Expon NT

In contrast to Euclid Alg, this unit does not revolve around a particular instinct or a philosophy. Problems in this unit were chosen according to the content of their statements, rather than the techniques that can tackles those problems.

Philosophy

Although this unit does not have a particular philosophy, there is a set of tools that you will learn in this unit.

Some of them are:

1. Orders: You should be comfortable with understanding and using orders to solve problems from this unit.
2. Fermat's Christmas theorem: which numbers can be written as a sum of two squares, which primes divide $x^2+1$ etc...
3. LTE (lifting the exponent lemma): Life is hard without LTE these days.

There are also some quotable theorems that can be useful in problems from this unit:

1. Zsigmondy's theorem: With some exceptions, for coprime $a,b$ and for any positive integer $n$, $a^n-b^n$ has a prime factor that any of the numbers $a^k-b^k, 0<k<n$ doesn't have.
2. Mihailescu theorem: The only consecutive perfect powers are $8$ and $9$.
3. Bertrand postulate: There is a prime between $n$ and $2n$
4. Dirichlet's theorem: An arithmetic sequence $a+(n-1)d$ contains infinitely many primes if $a$ and $d$ are coprime.
5. Kobayashi's theorem: If the prime set of an infinite set $M \subseteq \mathbb{Z}^+$ is finite, then the prime set of $M+a$ (elements of $M$ plus $a$) is infinite for any nonzero integer $a$.

Why you should do this unit

Exposing yourself into a set of number theory problems about equations with exponents is necessary to becoming comfortable with number theory since such problems appear often on contests (e.g., IMO 2022 had such a problem).

Notable problems

• SCHINZEL this a classical problem the source of which is unknown.
• IMO 1990/3: This problem is similar to the previous. But it is harder and it's source is known.
• IMO 2000/5: A classical problem from IMO.
• TSTST 2018/8: A rich problem from a more recent TSTST.
• China 2005/6: Get cooked.