Marco Ripà
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 245—260
DOI: 10.7546/nntdm.2020.26.3.245-260
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Authors and affiliations
Marco Ripà
sPIqr Society, World Intelligence Network
Rome, Italy
Abstract
Integer tetration, the iterated exponentiation ba for a ∈ ℕ − {0, 1}, is characterized by fascinating periodicity properties involving its rightmost figures, in any numeral system. Taking into account a radix-10 number system, in the book “La strana coda della serie n ^ n ^ … ^ n” (2011), the author analyzed how many new stable digits are generated by every unitary increment of the hyperexponent b, and he indicated this value as V(a) or “congruence speed” of a ≢ 0 (mod 10). A few conjectures about V(a) arose. If b is sufficiently large, the congruence speed does not depend on b, taking on a (strictly positive) unique value. We derive the formula that describes V(a) for every a ending in 5. Moreover, we claim that V(a) = 1 for any a (mod 25) ∈ {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and V(a) ≥ 2 otherwise. Finally, we show the size of the fundamental period P for any of the remaining values of the congruence speed: if V(a) ≥ 2, then P(V(a)) = 10V(a)+1.
Keywords
- Number theory
- Power tower
- Tetration
- Hyperoperation
- Charmichael function
- Euler’s totient function
- Primitive root
- Exponentiation
- Integer sequence
- Congruence speed
- Modular arithmetic
- Stable digit
- Rightmost digit
- Cycle
- Periodicity
2010 Mathematics Subject Classification
- 11A07
- 11F33
References
- Andreescu, T., Andrica, D., & Feng Z. (2006). 104 Number Theory Problems: From the Training of the USA IMO Team. Birkhäuser, Boston.
- Daccache, G. (2015). Climbing the ladder of hyper operators: tetration, Mathemathics Stack Exchange, Available online at: https://math.blogoverflow.com/2015/01/05/climbing-the-ladder-of-hyper-operators-tetration/.
- Elaqqad (2015). Last Digits of a Tetration, Mathemathics Stack Exchange (Version: 2015-04-01), Available online at: https://math.stackexchange.com/q/1216149.
- Germain, J. (2009). On the Equation ax ≡ x (mod b). Integers: Learning, Memory, and Cognition, 9 (6), 629–638.
- Jolly, N. (2008). Constructing the Primitive Roots of Prime Powers, arxiv.org, Available online at: https://arxiv.org/pdf/0809.2139.pdf.
- Ripà, M. (2011). La strana coda della serie n ^ n ^ … ^ n. UNI Service, Trento.
- Sloane, N. J. A. (2018). A317824. The Online Encyclopedia of Integer Sequences, Web. 10 Aug. 2018, Available online at: http://oeis.org/A317824.
- Sloane, N. J. A. (2018). A317903. The Online Encyclopedia of Integer Sequences. Web. 10 Aug. 2018, Available online at: http://oeis.org/A317903.
- Sloane, N. J. A. (2018). A317905. The Online Encyclopedia of Integer Sequences. Web. 10 Aug. 2018, Available online at: http://oeis.org/A317905.
- Urroz, J. J., & Yebra J. L. A. (2009) On the Equation ax ≡ x (mod bn), Journal of Integer Sequences, 8 (8).
- User26486, & Brandt, M. Do the last digits of exponential towers really converge to a fixed sequence? Mathematics Stack Exchange (Version: 22 Feb. 2015), Available online at https://math.stackexchange.com/questions/1159995/do-the-last-digits-of-exponential-towers-really-converge-to-a-fixed-sequence.
- Vladimir, & Doctor Jacques (2005). Modular Exponentiation, The Math Forum, Available online at: http://mathforum.org/library/drmath/view/51625.html.
- Yan, X.-Y., Wang, W.-X., Chen, G.-R., & Shi, D.-H. (2016). Multiplex congruence network of natural numbers, Scientific Reports, 6, Article No. 23714.
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Cite this paper
Ripà, M. (2020). On the constant congruence speed of tetration. Notes on Number Theory and Discrete Mathematics, 26 (3), 245-260, doi: 10.7546/nntdm.2020.26.3.245-260.