Friday, April 8, 2016

number theory - Show that anbn has a prime factor which does not divide ab for all n>1 .



I was asked to prove the following using the lifting the exponent lemma.




Show that anbn has a prime factor which does not divide ab 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(anbn)=Vp(ab)+Vp(n)




where Vp(x) is the highest power of p that divides x and p|ab but does not divide a or b.
I don't know how to approach this and would welcome some hints.


Answer



Hint:




anbnab=nk=1ak1bnk>np|(ab)pVp(n).




anbn>p|(ab)pVp(anbn)



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