View source for Tartaria.Math

[[Tartaria.Math]] is draft of the textbook on Mathematica from/for utopia [[Tartaria]]

At the beginning od 2020, yet, the only one example is loaded. However, it should not be first in the textbook.

==Example with [[tetration]] to integer base for integer arguments==

Problem by Yulya: 

Let the Yulya number

\(Y = \mathrm{tet}_7(5) - \mathrm{tet}_7(4) = 7^{7^{7^{7^{7}}}} - 7^{7^{7^{7}}} \)

Is \( Y \) an integer factor of 10?

Yulya asks for the simple deduction. For this simplicity, some notations should be defined and few theorems should be formulated.

===Definition===
Let \( \blacksquare \) denotes any sequence of decimals.

For example, any of numbers 0, 10, 20,30, 1230 can be expressed in form  \( \blacksquare 0 \)
Also,
<br>
\( 10= \blacksquare0 \) ,<br>
\( 132= \blacksquare2 \) ,<br>
\( 1435= \blacksquare35 \) , etc.

Consider powers of 7. Ignoring leading zeros, we may write<br>
\( 7^0=1=\blacksquare01  \) <br>
\( 7^1=7=\blacksquare07  \)<br>
\( 7^2=49=\blacksquare49 \)<br>
\( 7^3=343=\blacksquare43 \)<br>
\( 7^4=2401=\blacksquare01 \)<br>
\( 7^5=16807=\blacksquare07 \)<br>
\( 7^6=117649=\blacksquare49 \)<br>
\( 7^7=823543=\blacksquare43 \)<br>
\( 7^8=5764801 =\blacksquare01 \)<br>
\( 7^9=40353607 =\blacksquare07 \)<br>
\( 7^{10}= 282475249 =\blacksquare49 \)

and so on. We see periodicity in the last digits of each of the lines above:
after each four lines, the dasd digit reproduces.
This periodicity can be expressed with the following theorem.

=== Theorem 1 ===
For any non-negative integer  \( k \), the four relations below held:<br>
\( 7^{4k}=\blacksquare01\)<br>
\( 7^{4k+1}=\blacksquare07\)<br>
\( 7^{4k+2}=\blacksquare49\)<br>
\( 7^{4k+3}=\blacksquare43\)

===Theorem 2===
Theorem 1 above leads to the following statement. For any integer \( n \),

\( 7^{\blacksquare01+4n}=\blacksquare07\)<br>
\( 7^{\blacksquare03+4n}=\blacksquare43\) 

For the purpose of the problem above, now we are interested mainly in the even powers of seven.

In particular, 

\( 7^{\blacksquare43}=\blacksquare43\)

In such a way, 

\( \mathrm{tet}_7(n) = \blacksquare43\) for any integer \(n>1\).

Let the last statement be theorem 2.

===Solution===

Using the Theorem 2, we get:

\(
\mathrm{tet}_7(m) - 
\mathrm{tet}_7(n) = \blacksquare00 \)
for any integer \(m, n\) rarger than unity. 

This proves initial statement: in the special case 
\(m=5\),
\(n=4\)

we get

\(Y = \mathrm{tet}_7(5) - \mathrm{tet}_7(4) = 7^{7^{7^{7^{7}}}} - 7^{7^{7^{7}}} = 100J\)
for some integer (and, by the way, very large) \(J\).

In such a way, the Yulya Number \(Y\) happens to be not only integer factor of 10, but also integer factor of 100.

==References==
<references/>
[[Category:Fiction]]
[[Category:Mathemaics]]
[[Category:Tartaria]]
[[Category:Utopia]]