The prime numbers are an odd set of numbers.

The crux of the problem lies in teh following fact:

the function $P(n)$ that outputs the n'th prime is still only known to be defined using the idea "the nth prime is not divisible by any primes below it" and "the first prime is 2". This sort of definition means any question involving prime numbers cannot be solved easily. An example is the twin prime conjecture. The question quite simply asks:

are there infinitely many n such that

$$P(n) - P(n-1) = 2?$$

If we had a non-recursive (ie an explicit formula no matter how complicated) for describing the function then suddenly many many tools like calculus (think Newton's Method), functional analysis, theory of solvability of equations etc... can be used to tackle this problem.

But WE DONT HAVE AN EXPLICIT DEFINITION.

So all of that goes out the door and is essentially what makes Number Theory so brutally challenging yet so wonderfully rewarding.

The prime numbers are an odd set of numbers.

The crux of the problem lies in the following fact:

the function $P(n)$ that outputs the n'th prime is still only known to be defined using the idea "the nth prime is not divisible by any primes below it" and "the first prime is 2". This sort of definition means any question involving prime numbers cannot be solved easily. An example is the twin prime conjecture. The question quite simply asks:

are there infinitely many n such that

$$P(n) - P(n-1) = 2?$$

If we had a non-recursive (ie an explicit formula no matter how complicated) for describing the function then suddenly many many tools like calculus (think Newton's Method), functional analysis, theory of solvability of equations etc... can be used to tackle this problem.

But WE DONT HAVE AN EXPLICIT DEFINITION.

So all of that goes out the door and is essentially what makes Number Theory so brutally challenging yet so wonderfully rewarding.