Sum Of Primes (SOP) Conjecture

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Sum Of Primes (SOP) Conjecture states

(k + Δk)^2 + k + Δk = s / log(s)

where

s = ∑ (pi from i=1 to 2k ≥ 20)

and where prime, pi: p1=2, p2=3, ...

and where ∆k/k → 0 as k → ∞.

https://www.physforum.com/index.php?sho ... 0106&st=60

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David Cole
aka primework123

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The first links seems to be broken.

Please google 'primework123' and search for, 'On The Distribution of Prime Numbers' on physforum website.

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Sum Of Primes (SOP) Conjecture states

(k + Δk)^2 + k + Δk = s / log(s)

where

s = ∑ (pi from i=1 to 2k ≥ 20)

and where prime, pi: p1=2, p2=3, ...

and where ∆k/k → 0 as k → ∞.

https://www.gofundme.com/david_cole (Please support my research efforts.)

Thank you!

David Cole
aka primework123

Examples of SOP Conjecture:

If 2k = 750,000 then s=3,674,162,516,705 which implies ∆k =1357.

If 2k = 60,000 then s=21,410,141,065 which implies ∆k ≈ 0.71.

Reference: http://www.wolframalpha.com/

Please confirm results too. Thanks!

David Cole (aka primework123)

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...
Examples of SOP Conjecture:

If 2k = 750,000 (Wrong!) then s=3,674,162,516,705 which implies ∆k =1357.
...

Oops! Correction!

If 2k = 710,000 then s=3,674,162,516,705 which implies ∆k =1357.

David Cole
aka primework123

[Guest]

Sum Of Primes (SOP) Conjecture states

(k + Δk)^2 + k + Δk = s / log(s)

where

s = ∑ (pi from i=1 to 2k ≥ 20)

and where prime, pi: p1=2, p2=3, ...

and where ∆k/k → 0 as k → ∞.

https://www.physforum.com/index.php?sho ... 0106&st=60

https://www.gofundme.com/david_cole (Please support my research efforts.)

Thank you!

David Cole
aka primework123

Correction!

Instead of ∆k/k → 0 as k → ∞, it should be log(|∆k|/ k) → 0 k → ∞.

Therefore, the SOP Conjecture should be:

(k + Δk)^2 + k + Δk = s / log(s)

where

s =\sum_{k=1}^{2k\ge 20}p_{i }

and where prime, pi: p1=2, p2=3, ...

and where log(|∆k|) / k → 0 as k → [tex]\infty[/tex] in accordance with the Prime Number Theorem (PNT).

[Guest]

Sum Of Primes (SOP) Conjecture states

(k + Δk)^2 + k + Δk = s / log(s)

where

s = ∑ (pi from i=1 to 2k ≥ 20)

and where prime, pi: p1=2, p2=3, ...

and where ∆k/k → 0 as k → ∞.

https://www.physforum.com/index.php?sho ... 0106&st=60

https://www.gofundme.com/david_cole (Please support my research efforts.)

Thank you!

David Cole
aka primework123

Correction!

Sum Of Primes (SOP) Conjecture should be:

(k + Δk)^2 + k + Δk = s / log(s)

where

s =[tex]\sum_{k=1}^{2k }p_{i }[/tex] with 2k ≥ 20

and where prime, [tex]p_{i }[/tex]: [tex]p_{1 }[/tex]=2, [tex]p_{2 }[/tex]=3, ...,[tex]p_{2k }[/tex]

and where |log(∆k)|/k → 0 as k → [tex]\infty[/tex].

David Cole
aka primework123

[Guest]

Sum Of Primes (SOP) Conjecture should be:

(k + Δk)^2 + k + Δk = s / log(s)

where

s =[tex]\sum_{k=1}^{2k }p_{i }[/tex] with 2k ≥ 20

and where prime, [tex]p_{i }[/tex]: [tex]p_{1 }[/tex]=2, [tex]p_{2 }[/tex]=3, ...,[tex]p_{2k }[/tex]

and where |log(|∆k|)|/k → 0 as k → [tex]\infty[/tex].

If we let |∆k| = k^|x|, then |log(|∆k|)| = |x * log(k)| where x << k (another conjecture!)

Some Examples:

Table
k=100,000; s=264,129,169,599; x = .466287; ∆k = 214.50486; log(∆k)/k =.0000250318;
k=300,000; s=2,591,169,323,835; x = .554362; ∆k =1087.19; log(∆k)/k=.0000129191;
k=500,000; s=7,472,966,967,499; x = .5822944; ∆k =2099.81; log(∆k)/k=.00000891858;
k=750,000; s=17,304,581,201,103; x = .60233; ∆k =3457.19; log(∆k)/k=.00000654388;
k=1,000,000; s=31,381,137,530,481; x = .614695; ∆k =4877.19; log(∆k)/k=.00000522019;
(Please confirm all results. And please share your comments. Thank you! )

https://www.gofundme.com/david_cole (Please support my research efforts.)

