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# Prime Numbers

2, 3, 5, 7, ... The set of prime numbers is the alphabet of mathematics that allows us to communicate across the universe.

# Prime Numbers: Level 4 Challenges

A common flawed presentation of Euclid's proof of the infinitude of primes is as follows:

Assume there are only a finite number of primes $$p_1,p_2,\ldots,p_n$$. Let $$N$$ be the product of all of those primes, add to it $$1$$ and you get a new prime number since it isn't divisible by any of the primes we listed at first. Contradiction! $$\Rightarrow\Leftarrow$$ Therefore, there is an infinite number of primes.

Let $$q_1, q_2, q_3, \ldots$$ be the list of all primes (in ascending order). Your mission is to find the smallest value of $$q_1 q_2 q_3 \ldots q_n + 1$$ that is not a prime.

Let $$A={102^1, 102^2, 102^3, \cdots}$$. How many primes $$p$$ are there such that $$A$$ has at least one element $$a$$ such that $$a \equiv -1 \text{ (mod p)}$$?

For example, one such prime is $$103$$, because $$102^1 \equiv -1 \text{ (mod 103)}$$.

$\Large \frac{d}{dx} n \neq 0?$

In calculus, when you take the derivative of a constant you get zero as an answer. In number theory, there is something called the arithmetic derivative which allows you to differentiate a number and get a nonzero answer. The arithmetic derivative works as follows.

Where $$n'$$ denotes the arithmetic derivative of $$n$$:

$$p' = 1$$ for all primes $$p$$

$$(ab)'=a'b+ab'$$

$$0'=1'=0$$

For example, $$6'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5$$

The double arithmetic derivative, denoted by $$n''$$, is simply defined by $$n''=(n')'$$.

Find the sum of all positive integers $$n<100$$ such that $$n''=1$$

$\Huge { \sum_{m=1}^{2^{820}} } \large \left \lfloor \left \lfloor \frac{820}{1 + \displaystyle \sum_{j=2}^m \left \lfloor\frac{(j-1)!+1}{j} -\left \lfloor\frac{(j-1)!}{j}\right \rfloor \right \rfloor }\right \rfloor ^{1/820} \right \rfloor = \ ?$

You may use this List of Primes as a reference.

Note: By definition, $$\displaystyle \sum_{a=b+1}^b 1 = 0$$.

###### This summation is rather infamous.

What is the smallest prime number that cannot be written in the form $$\dfrac{pq + 1}{p + q}$$, where $$p$$ and $$q$$ are prime numbers?

For example, $$2$$ can be written as $$\dfrac{3*5 + 1}{3 + 5}$$.

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