<p>There are some results proven by graph theory, however I am not sure whether those results are good examples you want. Because number theory has offsprings by cross-breeding with some other fields, like arithmetic combinatorics and <a href="http://en.wikipedia.org/wiki/Additive_number_theory" rel="nofollow">additive number theory</a> </p>

<p>In number theory, Szemerédi's theorem that was formerly the Erdős–Turán conjecture is a good example proven by graph theory. (Although it is not necessary to prove by graph theory. Other proofs are using different approaches, e.g., Hillel Furstenberg uses ergodic theory, and Timothy Gowers uses both Fourier analysis and combinatorics)</p>

<p>And this result leads to <a href="http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem" rel="nofollow">Green–Tao theorem</a>.</p>

CC BY-SA 3.0
2011-04-10T20:25:26.297Z
2011-04-10T20:25:26.297Z
Abner Huang
<p>There's <a href="http://www.proofwiki.org/wiki/Schur%27s_Theorem_%28Ramsey_Theory%29" rel="nofollow">Schur's theorem</a> which is proved using graphical Ramsey theory.</p>

CC BY-SA 3.0
2011-04-10T16:34:02.773Z
2011-04-10T16:34:02.773Z
Yuval Filmus
<p>Perhaps start on <a href="http://en.wikipedia.org/wiki/Additive_number_theory" rel="nofollow">this wikipedia entry on additive number theory</a> and its cited references.</p>

CC BY-SA 3.0
2011-04-10T17:09:28.167Z
2011-04-11T10:47:06.687Z
2011-04-11T10:47:06.687Z
RJK
<p>Riemann Hypothesis is equivalent to a statement about Merten's function, which is in turn gotten by a determinant of a matrix given by RedHeffer.</p>

<p>If I recollect correctly, Graph Theory has been used to investigate the eigenvalues of the RedHeffer Matrix (you might find that in papers by Barrett and Jarvis).</p>

<p>Sorry, I don't have a link handy.</p>

CC BY-SA 3.0
2011-04-10T21:30:15.013Z
2011-04-10T21:30:15.013Z
Aryabhata
<p>i try to find if there was an attempt to prove any famous Number Theory conjectures like Goldbach conjecture or Twin Prime conjectures through Graph Theory.
I have in my mind something like the approach to Fermat Theorem by Andrew Wiles through the Taniyama-Shimura conjecture. 
Is there something like that (succesfull or not) using Graph Theory to attack Number Theory?</p>

CC BY-SA 3.0
2011-04-10T09:12:16.020Z
Not sure why this got two votes. it's highly open ended. close?
Suresh Venkat
CC BY-SA 3.0
2011-04-10T11:08:18.250Z
Or the OP may like to revise the question to become more concrete?
Hsien-Chih Chang 張顯之
CC BY-SA 3.0
2011-04-10T16:01:25.397Z
I think it should remain open and be made CW. For example, there were previous open-ended questions asking for uses of entropy in combinatorial arguments, topology in computer science, etc. The answers could be interesting.
arnab
CC BY-SA 3.0
2011-04-10T17:01:04.310Z
Here's a recent IAS talk titled [Expander Graphs: Why Number Theorists Might Care About Network Optimization](http://video.ias.edu/conversations/fuchs)
Mark Reitblatt
<graph-theory><co.combinatorics><big-picture><nt.number-theory>
Approaching Number Theory conjectures through Graph Theory
Van Gel Papadog
# Approaching Number Theory conjectures through Graph Theory

(Asked by: username_0 on 2011-04-10)

i try to find if there was an attempt to prove any famous Number Theory conjectures like Goldbach conjecture or Twin Prime conjectures through Graph Theory.
I have in my mind something like the approach to Fermat Theorem by Andrew Wiles through the Taniyama-Shimura conjecture.
Is there something like that (succesfull or not) using Graph Theory to attack Number Theory?

**Tags:** graph-theory, co.combinatorics, big-picture, nt.number-theory

> **Comments:**
> **username_1**: Not sure why this got two votes. it's highly open ended. close?  
> **username_2**: Or the OP may like to revise the question to become more concrete?  
> **username_3**: I think it should remain open and be made CW. For example, there were previous open-ended questions asking for uses of entropy in combinatorial arguments, topology in computer science, etc. The answers could be interesting.  
> **username_4**: Here's a recent IAS talk titled [Expander Graphs: Why Number Theorists Might Care About Network Optimization](http://video.ias.edu/conversations/fuchs)  

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## 4 answers

### Answer by username_5 on 2011-04-10. Score: 4

There are some results proven by graph theory, however I am not sure whether those results are good examples you want. Because number theory has offsprings by cross-breeding with some other fields, like arithmetic combinatorics and [additive number theory](http://en.wikipedia.org/wiki/Additive_number_theory)

In number theory, Szemerédi's theorem that was formerly the Erdős–Turán conjecture is a good example proven by graph theory. (Although it is not necessary to prove by graph theory. Other proofs are using different approaches, e.g., Hillel Furstenberg uses ergodic theory, and Timothy Gowers uses both Fourier analysis and combinatorics)

And this result leads to [Green–Tao theorem](http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem).


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### Answer by username_6 on 2011-04-10. Score: 3

There's [Schur's theorem](http://www.proofwiki.org/wiki/Schur%27s_Theorem_%28Ramsey_Theory%29) which is proved using graphical Ramsey theory.


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### Answer by username_7 on 2011-04-10. Score: 2

Perhaps start on [this wikipedia entry on additive number theory](http://en.wikipedia.org/wiki/Additive_number_theory) and its cited references.


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### Answer by username_8 on 2011-04-10. Score: 1

Riemann Hypothesis is equivalent to a statement about Merten's function, which is in turn gotten by a determinant of a matrix given by RedHeffer.

If I recollect correctly, Graph Theory has been used to investigate the eigenvalues of the RedHeffer Matrix (you might find that in papers by Barrett and Jarvis).

Sorry, I don't have a link handy.


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