On a Grand Hypothesis of Fundamental Number Theory

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Update: Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = Prob ( [tex]n_{m} \ne 1 \forall m \in \N | p_{m } - 2 \lambda = n_{m } *q_{m }[/tex] )
= [tex]\prod_{m=1}^{\infty }\frac{\pi (\sqrt{p_{m} - 2 \lambda})}{\pi (\sqrt{p_{m} - 2 \lambda}) + 1} = 0[/tex].

[Guest]

FYI: For the latest update of our proof of Polignac's conjecture, please refer to the link below.

'Randomness can be a useful tool for solving problems.'

https://www.math10.com/forum/viewtopic.php?f=1&t=8855&start=120.

Dave.

[Guest]

Update: Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = Prob ( [tex]n_{m} \ne 1 \forall m \in \N | p_{m } - 2 \lambda = n_{m } *q_{m }[/tex] )
= [tex]\prod_{m=1}^{\infty }\frac{\pi (\sqrt{p_{m} - 2 \lambda})}{\pi (\sqrt{p_{m} - 2 \lambda}) + 1} = 0[/tex].

Remark 1: [tex]p_{k+1 } - p_{k } \ne 2 \lambda[/tex] over E for some [tex]\lambda[/tex] such that [tex]1 \le \lambda < \frac{log^{2}(p_{k+1 }p_{k})}{2}[/tex].

Remark: We must redefine our current exceptional set, E, to comply with remark one.

Remark: This problem is a big headache! Ouch!

Oops! Our current proof of Polignac's conjecture is wrong!! The proof of Polignac's conjecture should be about the spacing between consecutive odd primes.

Example: Suppose we want to exclude [tex]2 \lambda_{0 }[/tex] over E.

We have [tex]p_{2 } - p_{1 } \ne 2 \lambda_{0 }[/tex] such that [tex]1 \le \lambda_{0 } < \frac{log^{2}(p_{1 }p_{2})}{2}[/tex].

The chance that [tex]2 \lambda_{0 }[/tex] is the wrong spacing between consecutive odd primes, [tex]p_{2 } > p_{1 }[/tex], is roughly

[tex]\frac{Floor( \frac{log^{2}(p_{1 }p_{2})}{2} - 1) }{Floor( \frac{log^{2}(p_{1 }p_{2})}{2}) }[/tex].

However, over E, we generate the infinite product of similar values because of independence so that the chance [tex]2 \lambda_{0 }[/tex] is the wrong spacing between consecutive odd primes over E equates to zero.

In short, Polignac's conjecture is still correct! But our previous reasoning was wrong! We hope we have it right this time. We will review it later.

Dave.

Go Blue!

[Guest]

Update: Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = Prob ( [tex]n_{m} \ne 1 \forall m \in \N | p_{m } - 2 \lambda = n_{m } *q_{m }[/tex] )
= [tex]\prod_{m=1}^{\infty }\frac{\pi (\sqrt{p_{m} - 2 \lambda})}{\pi (\sqrt{p_{m} - 2 \lambda}) + 1} = 0[/tex].

Update: Prob( [tex]p - q \ne 2 \lambda[/tex] over E )

= [tex]\prod_{m=1}^{\infty }\frac{Floor( \frac{log^{2}(p_{m }p_{m+1})}{2} - 1) }{Floor( \frac{log^{2}(p_{m }p_{m+1})}{2}) } = 0[/tex].

Dave.

[Guest]

Update:

Remark 1: [tex]p_{k+1 } - p_{k } \ne 2 \lambda[/tex] over E for some [tex]\lambda[/tex] such that [tex]1 \le \lambda < \frac{log^{2}(max(p_{k+1 }, p_{k}))}{2}[/tex].

Remark: We must redefine our current exceptional set, E, to comply with remark one.

Remark: This problem is a big headache! Ouch!

Oops! Our current proof of Polignac's conjecture is wrong!! The proof of Polignac's conjecture should be about the spacing between consecutive odd primes.

Example: Suppose we want to exclude [tex]2 \lambda_{0 }[/tex] over E.

Update:

We have [tex]p_{2 } - p_{1 } \ne 2 \lambda_{0 }[/tex] such that [tex]1 \le \lambda_{0 } < \frac{log^{2}(max((p_{1 }, p_{2}))}{2}[/tex].
_____________________________________________________________________________________________________________________________________________
Update:

The chance that [tex]2 \lambda_{0 }[/tex] is the wrong spacing between consecutive odd primes, [tex]p_{2 } > p_{1 }[/tex], is roughly

[tex]\frac{Floor( \frac{log^{2}(max(p_{1 }, p_{2}))}{2} - 1) }{Floor( \frac{log^{2}(max(p_{1}, p_{2}))}{2}) }[/tex].

Remark: "Roughly" indicates too large.

However, over E, we generate the infinite product of similar values because of independence so that the chance [tex]2 \lambda_{0 }[/tex] is the wrong spacing between consecutive odd primes over E equates to zero:

Prob( [tex]p - q \ne 2 \lambda[/tex] over E )

= [tex]\prod_{m=1}^{\infty }\frac{Floor( \frac{log^{2}(max(p_{m }, p_{m+1}))}{2} - 1) }{Floor( \frac{log^{2}(max(p_{m }, p_{m+1}))}{2}) } = 0[/tex].
_______________________________________________________________________________________________________________________________________________
In short, Polignac's conjecture is still correct! But our previous reasoning was wrong! We hope we have it right this time. We will review it later.

Remark: We apologized for the sloppy (flawed) math in previous posts.

Dave.

Go Blue!

[Guest]

Remark: "Math is hard work!"

[Guest]

Relevant Reference Link:

'LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS',

https://www.math10.com/forum/viewtopic.php?f=63&t=8263.

[Guest]

FYI: According to the Prime Number Theorem, primes have a zero density relative to the non-primes (composite integers), and we expect the gaps between some successive/consecutive primes will approach infinity.

Our sample space is based on the maximum gap possible between successive primes according to theory...

And we expect every gap (any positive even integer) between successive primes will repeat infinitely many times over all the primes according to theory...

Primes are a true wonder!

[Guest]

Go HLM! Go Blue! Montgomery's Pair Correlation Conjecture is true!

[Guest]

Go HLM! Go Blue! Montgomery's Pair Correlation Conjecture is true!

CONGRATULATIONS!!!