12:33 PM
Has this already been calculated ? Which is the smallest positive integer $a$ such that $$a^{2^n}+1$$ is prime for $n=0,1,\cdots,k$ , where $k$ is a positive integer ? I currently search the minimum for $k=5$. For $k\le 4$, it is of course $a=2$

1:18 PM
$$7\ 072\ 833\ 120$$

3 hours later…
4:00 PM
There is an OEIS sequence : [the smallest b>=2 such that b^2^n+1 is prime](https://oeis.org/A056993)

I dont't like the sequence, somehow incomplete.
By the definition of a generalised Fermat number follows:

$\large GF(n,b)=b^{2^{n}}+1$ if $b$ is even and $GF(n,b)=(b^{2^{n}}+1)/2$ for odd $b$.

Generalised Fermat numbers for odd $b$ are just as interesting. We only (!) know that they are divisible by $2$.

4:34 PM
=========================================================================
\\ Generalised Fermat prime test:
\\ Input 'n'>1 and 'b'>1 and 'b'<=2^(n+1) for GF(n,b)=b^2^n+1 to test.
\\ Returns '1' if GF(n,b) is prime, otherwise '0'.

GFprime(n,b)={my(F=(b^2^n+1)/(b%2+1),a=2,c=0,s=valuation(F-1,2));
if(n<2||b<2||b>2^(n+1),print("Input not valid for test.");return);
while(gcd(a,b)>1,a++);
if(1==a=Mod(a,F)^(F>>s),return(1));
while(a!=-1&&s--,a=a^2);a==-1
};

n=2;print1("2, 2, ");for(b=2,+oo,while(GFprime(n,b),n++;print1(b", ")));
returns the more interesting sequence:

2, 2, 2, 2, 2, 3, 3, 113, 278, 513, 824, 2117, ...

5:01 PM
@MartinHopf Hi. So, which terms are known in my sequence and what is the meaning of the last sequence you posted ?
In fact, for $n\ge 1$ and odd $b$ we have factor $2$ (with multiplicity $1$) and the other prime factors must have the form $k\cdot 2^{n+1}+1$
We could also ask for the generalized Mersenne numbers $$\frac{a^p-1}{a-1}$$ which are prime for $p=2,3,5,7,11,\cdots ,q$ , where $q$ is a prime number. These are perhaps easier to determine.

my sequence is not listed in OEIS :(

I knew someone did that. What is the value for $k=6$ ?
for $a=1347552$ , the expression $$\frac{a^p-1}{a-1}$$ is prime for $p=2,3,5,7,11$
I currently search for $a$ such that we also get a prime for $p=13$
Is your sequence the smallest base $b$ (maybe odd) such that $b^{2^n}+1$ or $\frac{b^{2^n}+1}{2}$ is prime ?

5:20 PM
Peter, are we talking about the same subject?

Not sure. 1) What is the OEIS-link ?

:55961299
@Peter was the subject

Sorry, this number does not tell me anything.
For generalized Mersenne, the smallest solutions are :
2 7
1347552 11
540007650 13

5:36 PM
How do you prove primality for generalized Mersenne?

To be honest, I searched the examples with "ispseudoprime" and verified them with factordb. With PARI/GP , you can either choose isprime(n,2) (Adleman-Pomerance-Rumely) or isprime(n,3) (Ellliptic curve primality proving). In the case , $n-1$ is easy to factor, you can also choose isprime(n,1) (Pocklington)
What does the posted number :55961299 mean ? I do not remember ...
@MartinHopf

Generalized Fermat numbers are easier to be proved prime than generalized Mersenne!

yes, because of the structure ($n-1$ can be factored easily)
Unless the base is odd

@Peter this number :55961299 means nothing!

5:55 PM
My $$7 072 833 120$$ has the following meaning : It is the smallest even number $b$ such that $b^{2^n}+1$ is prime for all $n\le 5$ (including $n=0$)

4 hours later…
10:23 PM