[seqfan] Re: (no subject)
israel at math.ubc.ca
israel at math.ubc.ca
Thu Dec 7 05:10:36 CET 2017
Of course, we don't know whether the Fermat primes are finite or not.
We do know that all primes of the form 2^n+1, and more generally a^n+b^n
where one of a and b is even and the other odd,
must be of the quite special form a^(2^k)+b^(2^k),
because if n is divisible by an odd number d, a^n + b^n would be
divisible by a^(n/d) + b^(n/d).
But all this says about 1+2^a*3^b being prime is that gcd(a,b)
must be a power of 2 (including 1).
Cheers,
Robert
On Dec 6 2017, Frank Adams-Watters via SeqFan wrote:
> A005109 includes primes of the form 2^n + 1 (i.e., Fermat primes), while
> A058383 includes them.
>
>I is the finitude of the Fermat primes that led to wonder about these.
>
>Franklin T. Adams-Watters
>
>
>-----Original Message-----
>From: Allan Wechsler <acwacw at gmail.com>
>To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>Sent: Wed, Dec 6, 2017 8:56 pm
>Subject: [seqfan] Re: (no subject)
>
>At the moment, I cannot reach OEIS. But coincidentally I ran into primes of
>this class in a completely different context, only yesterday. Wikipedia, at
>least, calls them "Pierpont primes", primes p such that p-1 is 3-smooth.
>The article https://en.wikipedia.org/wiki/Pierpont_prime attributes to
>Andrew Gleason the conjecture that there are infinitely many Pierpont
>primes. Curiously, the article refers to the sequence A005109, not A058383;
>I can't investigate this discrepancy until OEIS starts answering the phone
>for me.
>
>On Wed, Dec 6, 2017 at 1:45 PM, Hugo Pfoertner <yae9911 at gmail.com> wrote:
>> In the range 1<=a,b<=500 there are 2111 primes of this form. Large
>> examples: 1+(2^498)*(3^493) or 1+(2^994)*(3^993). Why should there be a
>> limit?
>>> Hugo Pfoertner>> On Wed, Dec 6, 2017 at 6:51 PM, Frank Adams-Watters
>>> via SeqFan <
>> seqfan at list.seqfan.eu> wrote:>
>> > Is https://oeis.org/A058383 (Primes of form 1+(2^a)*(3^b)) infinite?>
>> > > Franklin T. Adams-Watters> >> >> > --> > Seqfan Mailing list -
>> > http://list.seqfan.eu
>
>
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