What is the next Fermat prime beyond 65,537?

What is the next Fermat prime beyond 65,537?

Postby Guest » Fri May 08, 2020 12:51 pm

FYI: [tex]F_{k} = 2^{2^{k}} + 1 = 4^{2^{k-1}} + 1 = (4^{2^{k-2}})^{2} + 1 = (F_{k -1 } - 1)^{2} + 1[/tex] for [tex]k \ge 1[/tex] where [tex]F_{k}[/tex] is called a Fermat number.

Relevant Reference Links:

'Fermat Primes and Pascal’s Triangle',

https://johncarlosbaez.wordpress.com/2019/02/05/fermat-primes-and-pascals-triangle/;

'A NECESSARY AND SUFFICIENT CONDITION FOR THE PRIMALITY OF FERMAT NUMBERS',

https://www.emis.de/journals/MB/126.3/mb126_3_1.pdf.
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What is the next Fermat prime beyond 65,537?
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Re: What is the next Fermat prime beyond 65,537?

Postby Guest » Fri May 08, 2020 6:47 pm

Hmm. The next Fermat prime beyond 65,537 is [tex]F_{k+32 }[/tex] such that [tex]0 < k \le k_{max } < \infty[/tex]. :idea:

We must determine [tex]k_{max }[/tex] in accordance with the Fundamental Theorem of Arithmetic and in accordance with the distribution of primes between [tex]F_{32 }[/tex] and [tex]F_{k_{max } + 32 }[/tex]. :idea:

Furthermore, we must discover generally (via data analysis) how 'dense' composites (integers with many prime factors) give way (provide placement among integers) to almost primes (integers with two prime factors) and to primes. Moreover, we imagine a complex sequence of convergence from composites to primes. And among primes in general, we must also discover how Fermat primes are roughly distributed. :idea:

Our ideas are tentative... And our challenge is daunting. But we are confident we will solve our problem somehow.
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Re: What is the next Fermat prime beyond 65,537?

Postby Guest » Sat May 09, 2020 5:56 pm

Guest wrote:Hmm. The next Fermat prime beyond 65,537 is [tex]F_{k+32 }[/tex] such that [tex]0 < k \le k_{max } < \infty[/tex]. :idea:

We must determine [tex]k_{max }[/tex] in accordance with the Fundamental Theorem of Arithmetic and in accordance with the distribution of primes between [tex]F_{32 }[/tex] and [tex]F_{k_{max } + 32 }[/tex]. :idea:

Furthermore, we must discover generally (via data analysis) how 'dense' composites (integers with many prime factors) give way (provide placement among integers) to almost primes (integers with two prime factors) and to primes. Moreover, we imagine a complex sequence of convergence from composites to primes. And among primes in general, we must also discover how Fermat primes are roughly distributed. :idea:

Our ideas are tentative... And our challenge is daunting. But we are confident we will solve our problem somehow.


Let's estimate [tex]k_{max }[/tex].

First, we shall consider an ordered sequence of Fermat numbers, {[tex]F_{k }[/tex] | [tex]33 \le k \le k_{max } < \infty[/tex]}.

The chance that the sequence is devoid of primes is

1. [tex]\prod_{k = 33}^{ k_{max } }\frac{\pi(\sqrt{F_{k }})}{\pi(\sqrt{F_{k }}) + 1} =[/tex] ?

We should choose [tex]k_{max }[/tex] so that equation one is close to zero.

A Remark: [tex]\frac{\pi(\sqrt{F_{k }})}{\pi(\sqrt{F_{k }}) + 1} \rightarrow 1[/tex] as [tex]k \rightarrow \infty[/tex].

Relevant Reference Link:

'What is required for large integers to be prime?',

https://www.math10.com/forum/viewtopic.php?f=63&t=8827.
Guest
 

Re: What is the next Fermat prime beyond 65,537?

Postby Guest » Sat May 09, 2020 8:58 pm

Hmm. We need a supercomputer or a planetary computing platform because Fermat numbers become so big after k > 10. And equation one from the previous post requires great accuracy in regards to the prime count etc.

We are stuck with [tex]F_{31} \approx 1.761613051683963* 10^{646,456,993}[/tex]. :(

Guest wrote:Infinity has no limit, and to know the next Fermat prime beyond 65,537,

we cannot say.

Shall it be a math riddle unsolved?

We cannot say.

We are sure it exists.

But we cannot say.

The next Fermat prime beyond 65,537,

we cannot say...
Guest
 

Re: What is the next Fermat prime beyond 65,537?

Postby Guest » Tue May 12, 2020 3:13 pm

Could [tex]F_{54 }[/tex] be the next Fermat prime beyond [tex]F_{4 } = 65,537[/tex]?

And could [tex]F_{83}[/tex] be the next Fermat prime beyond [tex]F_{54 }[/tex]?
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