Goldbach's Conjecture and the totient function

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A while ago, I was somewhat bored, so I decided to plot the number of ways that each even could be expressed as the sum of two primes. The evens are on the x-axis, and the number of different ways (where ordering doesn't matter) is on the y-axis.



enter image description here
This graph is strikingly similar to that of the totient function
enter image description here



Is there any good explanation for the similarity of these graphs? Or am I seeing similarities where there are none?



Update



Upon seeing @RobertIsrael's answer, I decided to do some more plotting. This time, it has been colorized so that multiples of 3 are green, multiples of 5 are red, multiples of 7 are blue, multiples of 15 are yellow, multiples of 21 are cyan, and multiples of 35 are magenta, while multiples of 105 are gray, etc, etc.



enter image description here



My conjecture is that the graph here is more similar to $n-phi(n)$, as it seems to be related to the number of divisors $n$ has. This might also explain why the totient graph looks like an upside-down version of the Goldbach graph.






number-theory prime-numbers totient-function goldbachs-conjecture







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  • The general form is similar, but you didn't care to write down scales in the first, upper graph... So far, this looks like saying the graphs of $;x^2,,x^4;$ are "similar".
    – DonAntonio
    4 hours ago







  • 1




    I don't find them very similar since the first graph is more like a square root relation and the second graph has somewhat $mathbbR^2$-geodesics-like trends. I really do like the plots though.
    – Guus Palmer
    4 hours ago











  • @DonAntonio Oh the scales are nearly unreadable. I'll fix that momentarily.
    – Rushabh Mehta
    4 hours ago






  • 1




    @DonAntonio Please see edits!
    – Rushabh Mehta
    3 hours ago










  • @Downvoter is there something else you'd like to see?
    – Rushabh Mehta
    2 hours ago














up vote
3
down vote

favorite
1












A while ago, I was somewhat bored, so I decided to plot the number of ways that each even could be expressed as the sum of two primes. The evens are on the x-axis, and the number of different ways (where ordering doesn't matter) is on the y-axis.



enter image description here
This graph is strikingly similar to that of the totient function
enter image description here



Is there any good explanation for the similarity of these graphs? Or am I seeing similarities where there are none?



Update



Upon seeing @RobertIsrael's answer, I decided to do some more plotting. This time, it has been colorized so that multiples of 3 are green, multiples of 5 are red, multiples of 7 are blue, multiples of 15 are yellow, multiples of 21 are cyan, and multiples of 35 are magenta, while multiples of 105 are gray, etc, etc.



enter image description here



My conjecture is that the graph here is more similar to $n-phi(n)$, as it seems to be related to the number of divisors $n$ has. This might also explain why the totient graph looks like an upside-down version of the Goldbach graph.






number-theory prime-numbers totient-function goldbachs-conjecture







share|cite|improve this question























  • The general form is similar, but you didn't care to write down scales in the first, upper graph... So far, this looks like saying the graphs of $;x^2,,x^4;$ are "similar".
    – DonAntonio
    4 hours ago







  • 1




    I don't find them very similar since the first graph is more like a square root relation and the second graph has somewhat $mathbbR^2$-geodesics-like trends. I really do like the plots though.
    – Guus Palmer
    4 hours ago











  • @DonAntonio Oh the scales are nearly unreadable. I'll fix that momentarily.
    – Rushabh Mehta
    4 hours ago






  • 1




    @DonAntonio Please see edits!
    – Rushabh Mehta
    3 hours ago










  • @Downvoter is there something else you'd like to see?
    – Rushabh Mehta
    2 hours ago












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





A while ago, I was somewhat bored, so I decided to plot the number of ways that each even could be expressed as the sum of two primes. The evens are on the x-axis, and the number of different ways (where ordering doesn't matter) is on the y-axis.



enter image description here
This graph is strikingly similar to that of the totient function
enter image description here



Is there any good explanation for the similarity of these graphs? Or am I seeing similarities where there are none?



