On a Grand Hypothesis of Fundamental Number Theory

[Guest]

A Grand Hypothesis:

"The repetition and the growth of prime gaps between consecutive prime numbers are essential for the efficient generation of all composites (all positive integers that are not prime) in accordance with the Fundamental Theorem of Arithmetic and in accordance with the Prime Number Theorem." -- David Cole.

A Grand Claim:

"As a result of our grand hypothesis, we claim the Polignac Conjecture ( https://en.wikipedia.org/wiki/Polignac%27s_conjecture ) is true!" -- David Cole.

Relevant Reference Link:

'What great conjectures in mathematics combine additive theory of numbers with the multiplicative theory of numbers?',

https://www.quora.com/What-great-conjectures-in-mathematics-combine-additive-theory-of-numbers-with-the-multiplicative-theory-of-numbers/answer/David-Cole-146.

[Guest]

A Grand Hypothesis:

"The repetition and the growth of prime gaps between consecutive prime numbers are essential for the efficient generation of all composites (all positive integers that are not prime) in accordance with the Fundamental Theorem of Arithmetic and in accordance with the Prime Number Theorem." -- David Cole.

A Grand Claim:

"As a result of our grand hypothesis, we claim the Polignac Conjecture ( https://en.wikipedia.org/wiki/Polignac%27s_conjecture ) is true!"
-- David Cole.

Relevant Reference Link:

'What great conjectures in mathematics combine additive theory of numbers with the multiplicative theory of numbers?',

https://www.quora.com/What-great-conjectures-in-mathematics-combine-additive-theory-of-numbers-with-the-multiplicative-theory-of-numbers/answer/David-Cole-146.

Probability( Polignac Conjecture fails for some positive even integer, e. )

= Prob( [tex]p_{m} - e ≠ s_{m}[/tex] or [tex]r_{m}[/tex] over E*) = [tex]\prod_{m=1}^{\infty }\lambda_{m} = 0[/tex].


where [tex]p_{m}[/tex] and [tex]s_{m}[/tex] or [tex]r_{m}[/tex] are any appropriate consecutive prime numbers such that [tex]p_{m} > s_{m}[/tex] or [tex]p_{m} > r_{m}[/tex] and whose difference is not e, a positive even integer, and where E* is the exceptional set of such values of e for which Polignac's Conjecture fails.

Thus, we are very confident that Polignac's Conjecture is true!


Remarks: π() is the odd prime-counting function. And the positive even integers, [tex]e_{1}[/tex] and [tex]e_{2}[/tex], are appropriately chosen for our lambda function, [tex]\lambda_{m}[/tex].

[Guest]

We shall review our proof of the Polignac Conjecture (or Polignac's Conjecture). Hopefully, our proof is clear, concise, and complete. And we are ready to make the necessary changes so that our proof is clear, concise, and complete.

Dave,

https://www.researchgate.net/profile/David_Cole29.

[Guest]

Hmm. Where do we start our computations for [tex]\prod_{m=1}^{\infty }\lambda_{m}[/tex]?

Let's consider the case of [tex]e = 2[/tex] (the Twin Prime Conjecture), and we assume the conjecture fails after a very large prime number, [tex]s_{1}[/tex].

Let [tex]p_{1}[/tex] = NextPrime( [tex]s_{1}[/tex] ),

Let [tex]s_{2} = p_{1}[/tex] with [tex]p_{2}[/tex] = NextPrime( [tex]s_{2}[/tex] ),

Let [tex]s_{3} = p_{2}[/tex] with [tex]p_{3}[/tex] = NextPrime( [tex]s_{3}[/tex] ),

Let [tex]s_{4} = p_{3}[/tex] with [tex]p_{4}[/tex] = NextPrime( [tex]s_{4}[/tex] ),

...

as [tex]m \rightarrow \infty[/tex], [tex]s_{m} \rightarrow \infty[/tex] and [tex]p_{m} \rightarrow \infty[/tex].

And based on the above sequence, we compute [tex]\prod_{m=1}^{\infty }\lambda_{m}[/tex] for accordingly.

Dave,

Go Blue! PC (Polignac's Conjecture) is true! And Michigan Math (U-M Math Department) is great too!

[Guest]

Oops! We meant, "... And based on the above sequence, we compute [tex]\prod_{m=1}^{\infty }\lambda_{m}[/tex] accordingly ..."

