Other Prime Forms

There are a plethora of prime forms and by no means all are listed here. Some large forms are not provable by current methods and remain probable primes. Some are sequences or patterns in the primes, even picture primes which use numbers to form ascii art.

Twin primes are a pair of primes for the form $p$ and $p+2$. It is unknown whether there are an infinitude of these.

Fermat primes are of the form $2^{2^n}+1$. It is believed by most mathematicians that these are finite in number: $3,$ $5,$ $17,$ $257$ and $65537$. Generalized Fermat primes of the form $b^{2^n}+1$.

Rep-unit primes are of the form $111\ldots 111$ necessarily of prime length. Generalized rep-units (GRU) are of the form $\frac{b^p-1}{b-1}$. Mersenne primes are where the base $b$ is $2$. Near rep-digit primes are base $10$ numbers where all but one of the digits are the same e.g. $979\ldots999$.

Factorial primes are of the form $n!\pm 1$ where $n!$ $=$ $n\times (n-1)\times\ldots 2\times 1$. Associated multi-factorial primes take every $i^{th}$ term of the factorial expression. E.g. $8\times 6\times 4\times 2 + 1$. Primorial primes are similar to factorial but only primes are used e.g. $11\#$ $+$ $1$ $=$ $11\times 7\times 5\times 3\times 2$ $+$ $1$.

Cullen primes and Woodall primes are of the form $n2^n\pm 1$ respectively.

Palindrome primes read the same backwards as forwards, e,g. $151$.

Arithmetic progressions of primes are primes separated by a common difference e.g. $5,$ $11,$ $17,$ $23,$ $29$.