I was asked to prove the following using the lifting the exponent lemma.
Show that an−bn has a prime factor which does not divide a−b for all n>1 .
Using the first lemma, what I got was this:
if p is any prime greater than 2,
then we have
Vp(an−bn)=Vp(a−b)+Vp(n)
where Vp(x) is the highest power of p that divides x and p|a−b but does not divide a or b.
I don't know how to approach this and would welcome some hints.
Answer
Hint:
an−bna−b=n∑k=1ak−1bn−k>n≥∏p|(a−b)pVp(n).
⟹an−bn>∏p|(a−b)pVp(an−bn)
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