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].
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].
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.
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.