What can I do with the lifting the exponent lemma?

What can I do with LTE (lifting the exponent lemma)?

If you don't know LTE lemma visit here or here

Solution 1:

Assume some positive integers $a,b$ with $a \gt b$ and $gcd(a,b)=1$.
Then consider the primefactorization of $f(n) = a^n - b^n $ where $n$ is again positive integer, with $m$ primefactors $$ f(n) = p_1^{e_1} \cdot p_2^{e_2} \cdot \ldots p_m^{e_m} $$ Then the LTE gives you a formula, how to compute the exponents $e_k$ by a function depending on a given $n$ (and on properties of the prime $p_k$). Essentially this formula "algebraicalizes" the little theorem of Fermat and Euler's totient-theorem.
This little "LTE-algebra" can help to express exponential diophantine problems much more concise and possibly to find solutions (or a space of solutions) easier