Hydra function: Difference between revisions

Revision as of 09:43, 23 February 2025

The Hydra function is a Collatz-like function whose behavior is connected to the the unsolved halting problems for the Cryptids Hydra and Antihydra:

{\begin{array}{l}H(2n)&=&3n\\H(2n+1)&=&3n+1\\\end{array}}

which can alternatively be written as

H(n)={\begin{cases}{\frac {3n}{2}}&{\text{if }}n{\text{ even}}\\{\frac {3n-1}{2}}&{\text{if }}n{\text{ odd}}\\\end{cases}}

or simply

H(n)=\left\lfloor {\frac {3n}{2}}\right\rfloor

It has some connections to Mahler's 3/2 problem.

@@ Line 1: / Line 1: @@
-The '''Hydra function''' is a [[Collatz-like]] function whose behavior is connected to the unsolved halting problems for the [[Cryptids]] [[Hydra]] and [[Antihydra]]. It is defined as:
+The '''Hydra function''' is a [[Collatz-like]] function whose behavior is connected to the the unsolved halting problems for the [[Cryptids]] [[Hydra]] and [[Antihydra]]:
-<math display="block">\textstyle H(n)\equiv n+\left\lfloor\frac{1}{2}n\right\rfloor=\Big\lfloor\frac{3}{2}n\Big\rfloor=\begin{cases}
-  \frac{3n}{2}     & \text{if } n\equiv0\pmod{2} \\
-  \frac{3n-1}{2}  & \text{if } n\equiv1\pmod{2} \\
-\end{cases}</math>
-which can alternatively be written as
 <math display="block">\begin{array}{l}
    H(2n)   & = & 3n   \\
    H(2n+1) & = & 3n+1 \\
 \end{array}</math>
-It has some connections to [[wikipedia:Mahler's_3/2_problem|Mahler's 3/2 problem]].
-==Properties==
-Here, <math>s</math> and <math>t</math> are positive integers with <math>t</math> odd. Let <math>0\le k\le s</math> be an integer and <math>H^k</math> is the <math>k</math>th iterate of <math>H</math>.
-<math display="block">\begin{array}{l}
-  H^k(2^s t)   & = & 3^k2^{s-k}t  \\
-  H^k(2^s t+1)   & = & 3^k2^{s-k}t+1 \\
-\end{array}</math>
-==Relationship to Hydra and Antihydra==
-Both machines effectively track the progress of two varibles; one of them changes depending on its value modulo 2 but roughly multiplies itself by <math>\frac{3}{2}</math>, and the other increases by 2 or decreases by 1 depending on the parity of the first variable.
-In particular, Hydra halts if and only if the function
+which can alternatively be written as<math display="block">H(n) = \begin{cases}
-<math display="block">f(x)=\begin{cases}\frac{3x+6}{2}&\text{if }a\equiv0\pmod{2}\\8</math>
+  \frac{3n}{2}     & \text{if } n \text{ even} \\
+  \frac{3n-1}{2}  & \text{if } n \text{ odd} \\
+\end{cases}</math>or simply<math display="block">H(n) = \left\lfloor \frac{3n}{2} \right\rfloor</math>It has some connections to [[wikipedia:Mahler's_3/2_problem|Mahler's 3/2 problem]].

Hydra function: Difference between revisions

Revision as of 09:43, 23 February 2025

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