Prime number intercept
$begingroup$
Suppose I arrange my (infinite) list of prime numbers in the following way: begin{array}{c|c}x_i&2&5&11&17&23&31&cdots\hline y_i&3&7&13&19&29&37&cdotsend{array} so that if $p_k$ denotes the $k$th prime, $x_i$ contains $p_{2k-1}$ and $y_i$ contains $p_{2k}$. Note that $i=1,2,cdots$.
If $y_i=hat{alpha}+hat{beta}x_i$ is the line of best fit for these primes, does the intercept $hat{alpha}$ converge as $itoinfty$, and if so, to what value?
First, some preliminary thoughts.
Clearly $hatbeta>1$ since $y_i>x_i$ respectively, but $epsilon=hatbeta-1$ will be very small, due to the relatively small difference in $y_i-x_i$.
This may have connections with the twin prime conjecture, but this model only cannot capture all the twin primes due to the way they are arranged (for example, the pair $(29,31)$ does not exist here).
I have plotted the first $600$ primes and added a best fit line to them. During these additions, the intercept $hatalpha$ seems to sway in the interval $(2,4)$. However, this may not show much information as primes get sparser as they get larger, and there would be larger differences between $x_i$ and $y_i$. The plot below shows this; the $R^2$ value of $0.9997$ implies very good fit, and $$hatalpha^{(600)}=3.7909,quadhatbeta^{(600)}=1.00597.$$ I believe convergence of $hatalpha$ is likely as $mapprox1approx R^2$ consistently.
- Statistically, we can obtain expressions for the coefficient estimates. $$hatbeta=frac{isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{isum p_{2k-1}^2-(sum p_{2k-1})^2},quadhatalpha=frac{sum p_{2k}-hatbetasum p_{2k-1}}i$$ In a single formula, in terms of $p_{2k}$, $p_{2k-1}$ and $i$ only, $$hatalpha=frac{(isum p_{2k-1}^2-(sum p_{2k-1})^2)sum p_{2k}-isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{i(isum p_{2k-1}^2-(sum p_{2k-1})^2)}$$ although its direct use may be impractical.
convergence prime-numbers linear-regression
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
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add a comment |
$begingroup$
Suppose I arrange my (infinite) list of prime numbers in the following way: begin{array}{c|c}x_i&2&5&11&17&23&31&cdots\hline y_i&3&7&13&19&29&37&cdotsend{array} so that if $p_k$ denotes the $k$th prime, $x_i$ contains $p_{2k-1}$ and $y_i$ contains $p_{2k}$. Note that $i=1,2,cdots$.
If $y_i=hat{alpha}+hat{beta}x_i$ is the line of best fit for these primes, does the intercept $hat{alpha}$ converge as $itoinfty$, and if so, to what value?
First, some preliminary thoughts.
Clearly $hatbeta>1$ since $y_i>x_i$ respectively, but $epsilon=hatbeta-1$ will be very small, due to the relatively small difference in $y_i-x_i$.
This may have connections with the twin prime conjecture, but this model only cannot capture all the twin primes due to the way they are arranged (for example, the pair $(29,31)$ does not exist here).
I have plotted the first $600$ primes and added a best fit line to them. During these additions, the intercept $hatalpha$ seems to sway in the interval $(2,4)$. However, this may not show much information as primes get sparser as they get larger, and there would be larger differences between $x_i$ and $y_i$. The plot below shows this; the $R^2$ value of $0.9997$ implies very good fit, and $$hatalpha^{(600)}=3.7909,quadhatbeta^{(600)}=1.00597.$$ I believe convergence of $hatalpha$ is likely as $mapprox1approx R^2$ consistently.
- Statistically, we can obtain expressions for the coefficient estimates. $$hatbeta=frac{isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{isum p_{2k-1}^2-(sum p_{2k-1})^2},quadhatalpha=frac{sum p_{2k}-hatbetasum p_{2k-1}}i$$ In a single formula, in terms of $p_{2k}$, $p_{2k-1}$ and $i$ only, $$hatalpha=frac{(isum p_{2k-1}^2-(sum p_{2k-1})^2)sum p_{2k}-isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{i(isum p_{2k-1}^2-(sum p_{2k-1})^2)}$$ although its direct use may be impractical.
convergence prime-numbers linear-regression
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
$endgroup$
$begingroup$
Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ?
$endgroup$
– Peter
Jan 20 at 17:32
add a comment |
$begingroup$
Suppose I arrange my (infinite) list of prime numbers in the following way: begin{array}{c|c}x_i&2&5&11&17&23&31&cdots\hline y_i&3&7&13&19&29&37&cdotsend{array} so that if $p_k$ denotes the $k$th prime, $x_i$ contains $p_{2k-1}$ and $y_i$ contains $p_{2k}$. Note that $i=1,2,cdots$.