Update



Upon seeing @RobertIsrael's answer, I decided to do some more plotting. This time, it has been colorized so that multiples of 3 are green, multiples of 5 are red, multiples of 7 are blue, multiples of 15 are yellow, multiples of 21 are cyan, and multiples of 35 are magenta, while multiples of 105 are gray, etc, etc.



enter image description here



My conjecture is that the graph here is more similar to $n-phi(n)$, as it seems to be related to the number of divisors $n$ has. This might also explain why the totient graph looks like an upside-down version of the Goldbach graph.






number-theory prime-numbers totient-function goldbachs-conjecture







share|cite|improve this question















A while ago, I was somewhat bored, so I decided to plot the number of ways that each even could be expressed as the sum of two primes. The evens are on the x-axis, and the number of different ways (where ordering doesn't matter) is on the y-axis.



enter image description here
This graph is strikingly similar to that of the totient function
enter image description here



Is there any good explanation for the similarity of these graphs? Or am I seeing similarities where there are none?



Update



Upon seeing @RobertIsrael's answer, I decided to do some more plotting. This time, it has been colorized so that multiples of 3 are green, multiples of 5 are red, multiples of 7 are blue, multiples of 15 are yellow, multiples of 21 are cyan, and multiples of 35 are magenta, while multiples of 105 are gray, etc, etc.



enter image description here



My conjecture is that the graph here is more similar to $n-phi(n)$, as it seems to be related to the number of divisors $n$ has. This might also explain why the totient graph looks like an upside-down version of the Goldbach graph.






number-theory prime-numbers totient-function goldbachs-conjecture




number-theory prime-numbers totient-function goldbachs-conjecture






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 16 mins ago

























asked 4 hours ago









Rushabh Mehta

3,687329




3,687329











  • The general form is similar, but you didn't care to write down scales in the first, upper graph... So far, this looks like saying the graphs of $;x^2,,x^4;$ are "similar".
    – DonAntonio
    4 hours ago







  • 1




    I don't find them very similar since the first graph is more like a square root relation and the second graph has somewhat $mathbbR^2$-geodesics-like trends. I really do like the plots though.
    – Guus Palmer
    4 hours ago











  • @DonAntonio Oh the scales are nearly unreadable. I'll fix that momentarily.
    – Rushabh Mehta
    4 hours ago






  • 1




    @DonAntonio Please see edits!
    – Rushabh Mehta
    3 hours ago










  • @Downvoter is there something else you'd like to see?
    – Rushabh Mehta
    2 hours ago
















  • The general form is similar, but you didn't care to write down scales in the first, upper graph... So far, this looks like saying the graphs of $;x^2,,x^4;$ are "similar".
    – DonAntonio
    4 hours ago







  • 1




    I don't find them very similar since the first graph is more like a square root relation and the second graph has somewhat $mathbbR^2$-geodesics-like trends. I really do like the plots though.
    – Guus Palmer
    4 hours ago











  • @DonAntonio Oh the scales are nearly unreadable. I'll fix that momentarily.
    – Rushabh Mehta
    4 hours ago






  • 1




    @DonAntonio Please see edits!
    – Rushabh Mehta
    3 hours ago










  • @Downvoter is there something else you'd like to see?
    – Rushabh Mehta
    2 hours ago















The general form is similar, but you didn't care to write down scales in the first, upper graph... So far, this looks like saying the graphs of $;x^2,,x^4;$ are "similar".
– DonAntonio
4 hours ago





The general form is similar, but you didn't care to write down scales in the first, upper graph... So far, this looks like saying the graphs of $;x^2,,x^4;$ are "similar".
– DonAntonio
4 hours ago





1




1




I don't find them very similar since the first graph is more like a square root relation and the second graph has somewhat $mathbbR^2$-geodesics-like trends. I really do like the plots though.
– Guus Palmer
4 hours ago





I don't find them very similar since the first graph is more like a square root relation and the second graph has somewhat $mathbbR^2$-geodesics-like trends. I really do like the plots though.
– Guus Palmer
4 hours ago













@DonAntonio Oh the scales are nearly unreadable. I'll fix that momentarily.
– Rushabh Mehta
4 hours ago




@DonAntonio Oh the scales are nearly unreadable. I'll fix that momentarily.
– Rushabh Mehta
4 hours ago




1




1




@DonAntonio Please see edits!
– Rushabh Mehta
3 hours ago




@DonAntonio Please see edits!
– Rushabh Mehta
3 hours ago












@Downvoter is there something else you'd like to see?
– Rushabh Mehta
2 hours ago




@Downvoter is there something else you'd like to see?
– Rushabh Mehta
2 hours ago










1 Answer
1






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It appears that the numbers divisible by $6$ tend to have more representations than those $equiv 2$ or $4 mod 6$. This is because if
$n = x + y$ with $x <y$ odd and not divisible by $3$, $n equiv 2 mod 6$ requires $x, y equiv 1 mod 6$, $n equiv 4 mod 6$ requires $x, y equiv 5 mod 6$, but $n equiv 0 mod 6$ can
have either $x equiv 1$, $y equiv 5$ or $x equiv 5$, $y equiv 1 mod 6$.



