Number Theory

Prime Numbers

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 p1,p2,,pnp_1,p_2,\ldots,p_n. Let NN be the product of all of those primes, add to it 11 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 q1,q2,q3, 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 q1q2q3qn+1 q_1 q_2 q_3 \ldots q_n + 1 that is not a prime.

Let A=1021,1022,1023,A={102^1, 102^2, 102^3, \cdots}. How many primes pp are there such that AA has at least one element aa such that a1 (mod p)a \equiv -1 \text{ (mod p)}?

For example, one such prime is 103103, because 10211 (mod 103)102^1 \equiv -1 \text{ (mod 103)}.

ddxn0?\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 nn' denotes the arithmetic derivative of nn:

p=1p' = 1 for all primes pp

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

0=1=00'=1'=0

For example, 6=(2×3)=(2)(3)+(2)(3)=(1)(3)+(2)(1)=56'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5

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

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

m=128208201+j=2m(j1)!+1j(j1)!j1/820= ? \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, a=b+1b1=0 \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 pq+1p+q\dfrac{pq + 1}{p + q}, where pp and qq are prime numbers?

For example, 22 can be written as 35+13+5\dfrac{3*5 + 1}{3 + 5}.

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