Prove that for a fixed integer n, the equation (x) = n has only finitely many so

Question

Prove that for a fixed integer n, the equation (x) = n has only finitely many solutions x.

where (x) is the euler totient function.

Explanation / Answer

Dear Student Thank you for using Chegg !! Let n be a positive integer and let p be the leaast prime number greater than n+1 Let x be an integer such that (x) = n If q>p is a prime divisor of x, then x=(q^k).m for some k>1 and m with q not dividing m (x) = (q^k). (m) > q-1 > p-1 >n,    a contradiction Thus no prime divisor of x is greater than n+1 . In particular, the distinct prime divisors of x belong to a finite set; say these primes are . p1,p2,p3…………………………..pm Now we can write x= p1a1 . P2a2 ……………………………………………….pmam for some 0 (piai-1)(pi-1) >n for sufficiently large ai Thus for each pi, there exists only finite many permissible choices for the exponents ai, So the set of all x, with (x) = n . Thus as x approaches infinity, so does (x) Hence prooved

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.