On a Grand Hypothesis of Fundamental Number Theory

[Guest]

Relevant Reference Link:

https://www.wolframcloud.com/obj/openmind123omega/Published/primegap.nb.

Dave.

[Guest]

Hmm. If linear extrapolation works here, then linear interpolation should also work. What we need now is a proof for our important and fundamental rule...

Dave.

[Guest]

Hmm. If linear extrapolation works here, then linear interpolation should also work. What we need now is a proof for our important and fundamental rule...

Dave.

"I do not have a clear and complete explanation. I do not know ...!" -- Dave.

[Guest]

Hmm. We are still searching for that elusive gap of 13,128 between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Hopefully, we will have an affirmative result soon.

Dave.

[Guest]

Hmm. We are still searching for that elusive gap of 13,128 between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Hopefully, we will have an affirmative result soon.

Dave.

FYI: There are roughly [tex]6.270 * 10^{688}[/tex] primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Are you very lucky?

Dave.

[Guest]

Hmm. We are still searching for that elusive gap of 13,128 between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Hopefully, we will have an affirmative result soon.

Dave.

FYI: Prime Gap Expectations: [tex]2 \le[/tex] gap [tex]< 2,538,888[/tex] (the maximum gap number: l[tex]og^{2}(10^{692}) \approx 2,538,888[/tex] ) between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Right?

Dave.

[Guest]

FYI:

Input: see[10^689, 13128, 689, 692]

Output:

{
508054408734207633183342333742902665842198266634856338934925807459420109625556168436623135082410722378958013794939363328766
506847480538398996203304767706550125730311025108229647260340935023594680652315414176102503799808233184495374497811560714269
890122871694214229214134165209632283226942947038129521769245680190755389137914276202364911907324520900346363678404178876219
457796854468767545435778686417512783215659363761364150799015191198032927965171527130608725107602644997615381808626153284348
629330302083251142786767689285955866949309844108558017227350288725847837876901390735030673239528339802330263620411952280657
78643570140577107786504306722922893427154020070238529781143986155803563513291,

508054408734207633183342333742902665842198266634856338934925807459420109625556168436623135082410722378958013794939363328766
506847480538398996203304767706550125730311025108229647260340935023594680652315414176102503799808233184495374497811560714269
890122871694214229214134165209632283226942947038129521769245680190755389137914276202364911907324520900346363678404178876219
457796854468767545435778686417512783215659363761364150799015191198032927965171527130608725107602644997615381808626153284348
629330302083251142786767689285955866949309844108558017227350288725847837876901390735030673239528339802330263620411952280657
78643570140577107786504306722922893427154020070238529781143986155803563527303,

14012} where gap = 14012.

Remarks: So far, the gap of 13128 (M. L. H.) is still elusive!

Dave.

[Guest]

FYI:

Input: see{10^689, 13128, 689, 692]

Output:

{
8144100487256316370818351412465511831168854459092080353664170376476635149783787505902873954742026770160161847508051973676246658567292386524700097
3404058440655413277292974943350099189637520532427391716331752283024524751769739704886613349802991623799333218657912624328845080811289141488074509
4595406649804763272930127281584453952260875376981160869546496201315756043921614066657333780491377276302427012446369924637924688216566534446241056
2789447846313615988580385953213252773638502130861011265468904839645369564743844932983568044656046946129582259507846749127419447133612761018474414
3846593367183940223313550375053507336173257252148389395431270631805601740343569301522041151785724685476723258449,

8144100487256316370818351412465511831168854459092080353664170376476635149783787505902873954742026770160161847508051973676246658567292386524700097
3404058440655413277292974943350099189637520532427391716331752283024524751769739704886613349802991623799333218657912624328845080811289141488074509
4595406649804763272930127281584453952260875376981160869546496201315756043921614066657333780491377276302427012446369924637924688216566534446241056
2789447846313615988580385953213252773638502130861011265468904839645369564743844932983568044656046946129582259507846749127419447133612761018474414
3846593367183940223313550375053507336173257252148389395431270631805601740343569301522041151785724685476723273269,

14820} where gap = 14,820.

