• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Repetition of the last digits of a tetration of generic base Luknik Junior Fellow Posts: 8 Threads: 2 Joined: Oct 2021   10/15/2021, 03:56 PM (This post was last modified: 10/16/2021, 03:08 PM by Luknik.) Hi! I'm Luca Onnis, 19 years old. I would like to share with you my conjecture about the repetition of the last digits of a tetration of generic base. My paper investigates the behavior of those last digits. In fact, last digits of a tetration are the same starting from a certain hyper-exponent and in order to compute them we reduce those expressions $$\mod 10^{n}$$. Very surprisingly (although unproved) I think that the repetition of the last digits depend on the residue $$\mod 10$$ of the base and on the exponents of a particular way to express that base. In the paper I'll discuss about the results and I'll show different tables and examples in order to support my conjecture. Here's the link: https://arxiv.org/abs/2109.13679 . I also attached the pdf. You can find the proposition of my conjecture and also a lot of different examples. Of course you can ask me for more! And maybe we can try to prove this, maybe using some sort of iterated carmichael function. I want to summarize the results I got: If: $$f_{q}(x,y,n)=u$$ Then for $$m\geq u$$ $${^{m}\Bigl[q^{(2^{x}\cdot5^{y})\cdot a}\Bigr]} \equiv {^{u}\Bigl[q^{(2^{x}\cdot5^{y})\cdot a}\Bigr]} \mod (10^{n})$$ where $$x,y,n,q,a \in\mathbb{N}$$ , $$q\not=10h$$, $$a \not=2h$$ and $$a\not=5h$$ and $$u$$ is the minimum value such that this congruence is true.  Note Those formulas work for $$x\geq 2$$  I define $$\Delta_2$$ and $$\Delta_5$$ as: $$\Delta_2=\max[v_2(q+1),v_2(q-1)]$$ $$\Delta_5=\max[v_5(q+1),v_5(q-1)]$$ We'll have that: $$f_{q \equiv 1,9 \mod 10}(x,y,n)=\max\Biggl[\Bigl\lceil\frac{n}{x+\Delta_2}\Bigr\rceil,\Bigl\lceil\frac{n}{y+\Delta_5}\Bigr\rceil\Biggr]-1$$   I define $$\Gamma_2$$ and $$\Gamma_5$$ as: $$\Gamma_2=\max[v_2(q+1),v_2(q-1)]$$ $$\Gamma_5=\max[v_5(q^{2}+1),v_5(q^{2}-1)]$$ We'll have that: $$f_{q \equiv 3,7 \mod 10}(x,y,n)=\max\Biggl[\Bigl\lceil\frac{n}{x+\Gamma_2}\Bigr\rceil,\Bigl\lceil\frac{n}{y+\Gamma_5}\Bigr\rceil\Biggr]-1$$   When the last digit of the base is 5 we know that $y$ could be every integer number, so in our function we only consider the variable $x$.   $$f_{q \equiv 5 \mod 10}(x,n)= \Bigl\lceil\frac{n}{x+\Delta_2}\Bigr\rceil-1$$   When the last digit of the base is 2,4,6,8 we know that $x$ could be every integer number, so in our function we only consider the variable $y$.   $$f_{q \equiv 0 \mod 2}(y,n)= \Bigl\lceil\frac{n}{y+\Gamma_5}\Bigr\rceil-1$$   Where $$\lceil n\rceil$$ is the ceil function of $$n$$ and represents the nearest integer to $$n$$ , greater or equal to $$n$$;  and $${^{a}n}$$ represent the $$a$$-th tetration of $$n$$ , or $$n^{n^{n^{\dots}}}$$ $$a$$ times.   For example consider the infinite tetration of $$63^{2^{5}\cdot 5^{2}\cdot 3}$$ , or $$63^{2400}$$. We know from our second formula that the last 15 digits are the same starting from the 4-th tetration of that number. Indeed, $$63 \equiv 3 \mod 10$$ and $$\lceil\frac{n}{y+\Gamma_5}\rceil\geq\lceil\frac{n}{x+\Gamma_2}\rceil$$. In fact:   $$\Gamma_2=\max[v_2(63+1),v_2(63-1)]=\max[6,1]=6$$   $$\Gamma_5=v_5(63^{2}+1)=1$$   And: $$\Bigr\lceil\frac{15}{2+1}\Bigr\rceil\geq\Bigl\lceil\frac{15}{5+6}\Bigr\rceil$$ So we'll have that:   $$f_{63}(5,2,15)=\Bigl\lceil\frac{15}{2+v_5(3970)}\Bigl\rceil-1$$   $$f_{63}(5,2,15)=\Bigl\lceil\frac{15}{3}\Bigl\rceil-1=4$$   So we'll have that:   $${^{4}\Bigl[63^{(2^{5}\cdot5^{2}\cdot 3)}\Bigr]} \equiv 547909642496001 \mod (10^{15})$$   $${^{5}\Bigl[63^{(2^{5}\cdot5^{2}\cdot 3)}\Bigr]} \equiv 547909642496001 \mod (10^{15})$$ $$\vdots$$ And so on for every hyper-exponent greater or equal to 4. Attached Files Image(s)       2109.13679.pdf (Size: 115.77 KB / Downloads: 66) « Next Oldest | Next Newest »

