Goldbach's Conjecture and the totient function

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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












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






active

oldest

votes

















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










Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);













 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2975216%2fgoldbachs-conjecture-and-the-totient-function%23new-answer', 'question_page');

);

Post as a guest

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








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













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

















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2975216%2fgoldbachs-conjecture-and-the-totient-function%23new-answer', 'question_page');

);

Post as a guest
































































Comments

Post a Comment

Popular posts from this blog

Long meetings (6-7 hours a day): Being “babysat” by supervisor

Image
Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 31 down vote favorite 4 I hold a PhD in mechanical engineering with a 7 year background in self-guided, self-paced research including 2 years as a graduate level course instructor and a subject matter expert for various engineering projects in a university setting in the United States. For the last 10-11 months, I am working in a research industry in France where my task is to debug C++ spaghetti code written over a period of several years by my current supervisor. It is a team of 2: just I and my supervisor. Diurnally, I am posed with a "bug of the day" to get out of this C++ code. My supervisor generally gives me about 15-30 minutes to solve it and then accosts me about whether or not the task is completed and then rinses and repeats this until the end of the day. Off-late, my supervisor has been holding 6-7 hour long "meetin
Read more

Is the Concept of Multiple Fantasy Races Scientifically Flawed? [closed]

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Many fantasy worlds have multiple humanoid species (elves, orcs, humans, etc) living in the same world, in addition to that, they often have the ability to interbreed. And I can't help but question their believability... fantasy-races reproduction interspecies-relations share | improve this question edited Aug 9 at 14:46 Michael Kjörling ♦ 21.1k 10 85 161 asked Aug 9 at 13:13 Chlodio 118 6 closed as unclear what you're asking by MichaelK, Gryphon, John, Vincent, Frostfyre Aug 9 at 15:25 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 3 Stack Exchange exist to
Read more

Confectionery

Image
Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of
Read more