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I was curious about this equation involving hyperoperations:

$$H_a(H_b(x,y),n)=H_b(H_a(x,n),H_a(y,n)), a,b \in \mathbb{N}$$

where $H_1(x,y)=x+y, H_2(x,y)=xy$ and so on. I found that this leads both to some actual laws such as $(x+y)*n=(x*n)+(y*n)$ and other “laws” such as $(x+y)^n=x^n+y^n$. I am interested in the question: For which values of $a$ and $b$ can we get actual laws and for which values can we get these "pseudo-laws"? And for these pseudo-laws, is there any non zero values $x$, $y$, and $n$ can take?

Zerox
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James Smith
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  • You can use latex for the equation parts: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – user1176409 Aug 11 '23 at 15:17
  • In a field of prime characteristic $p$, the equation $(x+y)^p=x^p+y^p$ holds for all values $x,y$ in the field. If I recall correctly this identity is sometimes called the "Freshman's Dream"... – Marc-André Brochu Aug 11 '23 at 15:31
  • @Marc-AndréBrochu In a finite field $K$, $x^y$ is not defined for most $y \in K$. – GEdgar Aug 11 '23 at 17:01
  • What do you mean by "and so on"? Are $x,y$ restricted to positive integers? – GEdgar Aug 11 '23 at 17:04
  • The so on means the different hyperoperations. Restrict x and y to whatever constraints you want to make calculations easier. – James Smith Aug 26 '23 at 17:43

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