Here's a plot of the number of representations of even numbers up to $10000$, with those $equiv 2mod 6$ in green, $4 mod 6$ in blue, $0mod 6$ in red.



enter image description here






share|cite|improve this answer






















  • Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
    – vadim123
    3 hours ago










  • Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
    – Rushabh Mehta
    3 hours ago










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up vote
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It appears that the numbers divisible by $6$ tend to have more representations than those $equiv 2$ or $4 mod 6$. This is because if
$n = x + y$ with $x <y$ odd and not divisible by $3$, $n equiv 2 mod 6$ requires $x, y equiv 1 mod 6$, $n equiv 4 mod 6$ requires $x, y equiv 5 mod 6$, but $n equiv 0 mod 6$ can
have either $x equiv 1$, $y equiv 5$ or $x equiv 5$, $y equiv 1 mod 6$.



Here's a plot of the number of representations of even numbers up to $10000$, with those $equiv 2mod 6$ in green, $4 mod 6$ in blue, $0mod 6$ in red.



enter image description here






share|cite|improve this answer






















  • Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
    – vadim123
    3 hours ago










  • Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
    – Rushabh Mehta
    3 hours ago














up vote
3
down vote













It appears that the numbers divisible by $6$ tend to have more representations than those $equiv 2$ or $4 mod 6$. This is because if
$n = x + y$ with $x <y$ odd and not divisible by $3$, $n equiv 2 mod 6$ requires $x, y equiv 1 mod 6$, $n equiv 4 mod 6$ requires $x, y equiv 5 mod 6$, but $n equiv 0 mod 6$ can
have either $x equiv 1$, $y equiv 5$ or $x equiv 5$, $y equiv 1 mod 6$.



Here's a plot of the number of representations of even numbers up to $10000$, with those $equiv 2mod 6$ in green, $4 mod 6$ in blue, $0mod 6$ in red.



enter image description here






share|cite|improve this answer






















  • Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
    – vadim123
    3 hours ago










  • Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
    – Rushabh Mehta
    3 hours ago












up vote
3
down vote










up vote
3
down vote









It appears that the numbers divisible by $6$ tend to have more representations than those $equiv 2$ or $4 mod 6$. This is because if
$n = x + y$ with $x <y$ odd and not divisible by $3$, $n equiv 2 mod 6$ requires $x, y equiv 1 mod 6$, $n equiv 4 mod 6$ requires $x, y equiv 5 mod 6$, but $n equiv 0 mod 6$ can
have either $x equiv 1$, $y equiv 5$ or $x equiv 5$, $y equiv 1 mod 6$.



Here's a plot of the number of representations of even numbers up to $10000$, with those $equiv 2mod 6$ in green, $4 mod 6$ in blue, $0mod 6$ in red.



enter image description here






share|cite|improve this answer














It appears that the numbers divisible by $6$ tend to have more representations than those $equiv 2$ or $4 mod 6$. This is because if
$n = x + y$ with $x <y$ odd and not divisible by $3$, $n equiv 2 mod 6$ requires $x, y equiv 1 mod 6$, $n equiv 4 mod 6$ requires $x, y equiv 5 mod 6$, but $n equiv 0 mod 6$ can
have either $x equiv 1$, $y equiv 5$ or $x equiv 5$, $y equiv 1 mod 6$.



Here's a plot of the number of representations of even numbers up to $10000$, with those $equiv 2mod 6$ in green, $4 mod 6$ in blue, $0mod 6$ in red.



enter image description here







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 3 hours ago

























answered 3 hours ago









Robert Israel

311k23202447




311k23202447











  • Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
    – vadim123
    3 hours ago










  • Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
    – Rushabh Mehta
    3 hours ago
















  • Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
    – vadim123
    3 hours ago










  • Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
    – Rushabh Mehta
    3 hours ago















Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
– vadim123
3 hours ago




Is the upper cluster in the red region due to $nequiv 0pmod24$ or something similar?
– vadim123
3 hours ago












Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
– Rushabh Mehta
3 hours ago




Wow, excellent answer +1. I decided to add more to the question upon further investigation thanks to your insight?
– Rushabh Mehta
3 hours ago

















 

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