## Results and estimates on pseudopowers (1996)

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Venue: | Math. Comp |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Bach96resultsand,

author = {Eric Bach and Richard Lukes and Jeffrey Shallit and H. C. Williams},

title = {Results and estimates on pseudopowers},

journal = {Math. Comp},

year = {1996},

volume = {65},

pages = {1737--1747}

}

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### Abstract

Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an x-pseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.

### Citations

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(Show Context)
Citation Context ...nstra [11] has shown that for every t, the limit A(2,t) = lim x→∞ A(2,t;x) (5) exists, assuming the ERH. Thus, it is plausible that the sum in (4) has a limit as x →∞, and that the limit is c ′ 2 = � =-=(6)-=- A(2,t)logt. t≥1 (Murata [14] gives an estimate for the rate of convergence in (5), but it does not seem sharp enough to prove this.) We can compute c ′ 2 using results of Wagstaff [24], who expressed... |

312 |
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(Show Context)
Citation Context ...s the computation of c to the evaluation of � q (log q)/qn for various n. Using Möbius inversion, these sums can be rewritten in terms of the logarithmic derivative of the zeta function (see (5.1) of =-=[16]-=-). Doing all this, we find that c = − � µ(m) � � n � ′ ζ 2 ζ (mn). m≥1 n≥2 Integer values of the zeta function and its derivative are easy to obtain by EulerMaclaurin summation [4]. Numerically, we ha... |

290 | Algebraic Number Theory - Lang - 1994 |

236 |
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(Show Context)
Citation Context ...tion (see (5.1) of [16]). Doing all this, we find that c = − � µ(m) � � n � ′ ζ 2 ζ (mn). m≥1 n≥2 Integer values of the zeta function and its derivative are easy to obtain by EulerMaclaurin summation =-=[4]-=-. Numerically, we have c . =0.89846489937400140618. This argument assumed a randomly chosen base. We now consider the specific base 2. The actual average value of log p−1 |〈2〉p| ,foroddprimesp≤106 ,is... |

27 |
On Artin’s conjecture and Euclid’s algorithm in global fields
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(Show Context)
Citation Context ...For any given x, we can express the average as � (4) A(2,t;x)logt, t≥1 where A(2,t;x) denotes the fraction of odd p ≤ x with the index of 〈2〉 mod p equal to t. (Note that this sum is finite.) Lenstra =-=[11]-=- has shown that for every t, the limit A(2,t) = lim x→∞ A(2,t;x) (5) exists, assuming the ERH. Thus, it is plausible that the sum in (4) has a limit as x →∞, and that the limit is c ′ 2 = � (6) A(2,t)... |

23 |
A bound for the least prime ideal in the Chebotarev density theorem
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(Show Context)
Citation Context ...≤4q 3 log q + q 3 log(2n)=O((log n)(log log n) 3 ). If the ERH holds, there is a degree-1 prime P of K modulo which X q − 2 splits completely and X q − n is irreducible, of norm O((log |∆|) 2 ). (See =-=[8]-=-.) Taking p to be the norm of P, we find as before that n �∈ 〈2〉 mod p. Necessarily, p>x, so we have x = O((log n) 2 (log log n) 6 ). Recalling that n = P2(x), we obtain the result. The estimate of Th... |

22 |
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- Hilbert
(Show Context)
Citation Context ...1 where A(2,t;x) denotes the fraction of odd p ≤ x with the index of 〈2〉 mod p equal to t. (Note that this sum is finite.) Lenstra [11] has shown that for every t, the limit A(2,t) = lim x→∞ A(2,t;x) =-=(5)-=- exists, assuming the ERH. Thus, it is plausible that the sum in (4) has a limit as x →∞, and that the limit is c ′ 2 = � (6) A(2,t)logt. t≥1 (Murata [14] gives an estimate for the rate of convergence... |

17 | Explicit bounds for primes in residue classes
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- 1996
(Show Context)
Citation Context ...we have x = O((log n) 2 (log log n) 6 ). Recalling that n = P2(x), we obtain the result. The estimate of Theorem 3 could be made explicit by using a strong form of the generalized Linnik theorem; see =-=[2]-=-. 4. A heuristic estimate for P2(x) Theorem 1 implies that P2(x) →∞. The theorems of the last two sections give us bounds of the form Ax 1/2−ɛ ≤ log P2(x) ≤ Bx, in which A and B are certain positive c... |

16 |
Integer sequences having prescribed quadratic character
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- 1970
(Show Context)
Citation Context ...ke an odd square modulo all primes ≤ x (i.e., n ≡ 1(mod8),and � � n p = 1 for all primes p ≤ x), then n is said to be an x-pseudosquare. Pseudosquares were first studied by Lehmer, Lehmer, and Shanks =-=[10]-=-. Williams et al. [12, 21] have computed the least xpseudosquare for all x ≤ 271. It is possible to show, assuming √the Extended x/2 Riemann Hypothesis (ERH), that the least x-pseudosquare is >e [25].... |

16 |
Numerical sieving devices: their history and some applications, Nieuw Archief voor
- Lukes, Patterson, et al.
(Show Context)
Citation Context ...ne designed and built by the fourth author and his colleagues. This machine searches for the least integer satisfying a set of congruence conditions, such as (1), and is described in detail elsewhere =-=[12, 13]-=-. It will be noted that (2) is not a very good predictor of P2(pk) within the range of this table. For example, if we take k = 55, so that pk = 257, then e γ log pk e (k−1)c2 ≈ 6 × 10 22 ,whereasP2(pk... |