Thank you!

David Cole
aka primework123

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Table Update: (Please examine and compare all entries especially those marked by arrow.)

k=100,000; s=264,129,169,599; x = .466287; ∆k = 214.50486; log(∆k)/k =.0000250318;

k=300,000; s=2,591,169,323,835; x = .554362; ∆k =1087.19; log(∆k)/k=.0000129191;

k=500,000; s=7,472,966,967,499; x = .5822944; ∆k =2099.81; log(∆k)/k=.00000891858;

(k=750,000; s=17,304,581,201,103; x = .60233; ∆k =3457.19; log(∆k)/k=.00000654388;)

k=1,000,000; s=31,381,137,530,481; x = .614695; ∆k =4877.19; log(∆k)/k=.00000522019;

(k=1,500,000; s=72,562,860,163,907; x = .630583; ∆k =7843.93; log(∆k)/k=.0000037698;)

k=2,000,000; s=131,463,314,443,633; x = .6409; ∆k =10,922.7; log(∆k)/k=.0000029797;

(Please confirm all results. And please share your comments. Thank you! )

https://www.gofundme.com/david_cole (Please support my research efforts.)

Thank you!

David Cole
aka primework123

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Keywords: Linear regression and current data table (see previous post).

Tentatively, I have calculated that ∆k [tex]\approx[/tex] .00568272*k - 635.175 using current data.

David Cole
aka primework123[/quote]
https://www.gofundme.com/david_cole (Please support my research efforts.)
Thank you!

[primework123]

Update:

*****
Sum Of Primes (SOP) Conjecture states:

(k + Δk)^2 + k + Δk = s / log(s) with ∆k [tex]\approx[/tex] .00568272*k - 635.175 (tentative)

where

[tex]s_{2k } = \sum_{i=1}^{2k }p_{i }[/tex] with 2k ≥ 20

and where prime, [tex]p_{i }: p_{1 }=2, p_{i }=3, ...,p_{2k }[/tex];

In addition, |log(|∆k|)|/k → 0 as k → [tex]\infty[/tex].

If we let |∆k| = k^|x|, then |log(|∆k|)| = |x * log(k)| where x << k (another conjecture!)
*****

Now consider:

[tex]s_{2k } -s_{2k-2 }[/tex] = [tex]p_{2k } + p_{2k-1 }[/tex];

[tex]s_{2k-2 } -s_{2k-4 }[/tex] = [tex]p_{2k-2 } + p_{2k-3 }[/tex];

...

[tex]s_{4} -s_{2 }[/tex] = [tex]p_{4 } + p_{3 }[/tex] = 7 + 5 = 12;

[tex]s_{2 }[/tex] = [tex]p_{2 } + p_{1}[/tex] = 3 + 2 = 5;

Moreover, according to Prime Parity Law (PPL), the density of primes in the closed interval, [ 2, [tex]p_{k }[/tex] ], is greater than the density of primes in [ [tex]p_{k+1 }[/tex], [tex]p_{2k }[/tex] ] since [tex]p_{k } - 2 < p_{2k } - p_{k+1 }[/tex] for 10 [tex]\le k <\infty[/tex].


David Cole
aka primework123[/quote]
https://www.gofundme.com/david_cole (Please support my research efforts.)
Thank you!

[primework123]

Update:

*****
Sum Of Primes (SOP) Conjecture states:

[tex](k + Δk)^{2} + k + Δk = s / log(s)[/tex] with [tex]Δk \approx .00568272*k - 635.175[/tex] for [tex]100,000 ≤ k ≤ 2,000,000[/tex]

where

[tex]s_{2k } = \sum_{i=1}^{2k }p_{i }[/tex] with k ≥ 10;

and where prime, [tex]p_{i }: p_{1 }=2, p_{2 }=3,p_{3 }=5, ...,p_{2k }[/tex];

In addition, [tex]|log(|Δk|)|/k → 0[/tex] as [tex]k → \infty[/tex].

If we let [tex]|Δk| = k^{x}[/tex], then [tex]|log(|Δk|)| = |x * log(k)|[/tex] where [tex]x << k[/tex] (another conjecture!)
*****

Now consider:

[tex]s_{2k } -s_{2k-2 }[/tex] = [tex]p_{2k } + p_{2k-1 }[/tex];

[tex]s_{2k-2 } -s_{2k-4 }[/tex] = [tex]p_{2k-2 } + p_{2k-3 }[/tex];

...

[tex]s_{4} -s_{2 }[/tex] = [tex]p_{4 } + p_{3 }[/tex] = 7 + 5 = 12;

[tex]s_{2 }[/tex] = [tex]p_{2 } + p_{1}[/tex] = 3 + 2 = 5;

Moreover, according to Prime Parity Law (PPL), the density of primes in the closed interval, [ 2, [tex]p_{k }[/tex] ], is greater than the density of primes in [ [tex]p_{k+1 }[/tex], [tex]p_{2k }[/tex] ] since [tex]p_{k } - 2 < p_{2k } - p_{k+1 }[/tex] for 10 [tex]\le k <\infty[/tex].


