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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?

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    $\begingroup$ Not sure why this got two votes. it's highly open ended. close? $\endgroup$ – Suresh Venkat Apr 10 '11 at 9:12
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    $\begingroup$ Or the OP may like to revise the question to become more concrete? $\endgroup$ – Hsien-Chih Chang 張顯之 Apr 10 '11 at 11:08
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    $\begingroup$ 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. $\endgroup$ – arnab Apr 10 '11 at 16:01
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    $\begingroup$ Here's a recent IAS talk titled Expander Graphs: Why Number Theorists Might Care About Network Optimization $\endgroup$ – Mark Reitblatt Apr 10 '11 at 17:01
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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

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.

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There's Schur's theorem which is proved using graphical Ramsey theory.

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Perhaps start on this wikipedia entry on additive number theory and its cited references.

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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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