IIT JEE , AIEEE Forum - Mathematics , Algebra

little_genius

if p is a prime no . then prove that log_p(P+1) IS AN IRRATIONAL NO.

hsbhatt

Suppose $\log_p p+1 = \frac{m}{n}$ where m and n are natural numbers.

Then p^m = (p+1)^n

This means that p divides (p+1)^n

If a prime number p divides a^m then it divides a.

This means p divides p+1 which is absurd.

little_genius

i too had solved it like that except that in the relation p^m = (p+1)^n i used the arguement that p and p+1 being consecutve nos. so one of them is even and the other is odd .. hence there cannot exist natural nos. m ,n bcoz the lhs and rhs will be even and odd respectively or vice versa...

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