David Cole
aka primework123
https://www.gofundme.com/david_cole (Please support my research efforts.)
Thank you! Thank Lord GOD!

[primework123]

Update:

*****
Sum Of Primes (SOP) Conjecture states:

[tex](k + Δk)^{2} + k + Δk = s_{2k} / log(s_{2k})[/tex] with [tex]Δk \approx .00568272*k - 635.175[/tex]

for [tex]100,000 ≤ k ≤ 2,000,000[/tex]

where

[tex]s_{2k } = \sum_{i=1}^{2k }p_{i }[/tex] with k ≥ 10;

and where prime, [tex]p_{i }: p_{1 }=2, p_{2 }=3,p_{3 }=5, ...,p_{2k }[/tex];

In addition, [tex]|log(|Δk|)|/k → 0[/tex] as [tex]k → \infty[/tex].

If we let [tex]|Δk| = k^{x}[/tex], then [tex]|log(|Δk|)| = |x * log(k)|[/tex] where [tex]x << k[/tex] (another conjecture!)
*****

Now consider:

[tex]s_{2k } -s_{2k-2 }[/tex] = [tex]p_{2k } + p_{2k-1 }[/tex];

[tex]s_{2k-2 } -s_{2k-4 }[/tex] = [tex]p_{2k-2 } + p_{2k-3 }[/tex];

...

[tex]s_{4} -s_{2 }[/tex] = [tex]p_{4 } + p_{3 } = 7 + 5 = 12[/tex];

[tex]s_{2 }[/tex] = [tex]p_{2 } + p_{1} = 3 + 2 = 5[/tex].

Moreover, according to Prime Parity Law (PPL), the density of primes in the closed interval, [ 2, [tex]p_{k }[/tex] ], is greater than the density of primes in [ [tex]p_{k+1 }[/tex], [tex]p_{2k }[/tex] ] since [tex]p_{k } - 2 < p_{2k } - p_{k+1 }[/tex] for 10 [tex]\le k <\infty[/tex].


David Cole
aka primework123
https://www.gofundme.com/david_cole (Please support my research efforts.)
Thank you! Thank Lord GOD!

[Guest]

Update:

*****
Sum Of Primes (SOP) Conjecture states:

[tex](k + Δk)^{2} + k + Δk = s_{2k} / log(s_{2k})[/tex] with [tex]Δk \approx .00568272*k - 635.175[/tex]

for [tex]100,000 ≤ k ≤ 2,000,000[/tex]

where

[tex]s_{2k } = \sum_{i=1}^{2k }p_{i }[/tex] with k ≥ 10;

and where prime, [tex]p_{i }: p_{1 }=2, p_{2 }=3,p_{3 }=5, ...,p_{2k }[/tex];

In addition, [tex]|log(|Δk|)|/k → 0[/tex] as [tex]k → \infty[/tex].

If we let [tex]|Δk| = k^{x}[/tex], then [tex]|log(|Δk|)| = |x * log(k)|[/tex] where [tex]x << k[/tex] (another conjecture!)
*****

Now consider:

[tex]s_{2k } -s_{2k-2 }[/tex] = [tex]p_{2k } + p_{2k-1 }[/tex];

[tex]s_{2k-2 } -s_{2k-4 }[/tex] = [tex]p_{2k-2 } + p_{2k-3 }[/tex];

...

[tex]s_{4} -s_{2 }[/tex] = [tex]p_{4 } + p_{3 } = 7 + 5 = 12[/tex];

[tex]s_{2 }[/tex] = [tex]p_{2 } + p_{1} = 3 + 2 = 5[/tex].

Moreover, according to Prime Parity Law (PPL), the density of primes in the closed interval, [ 2, [tex]p_{k }[/tex] ], is greater than the density of primes in [ [tex]p_{k+1 }[/tex], [tex]p_{2k }[/tex] ] since [tex]p_{k } - 2 < p_{2k } - p_{k+1 }[/tex] for 10 [tex]\le k <\infty[/tex].


David Cole
aka primework123
https://www.gofundme.com/david_cole (Please support my research efforts.)
Thank you! Thank Lord GOD!

[tex]s_{2k } -s_{2k-1 }[/tex] = [tex]p_{2k }[/tex] ;

[tex]s_{2k-1 } -s_{2k-2 }[/tex] = [tex]p_{2k-1 }[/tex];

...

[tex]s_{3} -s_{2 }[/tex] = [tex]p_{3 } = 5[/tex];

[tex]s_{2 }-s_{1}[/tex]= [tex]p_{2 } = 3[/tex].

[tex]s_{1}[/tex]= [tex]p_{1 } = 2[/tex].