# Kuznetsova theorem

Kuznetsova theorem refers to residual of division of tetration to integer base by any integer number.

## Kuznetsova theorem

Let \( b>1 \) and \( q>1 \) be integers.

Then, there exist positive integer \( Q \) and integer \(r\) such that for any integer \( n > Q \) the equation holds:

\( \mathrm{tet}_b(n)\%q = r \)

## Notations

Here symbol tet veters to tetration. The base is indicated as subscript.

Character % refers to residual of division of the number at left (treated as numerator) by number at right (intepreted as denominator).

For example,

\(3 \%2=1\)

\( 14\%2=0 \)

\( 14\%10=4 \)

## References

## Keywords

Integer number, Tartaria, Tartaria.Math, Tetration, Yulya Kuznetsova