Number Theory

    

On a Modular Property of Tetration

Authors: Pranjal Jain

This paper generalizes problem 3 of the 2019 PROMYS exam, which asks to show that the last 10 digits (in base 10) of the n-th tetration of 3 are independent of ]n if n>10. The generalization shows that given any positive integers $a$ and b satisfying certain conditions, the last n digits (in base b) of the m-th tetration of a are independent of m if m>n. We use numerical patterns as a guide towards the solution and explore an additional numerical pattern which shows a relation between decimal expansions and multiplicative inverses of powers of 3 modulo powers of 10.

Comments: 12 Pages.

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Submission history

[v1] 2021-07-13 02:01:25

Unique-IP document downloads: 656 times

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