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