41.\] It is easy to see that $q(x)$ is prime for each integer $x \in \{-40, -39, \ldots, -1, 0, 1, \ldots, 39, 40\}$. Motivated by this observation, we can ask the following question: Does there exist a polynomial $q(x)$ with integer coefficients whose values are primes for every integer $x$ ? The answer is: YES! In fact, we can take the polynomial $q(x) = p$, where $p$ is a prime. Let us modify the above question slightly: Does there exist a non-constant polynomial $q(x)$ with integer coefficients whose values are primes for every integer $x$ ? This time, the answer is: NO! Why ? Let us prove this. Assume that $q(x)$ is a non-constant polynomial and is prime for every integer $x$. Then $q(0)$ is prime. Let $q(0) = p$. Now there are infinitely many integers $a$ such that $a \equiv 0 \pmod p$. Then by the theory of congruences, $q(a) \equiv q(0) \pmod p$ which says that $q(a) \equiv 0