[Guest]

FYI:

Input: see[10^689, 13128, 689, 692]

Output:

{
881614645566049907208803517191174939563477696447029517372229114789424509727411592807817421934442908760869918748217633181567676537111094524324
623669861085112840645633464400285018151493838923581690858816781520862355749241298255467984632658893381832766947159303314015394545356386873095
562477609416924951084698498101702571490766806610511135040010966317399867112009349810354821551351980855651926285405281361538716243174681101392
425000081743754438677863169132787650897084012770299651290621419849877832962099735112344707734490839588822422770466878187827554444861571949613
21506508473948638147284730118527678071401640464346643351927357658206358996408335623767403834307090957136504122621750510105669923,

881614645566049907208803517191174939563477696447029517372229114789424509727411592807817421934442908760869918748217633181567676537111094524324
623669861085112840645633464400285018151493838923581690858816781520862355749241298255467984632658893381832766947159303314015394545356386873095
562477609416924951084698498101702571490766806610511135040010966317399867112009349810354821551351980855651926285405281361538716243174681101392
425000081743754438677863169132787650897084012770299651290621419849877832962099735112344707734490839588822422770466878187827554444861571949613
21506508473948638147284730118527678071401640464346643351927357658206358996408335623767403834307090957136504122621750510105683059,

13136} where gap = 13,136 which is close to the gap of 13,128.

[Guest]

Hmm. We are still searching for that elusive gap of 13,128 between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Hopefully, we will have an affirmative result soon.

Dave.

FYI: Prime Gap Expectations: [tex]2 \le[/tex] gap [tex]< 2,538,888[/tex] (the maximum gap number: l[tex]og^{2}(10^{692}) \approx 2,538,888[/tex] ) between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Right?

Dave.

Thus far, we expect an average prime gap of roughly [tex]692 * log(10) \approx 1594[/tex] between consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex] according to theory.

Higher prime gaps than the average prime gap (1,594) are exceedingly difficult to discover according to theory. Thus, the distribution of prime gaps is skewed!

Dave.

[Guest]

Hmm. Our conclusion is not definitive! And we expect better results with more powerful computers and with better programming etc. -- Dave.

Wolfram Mathematica Revised Code:

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

SeedRandom[];

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

While [ NextPrime[start] - start < gap,

If[ RandomInteger[]==0, start = NextPrime[RandomInteger[{n,RandomInteger[{n, 10^u}]}]], start = NextPrime[RandomInteger[{RandomInteger[{n, 10^u}], 10^u}]]]];

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

_____________________end of code_____________________

Relevant Reference Link:

https://www.wolframcloud.com/.

FYI:

Input: see[10^689, 13128, 689, 692]

Output:

{
54631494895515698608190997633988086894451571322456760899683161503562683553728393668355442364309383347402594102156537808793288267194187570402741382588538410140279423596087568478
96123582075321088600842607923147165507251373893157667242002919386365972068302467776380660240890100708406358202406669571495428217973225321368484511699378958820874976741319412862
46794309947536228218297579895331460300996712165725547513120386554961620466423008706236787997521228954073007930601800766291467766146315548361473932855849330499286768791051218909
25201472771261673611400088747127540560834596496171928467307106121651508553700439526126528606546176216928066043519770876036616407321993817310368859181601997307538887,

546314948955156986081909976339880868944515713224567608996831615035626835537283936683554423643093833474025941021565378087932882671941875704027413825885384101402794235960875684789
612358207532108860084260792314716550725137389315766724200291938636597206830246777638066024089010070840635820240666957149542821797322532136848451169937895882087497674131941286246
794309947536228218297579895331460300996712165725547513120386554961620466423008706236787997521228954073007930601800766291467766146315548361473932855849330499286768791051218909252
01472771261673611400088747127540560834596496171928467307106121651508553700439526126528606546176216928066043519770876036616407321993817310368859181601997307552711,

13824} where gap = 13,824.

[Guest]

Hmm. It would great if we could also discover the maximum gap (< 2,538,888) between consecutive primes between [tex]10^{689}[/tex] and[tex]10^{692}[/tex]. That very unlikely result should confirm theory.

Dave.

[Guest]

Hmm. It would be great if we could also discover the maximum gap (< 2,538,888) between consecutive primes between [tex]10^{689}[/tex] and[tex]10^{692}[/tex]. That very unlikely result should confirm theory.

Dave.

[Guest]

FYI:

Input: see[10^689, 13128, 689, 692]

Output:

{
37493528313978692681793302027874973249582207757812212300816309618295819122717840530416729762020413471956827348293066968289455386253271327696755911023650236549254132724333356
48444733806289061835278288538962067510902395976684681286362719434441275267482281817797812138162975509028430092512123524868637148951855155082005688914070635410267535174919321
26579171347627051432582944638337030695705585298524291285500276383986382768740052234930206833407030501128185626101161055941619368080768450686102468195390566108463939566565352
73275950663272317253795510929627033628576311760577425762242326725386423766610230190131364044050762858777290193828023965402779913691040326000059828211640615772419991914211883,

37493528313978692681793302027874973249582207757812212300816309618295819122717840530416729762020413471956827348293066968289455386253271327696755911023650236549254132724333356
48444733806289061835278288538962067510902395976684681286362719434441275267482281817797812138162975509028430092512123524868637148951855155082005688914070635410267535174919321
26579171347627051432582944638337030695705585298524291285500276383986382768740052234930206833407030501128185626101161055941619368080768450686102468195390566108463939566565352
73275950663272317253795510929627033628576311760577425762242326725386423766610230190131364044050762858777290193828023965402779913691040326000059828211640615772419991914230363,

18480} with gap = 18,480.