[Guest]

"A journey on a path with heart, begins with the first step. Mistakes are common and expected, but friends are rare jewels along the way. Be thankful for all the good there is! Thank Lord GOD! Amen!" -- Dave.

Hmm. Suppose we want to know where to start our computations for [tex]\prod_{m=1}^{\infty }\lambda_{m}[/tex] when we have a large prime gap, e = G(X), where

G(X) [tex]\le log^{2} X[/tex] for large X where [tex]G(X) \rightarrow log^{2} X[/tex] as [tex]X \rightarrow \infty[/tex].

And therefore, we let [tex]s_{1} \approx Exp ( \sqrt{e} )[/tex] with [tex]p_{1}[/tex] = NextPrime( [tex]s_{1}[/tex] ), and so on.

Remarks: The notation, a = NextPrime(p), indicates the next prime, a, following prime, p.

Example: Suppose we let [tex]e = 13,128[/tex].

We compute [tex]s_{1} \approx Exp ( \sqrt{e} ) = 5.579 * 10^{49}[/tex]. (Note: We suspect a prime gap of 13,128 will probably occur well below that approximate number.)

And therefore, we let [tex]s_{1} = 57,590,000,000,000,000,000,000,000,000,000,000,000,000,000,000,107[/tex] with [tex]p_{1} = 57,590,000,000,000,000,000,000,000,000,000,000,000,000,000,000,237[/tex] and so on.

And we compute [tex]\prod_{m=1}^{\infty }\lambda_{m}[/tex] accordingly.


Relevant Reference Link:

'LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS',

https://www.math10.com/forum/viewtopic.php?f=63&t=8263&sid=7dcf8d9d145494736dcd85e846654485

Dave.

[Guest]

Oops! Our note for the previous example is not strong enough.

Revised Note: We strongly expect a prime gap of 13,128 will occur well below that approximate number.

[Guest]

Oops! Our note for the previous example is not strong enough.

Revised Note: We strongly expect a prime gap of 13,128 will occur well below that approximate number based on the maximum prime gap, G(X)..

We can employ the Prime Number Theorem (PNT) to get a better estimate when a prime gap of 13,128 would first occur among consecutive primes.

Let log( x ) = [tex]\sqrt{c * 13,128}[/tex] where .7 < c < .8 (our very rough estimate). If c = .75, we have [tex]x = 1.24 * 10^{43}[/tex].

So we expect to see a prime gap of 13,128 between some consecutive primes after [tex]10^{40}[/tex](our rough estimate since our 'Prime Force' is weakening).

May the Prime Force be with you forever...

[Guest]

Oops! Our note for the previous example is not strong enough.

Revised Note: We strongly expect a prime gap of 13,128 will occur well below that approximate number.

Hmm. So far, We cannot find the gap of 13,128 between consecutive primes when those primes are between 10^40 and 10^100. Therefore, our computations are tentatively wrong!

Dave.

[Guest]

Oops! Our note for the previous example is not strong enough.

Revised Note: We strongly expect a prime gap of 13,128 will occur well below that approximate number.

Hmm. So far, We cannot find the gap of 13,128 between consecutive primes when those primes are between 10^40 and 10^100. Therefore, our computations are tentatively wrong!

Dave.

Moreover, we cannot find a gap of 10,000 or more between some consecutive primes when those primes are between [tex]10^{1500}[/tex] and [tex]10^{2000}[/tex].

Hmm. We shall search that range again and again...

Dave.

[Guest]

Wolfram Mathematica Code (Our search for a gap of 10,000 between some consecutive primes between [tex]10^{1500}[/tex] and [tex]10^{2000}[/tex]):

see[n_]:=(

start=NextPrime[n];

While [NextPrime[start]-start <= 10000,

start = NextPrime[Random)Integer[{10^1500,10^2000}]]];

Return[{start, NextPrime[start], NextPrime[start] - start}]);

_________________end of code________________

Input: see[RandomInteger[{10^1500,10^2000}]]

Output: ?

Good Luck!

Dave.

Relevant Reference Link:

https://www.wolframcloud.com/.

[Guest]

Wolfram Mathematica Code (Our search for a gap of 10,000 between some consecutive primes between [tex]10^{1500}[/tex] and [tex]10^{2000}[/tex]):

see[n_]:=(

start=NextPrime[n];

While [NextPrime[start]-start <= 10000,

start = NextPrime[Random)Integer[{10^1500,10^2000}]]];

Return[{start, NextPrime[start], NextPrime[start] - start}]);

_________________end of code________________

Input: see[RandomInteger[{10^1500,10^2000}]]

Output: ?

