What is required for large integers to be prime?

[Guest]

The size of the sample space for primes, [tex]p_{i}[/tex], is [tex]\pi(\sqrt{F_{k }} )[/tex] where [tex]\pi()[/tex] is the exact odd prime-counting function.

Now let's consider the infinite ordered sequence of Fermat numbers, {[tex]F_{k }[/tex] over all k > 4}. The chance that the sequence is devoid of Fermat primes is again,

[tex]\prod_{k > 4}^{\infty }\frac{\pi(\sqrt{F_{k }})}{\pi(\sqrt{F_{k }}) + 1} = 0[/tex].

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

Hmm. We have a contradiction! And therefore, there are infinitely many Fermat primes!

Remark: While there may be many primes [tex]p_{i} \le \sqrt{F_{k }}[/tex], that divide [tex]F_{k }[/tex], only one is required.

Remark: Further investigation of the Fermat numbers is warranted. We still have some lingering doubts since the nature of the beast (Fermat numbers) is not fully understood.

Final Remark: After some meditation, we observe for k >> 4, each Fermat number, [tex]F_{k } = 2^{2^{k}} + 1 = n_{ij }*p_{i }[/tex], is simply a sum of a very large even integer and one. We expect [tex]n_{ij } = 1[/tex] to occur infinitely many times over all k > 4. And therefore, we are now convinced there are infintely many Fermat primes. No further investigation of Fermat numbers is warranted for this matter.

*****

[Guest]

FYI: Fermat Numbers, [tex]F_{k } = 2^{2^{k}} + 1[/tex], minus one and the Areas of Squares:

[tex]F_{1} -1 = 2^{2^{1}} - 1 = 2^{2} = 4[/tex];

[tex]F_{2} -1 = 2^{2^{2}} - 1 = F_{1}^{2} = 4^{2} = 16[/tex];

[tex]F_{3} -1 = 2^{2^{3}} - 1 = F_{2}^{2} = 16^{2} = 256[/tex];

[tex]F_{4} -1 = 2^{2^{4}} - 1 = F_{3}^{2} = 256^{2} = 65,536[/tex];

[tex]F_{5} -1 = 2^{2^{5}} - 1 = F_{4}^{2} = 65536^{2} = 4,294,967,296[/tex];

...

[Guest]

Oops!

FYI: Fermat Numbers, [tex]F_{k } = 2^{2^{k}} + 1[/tex], minus one and the Areas of Squares:

[tex]F_{1} -1 = 2^{2^{1}} = 2^{2} = 4[/tex];

[tex]F_{2} -1 = 2^{2^{2}} = F_{1}^{2} = 4^{2} = 16[/tex];

[tex]F_{3} -1 = 2^{2^{3}} = F_{2}^{2} = 16^{2} = 256[/tex];

[tex]F_{4} -1 = 2^{2^{4}} = F_{3}^{2} = 256^{2} = 65,536[/tex];

[tex]F_{5} -1 = 2^{2^{5}} = F_{4}^{2} = 65536^{2} = 4,294,967,296[/tex];

...

[Guest]

[tex]F_{0} -1 = 2^{2^0} = 2^{1} = \sqrt{2} * \sqrt{2} = 2[/tex];

[tex]F_{1} -1 = 2^{2^1} = 2^{2} = F_{0 }^{2}= 2^{2}= 4[/tex];

...

[Guest]

Oops! (Sorry about the sloppy math of previous posts!)

FYI: Fermat Numbers, [tex]F_{k } = 2^{2^{k}} + 1[/tex], minus one and the Areas of Squares:

[tex]F_{0} -1 = 2^{2^0} = 2^{1} = \sqrt{2} * \sqrt{2} = 2[/tex];

[tex]F_{1} -1 = 2^{2^1} = 2^{2} =(F_{0 } - 1)^{2} - 1= 2^{2}= 4[/tex];

[tex]F_{2} -1 = 2^{2^{2}} = (F_{1} - 1)^{2} = 4^{2} = 16[/tex];

[tex]F_{3} -1 = 2^{2^{3}} = (F_{2}^{2} - 1)^{2} = 16^{2} = 256[/tex];

[tex]F_{4} -1 = 2^{2^{4}} = (F_{3}^{2} - 1)^{2} = 256^{2} = 65,536[/tex];

[tex]F_{5} -1 = 2^{2^{5}} = (F_{4}^{2} - 1)^{2} = 65536^{2} = 4,294,967,296[/tex];

...

