- --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz <s_m_ruiz@...> wrote:
>

For which n>1 is 4*n!^8 + 1 prime ??

> Hello all:

> ï¿½

> 32(n!)**8+1 is composite for all nï¿½positive integerï¿½??

Maximilian - --- In primenumbers@yahoogroups.com,

Sebastian Martin Ruiz <s_m_ruiz@...> wrote:

> 32(n!)**8+1 is composite for all n positive integer ??

On the contrary, the PNT makes it probable (but not yet provable)

that there is an *infinite* number of primes of this form.

The probability that (2*(n!))^8+1 is prime

is (heuristically) asymptotic to exp(Euler)/(8*n).

[If I got the constant wrong, Chris will correct me.]

The integral of 1/n diverges.

Moral: just because you didn't find a small one,

doesn't mean there isn't an infinity of larger ones.

David - correcting a typo, which does not seem to me to

affect the argument:

The probability that ((2*(n!))^8)/2+1 is prime

is (heuristically) asymptotic to exp(Euler)/(8*n).

[If I got the constant wrong, Chris will correct me.]

The integral of 1/n diverges.

David - --- In primenumbers@yahoogroups.com,

"Maximilian Hasler" <maximilian.hasler@...> wrote:

> For which n>1 is 4*n!^8 + 1 prime ??

For none. Proof: Set x = (n!)^2 in the identity

4*x^4 + 1 = (2*x^2 + 1)^2 - (2*x)^2

David (per proxy Léon François Antoine) - --- In primenumbers@yahoogroups.com, "David Broadhurst" <d.broadhurst@...> wrote:
>

Right... of course! Bad example - that's not what I meant to say.

> > For which n>1 is 4*n!^8 + 1 prime ??

>

> For none. Proof: Set x = (n!)^2 in the identity

> 4*x^4 + 1 = (2*x^2 + 1)^2 - (2*x)^2

Maybe,

Is 8*n!^8+1 composite for all n>4 ?

Probably not. Actually I don't mind at all. It was just to express the same "Moral" given elsewhere more explicitely. But I should have chosen my example more carefully...

At least this shows that the "32" is in some sense simply the 3rd possibiliy, after 2 and 8. For 2^7 there happen to be 3 small primes; for 2^9 and 2^11 only two. Thus, whenever the first prime is found for 2^5, then the same post can be made replacing 32 with 2^13.)

Maximilian