%0 Journal Article
%T Base-X Conjecture and Collatz Conjecture Proof
%A Sheng Zhao
%J Applied Mathematics
%P 365-382
%@ 2152-7393
%D 2025
%I Scientific Research Publishing
%R 10.4236/am.2025.164020
%X A new Base-X Conjecture was introduced in this paper, and Collatz Conjecture is just one case of Base-X Conjecture - Base-3 (Ternary). Based on Base-X number system property and Collatz Conjecture iteration, it has been proved that for any positive integer <i>D</i>, there are <i>n</i> and <i>m</i> which exist for <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>D</mi>
   <mi>n</mi>
  </msub>
  <mo>+</mo><msub>
   <mi>Y</mi>
   <mi>n</mi>
  </msub>
  <mo>=</mo><msup>
   <mn>2</mn>
   <mi>m</mi>
  </msup>
  </mrow>
</math>

. <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>D</mi>
   <mi>n</mi>
  </msub>
  <mo>+</mo><msub>
   <mi>Y</mi>
   <mi>n</mi>
  </msub>
  </mrow>
</math>

 is just the result built up by collecting divided by 2 of Collatz Conjecture iteration. Divided by 2<i><sup>m</sup></i> will make the Collatz Conjecture get a result of 1 for any positive integer. Also, the Collatz Tree showed that for any odd positive number, there is only one route existing in the Collatz Tree down to 1 on Collatz Conjecture iteration.
%K Collatz Conjecture
%K Base-X Conjecture
%K Collatz Spiral
%K Collatz Ring
%K Collatz Tree
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=142323