Prove that for any given positive integer there exist at most finitely many integers with , where denotes the Euler totient function. Conclude in particular that tends to infinity as tends to infinity.
Let be a positive integer, and let be the least prime greater than . Let be an integer such that .
If is a prime divisor of , then for some and with not dividing . Then , a contradiction. Thus no prime divisor of is greater than . In particular, the distinct prime divisors of belong to a finite set; say these primes are .
Now we can write , for some , and thus . Thus we have . Note that for each prime , for sufficiently large . Thus, for each , there are only finitely many permissible choices for the exponents . So the set of all with is a subset of a finite set hence finite.
Now for each positive integer , there is a largest integer with . Thus as approaches infinity, so does .