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. es An gers. introduced e (1) (2) 9 8 5 2 1 the generalized fibonacci primitive roots 2 conditional on the generalized Riemann hypothesis, was established in [11], and [19]. The complete statement of the GRH appears in Theorem 7.1.This note introduces a general- ization to the subset of integers
= n N : ord (r)= λ(n) and r2 r +1 mod n N, (3) NF ∈ n ≡ ⊂ and provides the corresponding asymptotic counting function
N (x)=# n x : ord (r)= λ(n) and r2 r +1 mod n . (4) F ≤ n ≡
Theorem 1.1. Assume the generalized Riemann hypothesis. Then, are infinitely many composite integers n 1 which have Fibonacci primitive roots. Moreover, the number of ≥ such integers n x has the asymptotic formula ≤ eγF −γαF x 1 NF (x)= + o(1) 1−α 1 2 , (5) Γ (αF ) (log x) F − p pY∈A where
2 2 = p P : ord (r)= p 1, ord 2 (r) = p(p 1) or r r +1 mod p , A ∈ p − p 6 − 6≡ and αF = 0.265705 . . . is a constant. The constant α = (27/38) (1 1/p(p 1)) is the average density of the densities F p≥2 − − of the primes in the residue classes modulo 20 that have Fibonacci primitive roots, confer Q [18] for the calculations.
Sections 2 to 6 provide some essential background and the required basic results. The proof of Theorem 1.1 is settled in Section 7.
2 Some Arithmetic Functions
The Euler totient function counts the number of relatively prime integers ϕ(n)=# k : { gcd(k,n) = 1 . This counting function is compactly expressed by the analytic formula } ϕ(n)= n (1 1/p),n N. p|n − ∈ Q Lemma 2.1. (Fermat-Euler) If a Z is an integer such that gcd(a, n) = 1, then aϕ(n) ∈ ≡ 1 mod n. The Carmichael function is basically a refinement of the Euler totient function to the finite ring Z/nZ.
v1 v2 vt Definition 2.1. Given an integer n = p1 p2 pt , the Carmichael function is defined by · · · v1 v2 vt v λ(n) = lcm (λ (p1 ) , λ (p2 ) λ (pt )) = p , (6) · · · v p Y||λ(n) where the symbol pv n,ν 0, denotes the maximal prime power divisor of n 1, and || ≥ ≥ ϕ (pv) if p 3 or v 2, λ (pv)= ≥ ≤ (7) 2v−2 if p = 2 and v 3. ≥ the generalized fibonacci primitive roots 3
The two functions coincide, that is, ϕ(n) = λ(n) if n = 2, 4,pm, or 2pm,m 1. And ≥ ϕ (2m) = 2λ (2m). In a few other cases, there are some simple relationships between ϕ(n) and λ(n). In fact, it seamlessly improves the Fermat-Euler Theorem: The improve- ment provides the least exponent λ(n) ϕ(n) such that aλ(n) 1 mod n. | ≡
Lemma 2.2. ([2]) Let n N be any given integer. Then ∈ (i) The congruence aλ(n) 1 mod n is satisfied by every integer a 1 relatively prime ≡ ≥ to n, that is gcd(a, n) = 1.
(ii) In every congruence xλ(n) 1 mod n, a solution x = u exists which is a primitive ≡ root mod n, and for any such solution u, there are ϕ(λ(n)) primitive roots congruent to powers of u.
Proof. (i) The number λ(n) is a multiple of every λ (pv)= ϕ (pv) such that pv n. Ergo, | for any relatively prime integer a 2, the system of congruences ≥ aλ(n) 1 mod pv1 , aλ(n) 1 mod pv2 , . . . , aλ(n) 1 mod pvt , (8) ≡ 1 ≡ 2 ≡ t where t = ω(n) is the number of prime divisors in n, is valid.
Definition 2.2. An integer u Z is called a primitive root mod n if the least exponent ∈ min m N : um 1 mod n = λ(n). { ∈ ≡ } Lemma 2.3. (Primitive root test) An integer u Z is a primitive root modulo an integer ∈ n N if and only if ∈ uλ(n)/p 1 0 mod n (9) − 6≡ for all prime divisors p λ(n). | The primitive root test is a special case of the Lucas primality test, introduced in [9, p. 302]. A more recent version appears in [3, Theorem 4.1.1], and similar sources.
