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,
\( 10= \blacksquare0 \) ,
\( 132= \blacksquare2 \) ,
\( 1435= \blacksquare35 \) , etc.

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

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:
\( 7^{4k}=\blacksquare01\)
\( 7^{4k+1}=\blacksquare07\)
\( 7^{4k+2}=\blacksquare49\)
\( 7^{4k+3}=\blacksquare43\)

Theorem 2

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

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

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.