[Guest]

Relevant Reference Link:

'Prime Gap',

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

[Guest]

Hmm. We are still searching for that elusive gap of 13,128 between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Hopefully, we will have an affirmative result soon.

Dave.

FYI: Prime Gap Expectations: [tex]2 \le[/tex] gap [tex]< 2,538,888[/tex] (the maximum gap number: l[tex]og^{2}(10^{692}) \approx 2,538,888[/tex] ) between any consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

Right?

Dave.

Thus far, we expect an average prime gap of roughly [tex]692 * log(10) \approx 1594[/tex] between consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex] according to theory.

Higher prime gaps than the average prime gap (1,594) are exceedingly difficult to discover according to theory. Thus, the distribution of prime gaps is skewed!

Dave.

Oops! The average prime gap of 1,594 is wrong! It is actually greater than that value between consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex].

The average prime gap of 1,594 is approximately correct for all consecutive primes between 3 and [tex]10^{692}[/tex] according to theory.

[Guest]

Hmm. Our conclusion is not definitive! And we expect better results with more powerful computers and with better programming etc. -- Dave.

Wolfram Mathematica Revised Code:

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

SeedRandom[];

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

While [ NextPrime[start] - start < gap,

If[ RandomInteger[]==0, start = NextPrime[RandomInteger[{n,RandomInteger[{n, 10^u}]}]], start = NextPrime[RandomInteger[{RandomInteger[{n, 10^u}], 10^u}]]]];

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

_____________________end of code_____________________

Relevant Reference Link:

https://www.wolframcloud.com/.

FYI:

Input: see[10^689, 13128, 689, 692]

Output:

{
54631494895515698608190997633988086894451571322456760899683161503562683553728393668355442364309383347402594102156537808793288267194187570402741382588538410140279423596087568478
96123582075321088600842607923147165507251373893157667242002919386365972068302467776380660240890100708406358202406669571495428217973225321368484511699378958820874976741319412862
46794309947536228218297579895331460300996712165725547513120386554961620466423008706236787997521228954073007930601800766291467766146315548361473932855849330499286768791051218909
25201472771261673611400088747127540560834596496171928467307106121651508553700439526126528606546176216928066043519770876036616407321993817310368859181601997307538887,

546314948955156986081909976339880868944515713224567608996831615035626835537283936683554423643093833474025941021565378087932882671941875704027413825885384101402794235960875684789
612358207532108860084260792314716550725137389315766724200291938636597206830246777638066024089010070840635820240666957149542821797322532136848451169937895882087497674131941286246
794309947536228218297579895331460300996712165725547513120386554961620466423008706236787997521228954073007930601800766291467766146315548361473932855849330499286768791051218909252
01472771261673611400088747127540560834596496171928467307106121651508553700439526126528606546176216928066043519770876036616407321993817310368859181601997307552711,

13824} where gap = 13,824.

Recall: We are searching for the maximum gap between consecutive primes between [tex]10^{689}[/tex] and [tex]10^{692}[/tex]. And there are roughly [tex]2.70 *10^{688}[/tex] primes between those numbers. And expect that the maximum gap is significantly less than 2,538,888 according to theory.

The parameter, gap, in the above program, can be increased until the maximum gap between consecutive primes is found in the specified range...

If we had access to the Summit, IBM supercomputer, we could probably find that maximum gap over several days or less since the search of all specified primes is not required.

Relevant Reference Link:

'Summit',

https://en.wikipedia.org/wiki/Summit_(supercomputer).

[Guest]

FYI: 'Japan's coronavirus supercomputer named fastest in world.
The Fugaku supercomputer topped the world rankings at once for the first time in history. It has been put to work creating models to fight the coronavirus pandemic, but its designers now have bigger plans for it.'

https://www.dw.com/en/japans-coronavirus-supercomputer-named-fastest-in-world/a-53901703.

[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 is true!" -- David Cole.

Relevant Reference Link:

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

...

An Update:

"Remark: Polignac's conjecture is true! (...)"

Hah! We are not convinced that Polignac's conjecture is true! There may be exceptional sets such that Poignac's conjecture is generally false!

