From Wikipedia, the free encyclopedia
Conjecture in number theory
This article is about the conjecture of Emil Artin on primitive roots. For the conjecture of Artin on L-functions, see
Artin L-function.
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of 2025. In fact, there is no single value of a for which Artin's conjecture is proved.
Let a be an integer that is not a square number and not −1. Write a = a0b2 with a0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then the conjecture states
The positive integers satisfying these conditions are:
The negative integers satisfying these conditions are:
Similar conjectural product formulas[1] exist for the density when a does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of CArtin. If a is a square number or a = −1, then the density is 0; more generally, if a is a perfect pth power for prime p, then the number needs to be multiplied by p ( p − 2 ) p 2 − p − 1 ; {\displaystyle {\frac {p(p-2)}{p^{2}-p-1}};} if there is more than one such prime p, then the number needs to be multiplied by p ( p − 2 ) p 2 − p − 1 {\displaystyle {\frac {p(p-2)}{p^{2}-p-1}}} for all such primes p). Similarly, if a0 is congruent to 1 mod 4, then the number needs to be multiplied by p ( p − 1 ) p 2 − p − 1 {\displaystyle {\frac {p(p-1)}{p^{2}-p-1}}} for all prime factors p of a0.
For example, take a = 2. The conjecture is that the set of primes p for which 2 is a primitive root has the density CArtin. The set of such primes is (sequence A001122 in the OEIS)
It has 38 elements smaller than 500 and there are 95 primes smaller than 500. The ratio (which conjecturally tends to CArtin) is 38/95 = 2/5 = 0.4.
For a = 8 = 23, which is a power of 2, the conjectured density is 3 5 C {\displaystyle {\frac {3}{5}}C} , and for a = 5, which is congruent to 1 mod 4, the density is 20 19 C {\displaystyle {\frac {20}{19}}C} .
In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized Riemann hypothesis.[2]
Without the generalized Riemann hypothesis, there is no single value of a for which Artin's conjecture is proved. However, D. R. Heath-Brown proved in 1986 (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes p.[3] He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.
Some variations of Artin's problem[edit]An elliptic curve E {\displaystyle E} given by y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} , Lang and Trotter gave a conjecture for rational points on E ( Q ) {\displaystyle E(\mathbb {Q} )} analogous to Artin's primitive root conjecture.[4]
Specifically, they said there exists a constant C E {\displaystyle C_{E}} for a given point of infinite order P {\displaystyle P} in the set of rational points E ( Q ) {\displaystyle E(\mathbb {Q} )} such that the number N ( P ) {\displaystyle N(P)} of primes ( p ≤ x {\displaystyle p\leq x} ) for which the reduction of the point P ( mod p ) {\displaystyle P{\pmod {p}}} denoted by P ¯ {\displaystyle {\bar {P}}} generates the whole set of points in F p {\displaystyle \mathbb {F_{p}} } in E {\displaystyle E} , denoted by E ¯ ( F p ) {\displaystyle {\bar {E}}(\mathbb {F_{p}} )} , is given by N ( P ) ∼ C E ( x log x ) {\displaystyle N(P)\sim C_{E}\left({\frac {x}{\log x}}\right)} .[5] Here we exclude the primes which divide the denominators of the coordinates of P {\displaystyle P} .
Gupta and Murty proved the Lang and Trotter conjecture for E / Q {\displaystyle E/\mathbb {Q} } with complex multiplication under the Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field.[6]
Krishnamurty proposed the question how often the period of the decimal expansion 1 / p {\displaystyle 1/p} of a prime p {\displaystyle p} is even.
The claim is that the period of the decimal expansion of a prime in base g {\displaystyle g} is even if and only if g ( p − 1 2 j ) ≢ 1 mod p {\displaystyle g^{\left({\frac {p-1}{2^{j}}}\right)}\not \equiv 1{\bmod {p}}} where j ≥ 1 {\displaystyle j\geq 1} and j {\displaystyle j} is unique and p is such that p ≡ 1 + 2 j mod 2 j {\displaystyle p\equiv 1+2^{j}\mod {2^{j}}} .
The result was proven by Hasse in 1966.[4][7]
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4