Finite sequences of prime numbers

Question

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$.

Even primes are primes of the form $2n$. The only even prime is $2$. Is that anything else? I mean primes sequences which are finite sequences by proof, and not by conjecture.

Answer 1

An integer-matrix quadratic form represents all prime numbers if and only if it represents the primes $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73$.

Exposition: https://math.nd.edu/assets/20630/hahntoulouse.pdf. (See page 674.)

Answer 2

• $5$ is the only prime that has $5$ as the right-most digit.

• $(2,3)$ is the only pair whose difference is $1$.

• There are only two sets $(a,b,c,d)=(2,3,5,7),(3,2,5,13)$ such that $$a+b=c\ \ \text{and}\ \ ac=b+d$$ where $a,b,c,d$ are all primes.

Answer 3

Since the other answers are base-dependent too, try truncatable primes which are primes which, when written in base 10, remains prime when you successively delete one digit from the end (or beginning) of the digit "word"/string.

Answer 4

3 is the the only prime where the sum of all the digits is 3.

Since we can show that if the digits sum to 3 its a multiple of 3 and the only multiple of 3 that is prime is 3 itself.

2 is the prime that has a even right most digit

Answer 5

The only prime that can be written in the form $n^2-1$ is $3$.

Answer 6

The prime-generating polynomial $n^2 + n + 41$ is prime for $n=0, \ldots, 39$.