If $y_i=hat{alpha}+hat{beta}x_i$ is the line of best fit for these primes, does the intercept $hat{alpha}$ converge as $itoinfty$, and if so, to what value?
First, some preliminary thoughts.
Clearly $hatbeta>1$ since $y_i>x_i$ respectively, but $epsilon=hatbeta-1$ will be very small, due to the relatively small difference in $y_i-x_i$.
This may have connections with the twin prime conjecture, but this model only cannot capture all the twin primes due to the way they are arranged (for example, the pair $(29,31)$ does not exist here).
I have plotted the first $600$ primes and added a best fit line to them. During these additions, the intercept $hatalpha$ seems to sway in the interval $(2,4)$. However, this may not show much information as primes get sparser as they get larger, and there would be larger differences between $x_i$ and $y_i$. The plot below shows this; the $R^2$ value of $0.9997$ implies very good fit, and $$hatalpha^{(600)}=3.7909,quadhatbeta^{(600)}=1.00597.$$ I believe convergence of $hatalpha$ is likely as $mapprox1approx R^2$ consistently.
- Statistically, we can obtain expressions for the coefficient estimates. $$hatbeta=frac{isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{isum p_{2k-1}^2-(sum p_{2k-1})^2},quadhatalpha=frac{sum p_{2k}-hatbetasum p_{2k-1}}i$$ In a single formula, in terms of $p_{2k}$, $p_{2k-1}$ and $i$ only, $$hatalpha=frac{(isum p_{2k-1}^2-(sum p_{2k-1})^2)sum p_{2k}-isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{i(isum p_{2k-1}^2-(sum p_{2k-1})^2)}$$ although its direct use may be impractical.
convergence prime-numbers linear-regression
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
$endgroup$
Suppose I arrange my (infinite) list of prime numbers in the following way: begin{array}{c|c}x_i&2&5&11&17&23&31&cdots\hline y_i&3&7&13&19&29&37&cdotsend{array} so that if $p_k$ denotes the $k$th prime, $x_i$ contains $p_{2k-1}$ and $y_i$ contains $p_{2k}$. Note that $i=1,2,cdots$.
If $y_i=hat{alpha}+hat{beta}x_i$ is the line of best fit for these primes, does the intercept $hat{alpha}$ converge as $itoinfty$, and if so, to what value?
First, some preliminary thoughts.
Clearly $hatbeta>1$ since $y_i>x_i$ respectively, but $epsilon=hatbeta-1$ will be very small, due to the relatively small difference in $y_i-x_i$.
This may have connections with the twin prime conjecture, but this model only cannot capture all the twin primes due to the way they are arranged (for example, the pair $(29,31)$ does not exist here).
I have plotted the first $600$ primes and added a best fit line to them. During these additions, the intercept $hatalpha$ seems to sway in the interval $(2,4)$. However, this may not show much information as primes get sparser as they get larger, and there would be larger differences between $x_i$ and $y_i$. The plot below shows this; the $R^2$ value of $0.9997$ implies very good fit, and $$hatalpha^{(600)}=3.7909,quadhatbeta^{(600)}=1.00597.$$ I believe convergence of $hatalpha$ is likely as $mapprox1approx R^2$ consistently.
- Statistically, we can obtain expressions for the coefficient estimates. $$hatbeta=frac{isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{isum p_{2k-1}^2-(sum p_{2k-1})^2},quadhatalpha=frac{sum p_{2k}-hatbetasum p_{2k-1}}i$$ In a single formula, in terms of $p_{2k}$, $p_{2k-1}$ and $i$ only, $$hatalpha=frac{(isum p_{2k-1}^2-(sum p_{2k-1})^2)sum p_{2k}-isum p_{2k-1}p_{2k}-sum p_{2k-1}sum p_{2k}}{i(isum p_{2k-1}^2-(sum p_{2k-1})^2)}$$ although its direct use may be impractical.
convergence prime-numbers linear-regression
convergence prime-numbers linear-regression
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
asked Jan 20 at 14:20
TheSimpliFireTheSimpliFire
12.5k62460
12.5k62460
$begingroup$
Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ?
$endgroup$
– Peter
Jan 20 at 17:32
add a comment |
$begingroup$
Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ?
$endgroup$
– Peter
Jan 20 at 17:32
$begingroup$
Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ?
$endgroup$
– Peter
Jan 20 at 17:32
$begingroup$
Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ?
$endgroup$
– Peter
Jan 20 at 17:32
add a comment |
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$begingroup$
Nice question(+1) The $109$ th prime is $599$ , sure that the table is correct ($599$ should appear on the left) ?
$endgroup$
– Peter
Jan 20 at 17:32