Good Luck!

Dave.

Relevant Reference Link:

https://www.wolframcloud.com/.

Correction: " (Our search for a gap of more than 10,000 between some consecutive primes between [tex]10^{1500}[/tex] and [tex]10^{2000}[/tex]): "

[Guest]

Wolfram Mathematica Code (Our search for a gap of more than 10,000 between some consecutive primes between [tex]10^{1500}[/tex] and [tex]10^{2000}[/tex]):

see[n_]:=(

start=NextPrime[n];

While [NextPrime[start]-start <= 10000,

start = NextPrime[Random)Integer[{10^1500,10^2000}]]];

Return[{start, NextPrime[start], NextPrime[start] - start}]);

_________________end of code________________

Input: see[RandomInteger[{10^1500,10^2000}]]

Output: ?

Good Luck!

Dave.

Relevant Reference Link:

https://www.wolframcloud.com/.

A Comment:

If one finds a gap of more than 10,000 (very rare instances if they exist...) between two consecutive primes for primes between [tex]10^{1500}[/tex] and [tex]10^{2000}[/tex], then one is very lucky!!

And one ought to play the lottery to win millions of dollars!

[Guest]

Revised Wolfram Mathematica Code:

see[n_,gap_,l_,u_]:=(

start=NextPrime[n];

While [NextPrime[start]-start <= gap,


start = NextPrime[RandomInteger[{10^l, 10^u}]]];

Return[{start,NextPrime[start], NextPrime[start] - start}]);

_________________end of code________________

Input: see[RandomInteger[{10^106, 2000, 106,108}]]

Output:

{ 556,278,877,530,936,159,965,199,353,517,375,654,230,162,574,932,591,981,581,987,124,304,436,305,824,074,665,825,248,820,946,464,780,199,402,199,
556,278,877,530,936,159,965,199,353,517,375,654,230,162,574,932,591,981,581,987,124,304,436,305,824,074,665,825,248,820,946,464,780,199,404,763,
2564 }

where gap = 2564.

Relevant Reference Link:

https://www.wolframcloud.com/

[Guest]

Input: see[10^212, 4000, 212, 215];

Output:

{862308931739653211093928667118442299013609149791803665193149787825510225739722406570955403782066063962684998911953418851184905144339567435003580282534361176716527422
58377529272678488832229291631489391197428815669531,

862308931739653211093928667118442299013609149791803665193149787825510225739722406570955403782066063962684998911953418851184905144339567435003580282534361176716527422583
77529272678488832229291631489391197428815673877,

4346 }

where gap = 4346.

We are making some progress...

[Guest]

We shall try see[10^689, 13000, 689, 692] since [tex]212:4,000 \Leftrightarrow 689:13,000[/tex] (Linear Extrapolation).

Dave.

[Guest]

We shall try see[10^689, 13000, 689, 692] since [tex]212:4,000 \Leftrightarrow 689:13,000[/tex] (Linear Extrapolation).

Dave.

Eureka!

LINEAR EXTRAPOLATION WORKS!

Input: see[10^689, 13000, 689, 692];

Output:

{ 63414427789327671115409705289958794361707540522015552896483841045292338475665409464200968791675345978858436480400687789338862101889156188895345321186092063
485719965787214341766066076572970091513077149743412813932674397691123169312819178026127342581744356058913414896990733495374678311541679779286235926820887109334
069414720153553969620399303497471645295105818733206231015952005428393306899696323457199349641945785006063752636072317610745842993679285117797428263417582578532
805831802011099765608988125668113663036458894572577851249866574518886484499725337699121114849574580489802665416715434557761620127054829750346136135163241553552
702603393530955170538464075218498320565584615710550132558833,

634144277893276711154097052899587943617075405220155528964838410452923384756654094642009687916753459788584364804006877893388621018891561888953453211860920634857
199657872143417660660765729700915130771497434128139326743976911231693128191780261273425817443560589134148969907334953746783115416797792862359268208871093340694
147201535539696203993034974716452951058187332062310159520054283933068996963234571993496419457850060637526360723176107458429936792851177974282634175825785328058
318020110997656089881256681136630364588945725778512498665745188864844997253376991211148495745804898026654167154345577616201270548297503461361351632415535527026
03393530955170538464075218498320565584615710550132572729,

13896} where gap = 13,896.

