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Math Help - Twin Prime Conjecture - Visual Proof

  1. #1
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    Twin Prime Conjecture - Visual Proof



    There are infinitely many primes p such that p + 2 is also prime.

    First off, I'm a noob at maths and always reverse my logic!

    Now, a little glimpse into my madness.

    I saw this unsolved twin prime conjecture several days ago and tried to solve it. After 15 mins I constructed the above visual proof.

    Here's how I thought it should work: Let line P represent the Infinitude of Primes. To be a member of P, the number must be prime. All other numbers are excluded. Next, pick a prime p and draw a 2D lattice L with sides sqrt(2). Notice how P is the on the lattice diagonal.

    At this stage, I knew at least one lattice square would intersect P at two distinct primes, say p and p+2 (if it exists!?). Now to prove that p+2 exists, I decided to use Polya's 2D random walk which proves that any point is reachable on a 2D lattice. I intentionally allowed the lattice point to represent p+2. So, let the random walk (shown in red) proceed and eventually it'll reach the lattice prime point p+2. Thus, if it's possible to reach p+2 from p, then p+2 must exist and is prime since it lies on P. Q.E.D.

    Okay, where did I go wrong?

    Thanks.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Are you serious?
    I don't see you using a single property of primes. Try modifying the argument to show that there are infinitely many primes p for which p+1 is also prime; if you succeed, and I don't doubt you will, then certainly your proof is flawed.
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  3. #3
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    Hi Bruno,

    p+1 would not be prime. Isn't p+2 the smallest except for 2,3. I do use Euclid's theorem that there are an infinite number of primes.

    Think of it this way ... "line" P is simply a representation of a prime boundary P={2,3,5,7,11,13,...}, so that superimposing a 2D grid will cross that boundary at two prime points, p and p+2 (if it exists?)

    The primary goal is to prove that p+2 exists. I tried to prove p+2 exists by finding a path from point p to p+2 by using a random walk by Polya.

    The key "trick" is to align prime p and a lattice point to create a "prime lattice point" p+2 so that I could say p+2 is prime (since it lies on P) and also simultaneously state that p+2 exists as it's a lattice point.
    Combining them together, p+2 is prime and exists!

    Perhaps my proof is flawed by a construction argument?

    Thanks for the reply.
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  4. #4
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    Quote Originally Posted by vengy View Post
    Hi Bruno,

    p+1 would not be prime. Isn't p+2 the smallest except for 2,3. I do use Euclid's theorem that there are an infinite number of primes.

    Think of it this way ... "line" P is simply a representation of a prime boundary P={2,3,5,7,11,13,...}, so that superimposing a 2D grid will cross that boundary at two prime points, p and p+2 (if it exists?)

    The primary goal is to prove that p+2 exists. I tried to prove p+2 exists by finding a path from point p to p+2 by using a random walk by Polya.

    The key "trick" is to align prime p and a lattice point to create a "prime lattice point" p+2 so that I could say p+2 is prime (since it lies on P) and also simultaneously state that p+2 exists as it's a lattice point.
    Combining them together, p+2 is prime and exists!

    Perhaps my proof is flawed by a construction argument?

    Thanks for the reply.
    See reply #2. Thread closed.
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