What are the implications of binomials with tetration?

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SUMMARY

The discussion centers on the implications of binomials when applied to tetration, particularly how tetration fails to distribute over multiplication, similar to the behavior of exponentiation. Participants explore the expression (xy) tetra squared and question whether there are elegant patterns to study within this framework or if the problem is trivial or unsolvable. The conversation references hyperoperations, including addition, multiplication, and exponentiation, as foundational concepts for understanding tetration's properties.

PREREQUISITES
  • Understanding of hyperoperations, including addition, multiplication, and exponentiation.
  • Familiarity with tetration and its mathematical notation.
  • Knowledge of binomials and their properties in algebra.
  • Basic comprehension of Knuth's up-arrow notation for expressing large numbers.
NEXT STEPS
  • Research the properties of tetration and its relationship with other hyperoperations.
  • Explore Knuth's up-arrow notation and its applications in expressing tetration.
  • Investigate mathematical literature on the distribution properties of tetration over multiplication.
  • Study advanced algebraic structures that involve binomials and their interactions with hyperoperations.
USEFUL FOR

Mathematicians, students of advanced algebra, and anyone interested in the theoretical implications of hyperoperations and tetration.

Ramanujan
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"Binomials" with tetration

With the hyperoperations of addition, iterated addition (multiplcation), which distributes over addition, and iterated multiplication (exponentiation), which distributes over multiplcation, we can study how exponentiation fails to distribute over the operation which is two iterative operationsbelow it, addition...Now in extending to tetration, how would tetration fail to distribute over multiplication ? What is (xy) tetra squared? Is there a system of elegant patterns to study, or is it trivial, unsolvable, etc.?
 
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