[Guest]

Oops! (Sorry about the sloppy math of previous posts!)

FYI: Fermat Numbers, [tex]F_{k } = 2^{2^{k}} + 1[/tex], minus one and the Areas of Squares:

[tex]F_{0} -1 = 2^{2^0} = 2^{1} = \sqrt{2} * \sqrt{2} = 2[/tex];

[tex]F_{1} -1 = 2^{2^1} = 2^{2} =(F_{0 } - 1)^{2} - 1= 2^{2}= 4[/tex];

[tex]F_{2} -1 = 2^{2^{2}} = (F_{1} - 1)^{2} = 4^{2} = 16[/tex];

[tex]F_{3} -1 = 2^{2^{3}} = (F_{2} - 1)^{2} = 16^{2} = 256[/tex];

[tex]F_{4} -1 = 2^{2^{4}} = (F_{3} - 1)^{2} = 256^{2} = 65,536[/tex];

[tex]F_{5} -1 = 2^{2^{5}} = (F_{4} - 1)^{2} = 65536^{2} = 4,294,967,296[/tex];

...

Finally, we have it right!

[Guest]

FYI: Fermat Numbers, [tex]F_{k } = 2^{2^{k}} + 1[/tex], minus one and the Areas of Squares:

[tex]F_{0} -1 = 2^{2^0} = 2^{1} = \sqrt{2} * \sqrt{2} = 2[/tex];

[tex]F_{1} -1 = 2^{2^1} = 2^{2} =(F_{0 } - 1)^{2} = 2^{2}= 4[/tex];

[tex]F_{2} -1 = 2^{2^{2}} = (F_{1} - 1)^{2} = 4^{2} = 16[/tex];

[tex]F_{3} -1 = 2^{2^{3}} = (F_{2} - 1)^{2} = 16^{2} = 256[/tex];

[tex]F_{4} -1 = 2^{2^{4}} = (F_{3} - 1)^{2} = 256^{2} = 65,536[/tex];

[tex]F_{5} -1 = 2^{2^{5}} = (F_{4} - 1)^{2} = 65536^{2} = 4,294,967,296[/tex];

Hopefully, we have it right!

[Guest]

Moreover, [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].

[Guest]

Moreover, [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].

FYI: "With the exception of [tex]F_{0 }[/tex] and [tex]F_{1 }[/tex], the last digit of a Fermat number is 7."

Source Link:

https://en.wikipedia.org/wiki/Fermat_number.

Yes! Even Powers of 6 (the last digit of [tex]F_{k -1 }[/tex] for [tex]k > 1[/tex]) is always 6. And the sum of six and one is always seven.

Do we believe one-quarter of all primes have the last digit seven? Yes!

[Guest]

Moreover, [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].

FYI: "With the exception of [tex]F_{0 }[/tex] and [tex]F_{1 }[/tex], the last digit of a Fermat number is 7."

Source Link:

https://en.wikipedia.org/wiki/Fermat_number.

Oops!

Yes! Even Powers of 6 (the last digit of [tex]F_{k -1 }[/tex] for k > 2) is always 6. And the sum of six and one is always seven.

Do we believe one-quarter of all primes have the last digit seven? Yes!

[Guest]

Moreover, do we believe that most of all odd integers with the last digit seven are not primes? Yes!

And yet, there are infinitely many Fermat primes! Go figure!

[Guest]

Moreover, [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].

FYI: "With the exception of [tex]F_{0 }[/tex] and [tex]F_{1 }[/tex], the last digit of a Fermat number is 7."

Source Link:

https://en.wikipedia.org/wiki/Fermat_number.

Oops! Oops!

Yes! Even Powers of 6 (the last digit of the difference, [tex]F_{k -1 } - 1[/tex], for k > 2) is always 6. And the sum of six and one is always seven.

Do we believe one-quarter of all primes have the last digit seven? Yes!

[Guest]

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]

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...

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]?

[Guest]

lampoonization