Bad induction proof how do I point out the flaw Claim log15

Bad induction proof, how do I point out the flaw?

Claim log_15 n = log_251 n for all natural numbers n. Proof (by strong induction) The inductive hypothesis is \"log_15n = log _251 n\'\'. Base case: log_15 1 = 0 = log_251 1. Induction Hypothesis: log_15 k = log_251 k for all natural numbers k lessthanorequalto n Inductive step: We wish to show that the claim is true for n + 1. Write n + 1 as a product of two natural numbers p and q so that we have: log_15 (n+1) = log_15 (pq) = log_15 q =log_15 g = log_251 p+log_251 q = log_251 (pq) = log _251 (n +1) which is true by the inductive hypothesis.

Solution

Base case and induction hypothesis have no error. The inductive step has a small error in it.

If n+1 is a prime number then it is not necessary that both the number must be less than n+1. That is, if one of the value either p and q becomes equal to n+1 the in this case, we will reach a dead end and the induction step failes.

For example,

Take the case of n=6, in this case, the value of n+1 become 7, and if the either of the value of p and q becomes 7 then in this case the inductive step will fail.