Lemma 2.4. Let n, and u N be integers, gcd(u, n) = 1. If u is a primitive root modulo ∈ pk for each prime power divisor pk n, then, the integer u = 1, v2 is a primitive root | 6 ± modulo n.
Proof. Without loss in generality, let n = pq with p 2 and q 2 primes. Let u be a ≥ ≥ primitive root modulo p and modulo q respectively. Then
u(p−1)/r 1 0 mod p and u(q−1)/s 1 0 mod q, (10) − 6≡ − 6≡ for every prime r p 1, and every prime s q 1 respectively, see Lemma 2.3. Now, | − | − suppose that u is not a primitive root modulo n. In particular,
uλ(n)/t 1 0 mod n (11) − ≡ for some prime divisor t λ(n). | Let v (λ(n)), v (p 1), and v (q 1) be the t-adic valuations of these integers. Since t t − t − λ(n) = lcm(ϕ(p 1), ϕ(q 1)), it follows that at least one of the relations − − v (λ(n)) = v (p 1) or v (λ(n)) = v (q 1) (12) t t − t t − the generalized fibonacci primitive roots 4 is valid. As consequence, at least one of the congruence equations
uλ(n)/t 1 0 mod n uλ(n)/t 1 mod p (13) − ≡ ⇐⇒ − ≡ or uλ(n)/t 1 0 mod n uλ(n)/t 1 mod q (14) − ≡ ⇐⇒ − ≡ fails. But, this in turns, contradicts the relations in (10) that u is a primitive root modulo both p and q. Therefore, u is a primitive root modulo n.
Example 2.1. The integer 2 is a primitive root modulo both p = 37 and q = 61. Let n = p q = 37 61, ϕ(p 1) = 22 32, ϕ(q 1) = 22 3 5, and λ(n) = lcm(ϕ(p 1), ϕ(q 1)) = · · − · − · · − − 22 32 5. The corresponding congruences and t-adic valuations of these integers are these. · · For t = 2, the valuations are: v (λ(n)) = v (p 1) = v (q 1) = 2. The assumption • 2 2 − 2 − that 2 is not a primitive root modulo n is not valid:
uλ(n)/2 1 0 mod n (15) − ≡ fails because at least one
2λ(n)/2 1 0 mod p or 2λ(n)/2 1 0 mod q (16) − 6≡ − 6≡ contradicts it.
For t = 3, the valuations are: v (λ(n)) = v (p 1) = 2, and v (q 1) = 1. The • 3 3 − 3 − assumption that 2 is not a primitive root modulo n is not valid:
uλ(n)/3 1 0 mod n (17) − ≡ fails because at least one
2λ(n)/3 1 0 mod p or 2λ(n)/3 1 0 mod q (18) − 6≡ − ≡ contradicts it.
For t = 5, the valuations are: v (λ(n)) = v (q 1) = 1, and v (p 1) = 0. The • 5 5 − 5 − assumption that 2 is not a primitive root modulo n is not valid:
uλ(n)/5 1 0 mod n (19) − ≡ fails because at least one
2λ(n)/5 1 0 mod p or 2λ(n)/5 1 0 mod q (20) − ≡ − 6≡ contradicts it.
Since the congruence (15) fails for every prime divisor t = 2, 3, 5 of λ(n) = 22 32 5, it · · implies that 2 is primitive root modulo n = 37 61. · the generalized fibonacci primitive roots 5
3 Characteristic Function For Fibonacci Primitive Roots
k × The symbol ord k (u) denotes the order of an element r Z p Z in the multi- p ∈ plicative group of the integers modulo pk. The order satisfies the divisibility condition ord k (r) λ(n), and primitive roots have maximal orders ord k (r) = λ(n). The basic p | p properties of primitive root are explicated in [1], [16], et cetera. The characteristic func- tion f : N 0, 1 of a fixed primitive root r in the finite ring Z pk Z, the integers −→ { } modulo pk, is determined here.