Let's assume [tex]1 \le \lambda < \frac{log^{2}(pq)}{2}[/tex] is in accordance with the Prime Number Theorem when p > q are consecutive odd primes with [tex]p - q = 2 \lambda[/tex].

Are there infinitely many consecutive prime pairs, p and q, such that [tex]\sqrt{pq + \lambda^{2}}[/tex] is a positive integer when [tex]\lambda =1[/tex]?

Are there infinitely many consecutive prime pairs, p and q, such that [tex]\sqrt{pq + \lambda^{2}}[/tex] is a positive integer when [tex]\lambda =2[/tex]?

Are there infinitely many consecutive prime pairs, p and q, such that [tex]\sqrt{pq + \lambda^{2}}[/tex] is a positive integer when [tex]\lambda =3[/tex]?
...

What is the probability that Polignac's conjecture is generally false for any positive even integer, [tex]2 \lambda[/tex]?

Remark: The exponent,[tex]\frac{1}{2}[/tex], in the equation, [tex]\sqrt{pq + \lambda^{2}} = (pq + \lambda^{2})^{\frac{1}{2}}[/tex], is a big indicator of the truth of the Riemann Hypothesis!

Assumption: Polignac's conjecture is generally false for any positive even integer, [tex]2 \lambda[/tex].

To indicate that the Polignac's conjecture is false or [tex]p - q \ne 2 \lambda[/tex] for for all or almost all consecutive primes p > q, we define an exceptional and infinite set, E.

E = {(p, q) |[tex]p > q > 2 \lambda[/tex] are consecutive primes with [tex]p - q \ne 2 \lambda[/tex]}.

The inequality, [tex]p - q \ne 2 \lambda[/tex], indicates [tex]n_{m } *q_{m } = p - 2 \lambda[/tex] where [tex]n_{m} > 1[/tex] is an odd integer associated with some odd prime, [tex]q_{m }[/tex].

Remark: [tex]n_{m} \ge q_{m}[/tex].

Remark: [tex]n_{m} = 1[/tex] indicates that Polignac's conjecture is true!

Therefore, as a result of E, we generate an infinite system of independent Diophantine equations with odd integer, [tex]n_{k } > 1[/tex], for appropriate odd primes, [tex]p_{k}[/tex] and [tex]q_{k}[/tex]:

[tex]p_{1 } - 2 \lambda = n_{1 } *q_{1 }[/tex];

[tex]p_{2 } - 2 \lambda = n_{2 } *q_{2 }[/tex];

[tex]p_{3 } - 2 \lambda = n_{3 } *q_{3 }[/tex];

...

[tex]p_{\infty } - 2 \lambda = n_{\infty } *q_{\infty}[/tex];

Remark: [tex]p_{k} < p_{k+1}[/tex] are consecutive primes over E.

Remark: [tex]3 \le q_{k} \le \sqrt{ n_{k } *q_{k}} \le n_{k }[/tex].

Remark: [tex]\pi (x)[/tex] indicates the odd prime-counting function.

Remark: We assume the Riemann Hypothesis since it is true! Go Blue!

Remark: "The Riemann Hypothesis is equivalent to a much tighter bound on the error in the estimate for [tex]\pi (x)[/tex], and hence to a more regular distribution of prime numbers..." Source Link: https://en.wikipedia.org/wiki/Prime-counting_function#The_Riemann_hypothesis.

Remark: [tex]\pi ( \sqrt{p_{m } - 2})[/tex] is the maximum number of primes, [tex]q_{m }[/tex], that may divide [tex]p_{m } - 2[/tex].

The probability that Polignac's conjecture is false over E is Probability ([tex]p - 2 \lambda \ne q[/tex] over E ) or Prob( [tex]p - 2 \lambda \ne q[/tex] over E ).

Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = Prob ( [tex]n_{m} \ne 1 | p_{m } - 2 \lambda = n_{m } *q_{m }[/tex] ) * Prob ( [tex]p_{m } - 2 \lambda = n_{m } *q_{m }[/tex]).

Remark: Prob ( [tex]p_{m } - 2 \lambda = n_{m } *q_{m }[/tex] ) = 1 since [tex]n_{m } \ne 1[/tex].

Therefore,

Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = Prob ( [tex]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: We count [tex]n_{m} = 1[/tex], the exception that violates our assumption.

That result [ Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = 0 ] contradicts our assumption.

Remark: Polignac's conjecture is true!

Remark: The tighter error bound associated with the odd prime-counting function does not violate our final result.

Dave.

Go Blue!

Relevant Reference Link:

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

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

[Guest]

Update: Prob( [tex]p - 2 \lambda \ne q[/tex] over E ) = Prob ( [tex]n_{m} \ne 1 | 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].