# 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.