Lemma 3.1. Let pk, k 1, be a prime power, and let r Z be an integer such that ≥ ∈ gcd r, pk = 1. Then
(i) The characteristic f function of the primitive root (r mod pk) is given by
0 if p 3, k 1, ≤ ≤ k−1 2 k 1 if ord k (r)= p (p 1) and r r +1 mod p , f pk = p − ≡ (21) for any p 5, k 1, ≥ k≥−1 0 if ord k (r) = p (p 1), and p 5, k 2. p 6 − ≥ ≥ (ii) The function f is multiplicative, but not completely multiplicative since
(iii) f(pq)= f(p)f(q), gcd(p,q) = 1,
2 (iv) f p = f(p)f(p), if ord 2 (r) = p(p 1). 6 p 6 − Proof. The congruences x2 x 1 0 mod 2 and x2 x 1 0 mod 3 have no roots, so − − ≡ − − ≡ f 2k = 0and f 3k = 0 for all k 1. Moreover, for primes p 5 and k 1, function has ≥ × ≥ ≥ the value f pk = 1 if and only if the element r Z pk Z is a Fibonacci primitive root ∈ modulo pk. Otherwise, it vanishes: f pk = 0. The completely multiplicative property fails for p = 5. Specifically, 0 = f 52 = f(5)f(5) = 1. 6 Observe that the conditions ord (r) = p 1 and ord 2 (r) = p(p 1) imply that the p − p 6 − integer r = 1,s2 cannot be extended to a primitive root modpk, k 2. But that the 6 ± ≥ condition ord 2 (r) = p(p 1) implies that the integer r can be extended to a primitive p − root modpk, k 2. ≥ 4 Wirsing Formula
This formula provides decompositions of some summatory multiplicative functions as prod- ucts over the primes supports of the functions. This technique works well with certain multiplicative functions, which have supports on subsets of primes numbers of nonzero densities.
Lemma 4.1. ([23, p. 71]) Suppose that f : N C is a multiplicative function with the following properties. −→
(i) f(n) 0 for all integers n N. ≥ ∈ (ii) f pk ck for all integers k N, and c< 2 constant. ≤ ∈ (iii) f(p) = (τ + o(1))x/ log x, where τ > 0 is a constant as x . −→ ∞ Xp≤x the generalized fibonacci primitive roots 6
Then 1 x f(p) f p2 f(n)= + o(1) 1+ + + . (22) eγτ Γ(τ) log x p p2 · · · n≤x p≤x ! X Y The gamma function appearing in the above formula is defined by Γ(s)= ∞ ts−1e−stdt,s 0 ∈ C. The intricate proof of Wirsing formula appears in [23]. It is also assembled in various R papers, such as [6], [14, p. 195], and discussed in [13, p. 70], [21, p. 308]. Various appli- cations are provided in [12], [15], [24], et alii.
5 Harmonic Sums And Products Over Fibonacci Primes
The subset of primes = p P : ord(r)= ϕ(p) and r2 r +1 mod p P consists PF ∈ ≡ ⊂ of all the primes with Fibonacci primitive roots r Z. By Theorem 7.1, it has nonzero ∈ density α = δ ( ) > 0. The real number α > 0 is a rational multiple of the well known F PF F Artin constant, see [18], and [11] for the derivation. The proof of the next result is based on standard analytic number theory methods in the literature, refer to [15, Lemma 4].