So we are quite certain to find a gap of 13,128 between some consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Dave.

[Guest]

We shall try see[10^689, 13000, 689, 692] since [tex]212:4,000 \Leftrightarrow 689:13,000[/tex] (Linear Extrapolation).

Dave.

We can solve [tex]212:4,000 \Leftrightarrow 689:13,000 \Leftrightarrow x:10^{12} ...[/tex] (Linear Extrapolation)


If [tex]\frac{689}{13,000} = \frac{x}{10^{12}}[/tex], then [tex]x = 5.3 * 10^{10}[/tex].

Therefore, if we are seeking a gap of more than a trillion between some consecutive primes, then we must search between [tex]10^{5.3 * 10^{10}}[/tex] and [tex]10^{5.3 * 10^{10} + 3}[/tex].

Wow! Math is fun! Math works! And math is powerful!

Dave.

P.S. We believe we have discovered an important and fundamental rule on large gaps between some consecutive primes!

[Guest]

"Therefore, if we are seeking a gap of more than a trillion between some consecutive primes, then we must search between [tex]10^{5.3 * 10^{10}}[/tex] and [tex]10^{5.3 * 10^{10} + 3}[/tex] (or beyond).",

Dave.

[Guest]

We shall try see[10^689, 13000, 689, 692] since [tex]212:4,000 \Leftrightarrow 689:13,000[/tex] (Linear Extrapolation).

Dave.

Eureka!

LINEAR EXTRAPOLATION WORKS!

Input: see[10^689, 13000, 689, 692];

Output:

{ 63414427789327671115409705289958794361707540522015552896483841045292338475665409464200968791675345978858436480400687789338862101889156188895345321186092063
485719965787214341766066076572970091513077149743412813932674397691123169312819178026127342581744356058913414896990733495374678311541679779286235926820887109334
069414720153553969620399303497471645295105818733206231015952005428393306899696323457199349641945785006063752636072317610745842993679285117797428263417582578532
805831802011099765608988125668113663036458894572577851249866574518886484499725337699121114849574580489802665416715434557761620127054829750346136135163241553552
702603393530955170538464075218498320565584615710550132558833,

634144277893276711154097052899587943617075405220155528964838410452923384756654094642009687916753459788584364804006877893388621018891561888953453211860920634857
199657872143417660660765729700915130771497434128139326743976911231693128191780261273425817443560589134148969907334953746783115416797792862359268208871093340694
147201535539696203993034974716452951058187332062310159520054283933068996963234571993496419457850060637526360723176107458429936792851177974282634175825785328058
318020110997656089881256681136630364588945725778512498665745188864844997253376991211148495745804898026654167154345577616201270548297503461361351632415535527026
03393530955170538464075218498320565584615710550132572729,

13896} where gap = 13,896.

So we are quite certain to find a gap of 13,128 between some consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Dave.

FYI:

Input: see[10^689, 13000, 689, 692];

Output:

{
2427621181023042167133280641549601396238158866067639433445410858019664822748923004816477275874448998831903224005310167415219109353516453214997256888932052
0755149929990102304386661147368791940966310181487074419863931866705679374025177614248560995544529030123461462170005044172893764573659512784023380571408047
6414663819475636891404600465133286691025560088509237070387978822368863383905319152953995362246319909973411185551040468632037980251819812816133296574456673
3612803905264183755083671137376033674279710591804739922722021574112047962226565104537509492893293190467963021324580678802566262979289025582092781680675572
1125373224608927425422487669660538695996444072631216851796336994704504664261,

2427621181023042167133280641549601396238158866067639433445410858019664822748923004816477275874448998831903224005310167415219109353516453214997256888932052
0755149929990102304386661147368791940966310181487074419863931866705679374025177614248560995544529030123461462170005044172893764573659512784023380571408047
6414663819475636891404600465133286691025560088509237070387978822368863383905319152953995362246319909973411185551040468632037980251819812816133296574456673
3612803905264183755083671137376033674279710591804739922722021574112047962226565104537509492893293190467963021324580678802566262979289025582092781680675572
1125373224608927425422487669660538695996444072631216851796336994704504677637,

13376} where gap = 13,376.