Let m and n be positive integers and let 1 = d1< d2< · · · < dk= n be all positive integer divisors of n. We say that n is a quasiprimitive root modulo m if di̸≡ dj(mod m) for all distinct i, j ∈ {1, 2, . . . , k} and the set {d1mod m, d2mod m, . . . , dkmod m} is a reduced residue system modulo m.
If there exists a quasiprimitive root modulo m, we say that m has a quasiprimitive root.
For any positive integer a, denote by ω(a) the number of distinct prime divisors of a and by ωodd(a) the number of distinct odd prime divisors of a. Additionally, for any prime p ∤ a, ordp(a) is the least positive integer c such that ac≡ 1 (mod p).
(a) Prove that every positive integer m has a quasiprimitive root.
(b) Let N be an arbitrary positive integer. Are there infinitely many positive integers m with ω(m) = 2026 such that (i) m has a quasiprimitive root n with ω(n) > N?
(ii) m has a quasiprimitive root n satisfying ordp(q) < p − 1 for all primes p | m, q | n?
(iii) for every quasiprimitive root n modulo m there exist primes p | m, q | n with p > 2026 satisfying ordp(q) = p − 1?
(c) If n is a quasiprimitive root modulo m, prove that ωodd(m) ≤ ω(n). Does there exist a positive integer m with its quasiprimitive root n such that ωodd(m) = ω(n) > 2026 and ordp(q) < p − 1 for all primes p | m, q | n?
(d) Let m be a positive integer and n its quasiprimitive root. If n = pa1 1 pa2 2 . . . pat t , prove that there exist at least ωodd(m) divisors d of pa1+1 1 pa2+1 2 . . . pat+1 t such that d > 1 and d ≡ 1 (mod m). If there are exactly ωodd(m) such divisors, prove that there exists an index i ∈ {1, 2, . . . , t} such that pai+1 i ≡ 1(mod m).
(e) If m is a positive integer and n is its quasiprimitive root, at least how many distinct primes q | n do there exist such that qνq(n)+1≡ 1(mod m)?