 Messages In This Thread Repetition of the last digits of a tetration of generic base - by Luknik - 10/15/2021, 03:56 PM RE: Repetition of the last digits of a tetration of generic base - by Daniel - 10/16/2021, 09:47 AM RE: Repetition of the last digits of a tetration of generic base - by Luknik - 10/16/2021, 10:08 AM RE: Repetition of the last digits of a tetration of generic base - by Daniel - 10/16/2021, 12:31 PM RE: Repetition of the last digits of a tetration of generic base - by Luknik - 10/16/2021, 02:51 PM RE: Repetition of the last digits of a tetration of generic base - by Daniel - 10/16/2021, 12:40 PM RE: Repetition of the last digits of a tetration of generic base - by JmsNxn - 10/18/2021, 07:06 AM RE: Repetition of the last digits of a tetration of generic base - by Luknik - 10/18/2021, 09:09 AM RE: Repetition of the last digits of a tetration of generic base - by MphLee - 10/24/2021, 12:49 PM RE: Repetition of the last digits of a tetration of generic base - by Luknik - 10/25/2021, 05:39 PM RE: Repetition of the last digits of a tetration of generic base - by tommy1729 - 10/26/2021, 10:02 PM RE: Repetition of the last digits of a tetration of generic base - by marcokrt - 12/16/2021, 12:17 AM RE: Repetition of the last digits of a tetration of generic base - by marcokrt - 12/16/2021, 12:26 AM

 Possibly Related Threads... Thread Author Replies Views Last Post On the $$2 \pi i$$-periodic solution to tetration, base e JmsNxn 0 344 09/28/2021, 05:44 AM Last Post: JmsNxn Complex Tetration, to base exp(1/e) Ember Edison 7 10,338 08/14/2019, 09:15 AM Last Post: sheldonison Is bounded tetration is analytic in the base argument? JmsNxn 0 3,131 01/02/2017, 06:38 AM Last Post: JmsNxn tetration base sqrt(e) tommy1729 2 6,305 02/14/2015, 12:36 AM Last Post: tommy1729 Explicit formula for the tetration to base $$e^{1/e}$$? mike3 1 5,568 02/13/2015, 02:26 PM Last Post: Gottfried tetration base > exp(2/5) tommy1729 2 5,988 02/11/2015, 12:29 AM Last Post: tommy1729 regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 16,635 06/25/2013, 01:37 PM Last Post: sheldonison tetration base conversion, and sexp/slog limit equations sheldonison 44 95,229 02/27/2013, 07:05 PM Last Post: sheldonison simple base conversion formula for tetration JmsNxn 0 4,777 09/22/2011, 07:41 PM Last Post: JmsNxn Does anyone have taylor series approximations for tetration and slog base e^(1/e)? JmsNxn 18 33,614 06/05/2011, 08:47 PM Last Post: sheldonison

Users browsing this thread: 2 Guest(s)