Lemma 5.1. Assume the generalized Riemann hypothesis, and let x 1 be a large number. ≥ Then, there exists a pair of constants βF > 0, and γu > 0 such that 1 log log x (i) = αF log log x + βF + O . p log2 x p≤x pX∈PF
log p log log x (ii) = αF log x γF + O . p 1 − log2 x p≤x − pX∈PF
Proof. (i) Let π (x)=# p x : ord(r)= ϕ(p) and r2 r +1 mod p be the counting F ≤ ≡ measure of the corresponding subset of primes . To estimate the asymptotic order of PF the prime harmonic sum, use the Stieltjes integral representation:
1 x 1 π (x) x π (t) = dπ (t)= F + c x )+ u dt, (23) p t F x ( 0 t2 p≤x Zx0 Zx0 pX∈PF
2 where x0 > 0 is a constant. By the GRH, πF (x)= αF π(x)= αF x/ log x+O log log x/ log x , see Theorem 7.1. This yields x 1 αF log log x dπF (t) = + O 2 + c0(x0) 0 t log x log (x) Zx x 1 log log t +αF + O dt (24) t log t t log2(t) Zx0 log log x = αF log log x log log x0 + c0 (x0)+ O , − log2 x where β = loglogx + c (x ) is a generalized Mertens constant. The statement (ii) F − 0 0 0 follows from statement (i) and partial summation. the generalized fibonacci primitive roots 7
A generalized Mertens constant βF and a generalized Euler constant γF have other equiv- alent definitions such as
1 1 βF = lim αu log log x and βu = γu , (25) x→∞ p − − kpk p≤x p∈P , k≥2 X XF X p∈PF respectively. These constants satisfy βF = β1αF and γF = γαF . If the density αF = 1, these definitions reduce to the usual Euler constant and the Mertens constant, which are defined by the limits
log p 1 γ = lim log x and β1 = lim log log x , (26) x→∞ p 1 − x→∞ p − Xp≤x − Xp≤x or some other equivalent definitions, respectively. Moreover, the linear independence re- −1 lation in (25) becomes β = γ kpk , see [7, Theorem 427]. − p≥2 k≥2 P P A numerical experiment utilizing a small subset of primes with Fibonacci primitive roots gives the followings approximate values: 27 1 1. α = 1 = 0.265705484288843681890137 ..., F 38 − p(p 1) pY≥2 − 1 2. β α log log x = 0.05020530308647012230491, and F ≈ p − F p≤1301 pX∈PF log p 3. γ α log x = 0.0221594862523476326826286 .... F ≈ p 1 − F p≤1301 − pX∈PF Lemma 5.2. Assume the generalized Riemann hypothesis, and let x 1 be a large number. ≥ Then, there exists a pair of constants γF > 0 and νF > 0 such that 1 −1 log log x (i) 1 = eγF log(x)αF + O . − p log2 x p≤x pY∈PF 1 log log x (ii) 1+ = eγF 1 p−2 log(x)αF + O . p − log2 x p≤xp∈PF p∈PF Y Y log p −1 xαF log log x (iii) 1 = eνF −γF xαF + O . − p 1 log2 x p≤x − pY∈PF Proof. (i) Express the logarithm of the product as
1 −1 1 log 1 = (27) − p kpk p≤x p≤x,p∈PF , k≥1 pX∈PF X X 1 1 = + . (28) p kpk p≤x p≤x,p∈PF , k≥2 pX∈PF X X the generalized fibonacci primitive roots 8
Apply Lemma 5.1 and relation (25) to complete the verification. For (ii) and (iii), use similar methods as in the first one.
The constant νF > 0 is defined by the double power series (an approximate numerical value for p x = 1301 is shown): ≤ 1 log p k ν = 0.188622600886988493134287... . (29) 2 k p 1 ≈ p∈PXF , Xk≥2 −
6 Density Correction Factor
The subsets of primes = p P : ord(r)= p 1 and r2 = r + 1 has an important PF ∈ − partition as a disjoint union = , where PF A∪B k−1 2 2 = p P : ord(r)= p 1, ord k r = p (p 1), or r r +1 mod p (30) A ∈ − p 6 − 6≡ n o and
k−1 2 2 = p P : ord(r)= p 1, ord k r = p (p 1), and r r +1 mod p . (31) B ∈ − p − ≡ n o This partition has a role in the determination of the density of the subset of integers = NF n N : ord(r)= λ(n) and r2 r +1 mod n with Fibonacci primitive roots r Z. The ∈ ≡ ∈ finite prime product occurring in the proof of Theorem 1.1 splits into two finite products over these subsets of primes. These are used to reformulate an equivalent expression suitable for this analysis: the generalized fibonacci primitive roots 9
1 P (x) = 1+ p k pY≤x, ordp(r)=p−1, 2 2 ordp2 (r)6=p(p−1) or r 6≡r+1 mod p 1 1 1+ + + × p p2 · · · pk≤x, Y 2 2 ordp2 (r)=p(p−1) and r ≡r+1 mod p 1 = 1 2 (32) k − p pY≤x, ordp(r)=p−1, 2 2 ordp2 (r)6=p(p−1) or r 6≡r+1 mod p 1 −1 1 × − p pk≤x, Y 2 2 ordp2 (r)=p(p−1) and r ≡r+1 mod p 1 1 −1 = 1 1 − p2 − p p≤x p≤x pY∈A pY∈PF 1 1 −1 log x = 1 1 + O , − p2 − p x p∈A p≤x Y pY∈PF where the convergent partial product is replaced with the approximation
1 1 1 1 = 1 + O . (33) − p2 − p2 x p≤x p∈A pY∈A Y
The product 1 p−2 reduces the density to compensate for those primes for which p∈A − the Fibonacii primitive root r mod p that cannot be extended to a Fibonacci primitive root Q r mod p2. This seems to be a density correction factor similar to case for primitive roots over the prime numbers. The correction required for certain densities of primes with respect to fixed primitive roots over the primes was discovered by the Lehmers, see [20].
7 The Proof Of The Theorem
The algebraic constraint r2 = r+1 restricts the Fibonacci primitive roots r modulo a prime p 5 to the algebraic integers 1 √5 2 in the finite field F . This in turns forces the ≥ ± p corresponding primes to the residue classes p 1 mod 10, the analysis appears in [18]. ≡ ± A conditional proof for the conjecture of Shank was achieved in [11] and [19].
Theorem 7.1. ([11]) If the extended Riemann hypothesis hold for the Dedekind zeta function over Galois fields of the type Q √d r, √n 1 , where n is a squarefree integer, and d n. | Then, the density of primes which have Fibonacci primitive roots is α = (27/38) (1 F p≥2 − 1/p(p 1)), and the number of such primes is − Q x x log log x PF (x)= αF + O as x . (34) log x log2 x −→ ∞ the generalized fibonacci primitive roots 10
In terms of the subsets and , see Section 6, the characteristic function f(n) of Fibonacci A B primitive roots has the simpler description 1 if p , and k = 1, 1 if p , f pk = ∈A and f pk = ∈B (35) 0 if p , and k 2, 0 otherwise. ∈A ≥ Proof. By Theorem 7.1, the density α = δ ( ) > 0 of the subset of primes is nonzero. F PF PF Put τ = αF in Wirsing formula, Lemma 4.1, and replace the characteristic function f(n) of Fibonacci primitive roots in the finite ring Z pk Z, k 1, see Lemma 3.1, to produce ≥ 1 x f(p) f p2 f(n) = + o(1) 1+ + + eγτ Γ(τ) log x p p2 · · · n≤x p≤x ! X Y 1 x = + o(1) eγαF Γ (α ) log x F 1 1+ (36) × k p pY≤x, ordp(r)=p−1, 2 2 ordp2 (r)6=p(p−1) or r 6≡r+1mod p 1 1 1+ + + . × p p2 · · · pk≤x, Y 2 2 ordp2 (r)=p(p−1) and r ≡r+1mod p Replacing the equivalent product, see (32) in Section 5, and using Lemma 5.2, yield
1 x f(n) = γα + o(1) e F Γ (αF ) log x nX≤x 1 1 −1 = 1 1 (37) − p2 − p p∈A p≤x Y pY∈PF eγF −γαF x 1 = + o(1) 1−α 1 2 Γ (αF ) (log x) F − p pY∈A where αF > 0 is a constant, and γF is a generalized Euler constant, see Lemma 6 for detail.
As stated before, the constant α = (27/38) (1 1/p(p 1)) = 0.265705... is the F p≥2 − − average density of the densities of the primes in the residue classes mod 20 that have Q Fibonacci primitive roots, confer [18] for the calculations.
The Fibonacci primitive root problem is a special case of a more general problem that investigates the existence of infinitely many primes such that the polynomial congruence f(n) 0 mod p has primitive root solutions. The algebraic and analytic analysis for these ≡ more general polynomials are expected to be interesting.
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