15 | Some results on pseudosquares
- Lukes, Patterson, et al.
(Show Context)
Citation Context ...o all primes ≤ x (i.e., n ≡ 1(mod8),and � � n p = 1 for all primes p ≤ x), then n is said to be an x-pseudosquare. Pseudosquares were first studied by Lehmer, Lehmer, and Shanks [10]. Williams et al. =-=[12, 21]-=- have computed the least xpseudosquare for all x ≤ 271. It is possible to show, assuming √the Extended x/2 Riemann Hypothesis (ERH), that the least x-pseudosquare is >e [25]. In this paper, we conside... |

10 |
A problem analogous to Artin’s conjecture for primitive roots and its applications
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- 1991
(Show Context)
Citation Context ...r every t, the limit A(2,t) = lim x→∞ A(2,t;x) (5) exists, assuming the ERH. Thus, it is plausible that the sum in (4) has a limit as x →∞, and that the limit is c ′ 2 = � (6) A(2,t)logt. t≥1 (Murata =-=[14]-=- gives an estimate for the rate of convergence in (5), but it does not seem sharp enough to prove this.) We can compute c ′ 2 using results of Wagstaff [24], who expressed A(2,t)asa rational number ti... |

9 |
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- Pritchard
- 1983
(Show Context)
Citation Context ... ON PSEUDOPOWERS 1743 By comparison with the Euler ϕ-function, it can be shown that g(t)=O((log log t)/t 2 ), so that � t≥1 A(2,t)logtconverges. Using a segmented version of the Sieve of Eratosthenes =-=[3, 15]-=- we were able to compute g(t)fort<109 , and obtain the approximation c ′ . 2 =0.927346 (correct to six figures). Within the limits of this calculation, we have c ′ . 2 = c2. We conjecture that this is... |

6 |
The segmented sieve of Eratosthenes and primes in arithmetic progressions to 1012
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- 1977
(Show Context)
Citation Context ... ON PSEUDOPOWERS 1743 By comparison with the Euler ϕ-function, it can be shown that g(t)=O((log log t)/t 2 ), so that � t≥1 A(2,t)logtconverges. Using a segmented version of the Sieve of Eratosthenes =-=[3, 15]-=- we were able to compute g(t)fort<109 , and obtain the approximation c ′ . 2 =0.927346 (correct to six figures). Within the limits of this calculation, we have c ′ . 2 = c2. We conjecture that this is... |

5 |
bounds for the Chebyshev functions `(x) and /(x
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- 1976
(Show Context)
Citation Context ..., since n is odd. Hence we find P 2 (x)sp 1 p 2 \Delta \Delta \Delta p k . The prime number theorem tells us that P 2 (x)sY px p = e #(x) = e x(1+o(1)) : Furthermore, a result announced by Schoenfeld =-=[20]-=- provides the more explicit upper bound e 1:000081x . 2 We can obtain a lower bound on P 2 (x) if we assume the ERH: Theorem 3. If the Riemann hypothesis holds for Dedekind zeta functions, then there ... |

3 |
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- 1960
(Show Context)
Citation Context ...ger ep ≥ 0 such that n ≡ 2 ep (mod p). Then n is a power of 2. As Armand Brumer kindly pointed out to us (personal communication), this theorem is a special case of a more general theorem of Schinzel =-=[17]-=-. (See also [18, Thm. 2]; [19, Thm. 2].) Since our proof seems to be simpler than Schinzel’s, we give it here. Proof. We prove the contrapositive. Assume n is not a power of 2. Let q be the least prim... |

3 | On Power Residues and Exponential Congruences - Schinzel |

2 |
A conjecture of Chowla
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- 1953
(Show Context)
Citation Context ... higher powers. He proved that if x n ≡ a (mod p) has a solution for all but finitely many primes p, then either (i) there exists an integer b with a = b n , or (ii) 8 | n and a =2 n/8 b n . Also see =-=[1, 7]-=-. Let n be a positive integer. If n is a nonsquare that looks like an odd square modulo all primes ≤ x (i.e., n ≡ 1(mod8),and � � n p = 1 for all primes p ≤ x), then n is said to be an x-pseudosquare.... |

2 |
Eisenstein reciprocity and n-th power residues
- Kraft, Rosen
- 1981
(Show Context)
Citation Context ... higher powers. He proved that if x n ≡ a (mod p) has a solution for all but finitely many primes p, then either (i) there exists an integer b with a = b n , or (ii) 8 | n and a =2 n/8 b n . Also see =-=[1, 7]-=-. Let n be a positive integer. If n is a nonsquare that looks like an odd square modulo all primes ≤ x (i.e., n ≡ 1(mod8),and � � n p = 1 for all primes p ≤ x), then n is said to be an x-pseudosquare.... |

2 | A refinement of a theorem of Gerst on power residues - Schinzel - 1970 |

2 |
An open architecture number sieve
- Stephens, Williams
- 1990
(Show Context)
Citation Context ...o all primes ≤ x (i.e., n ≡ 1(mod8),and � � n p = 1 for all primes p ≤ x), then n is said to be an x-pseudosquare. Pseudosquares were first studied by Lehmer, Lehmer, and Shanks [10]. Williams et al. =-=[12, 21]-=- have computed the least xpseudosquare for all x ≤ 271. It is possible to show, assuming √the Extended x/2 Riemann Hypothesis (ERH), that the least x-pseudosquare is >e [25]. In this paper, we conside... |

1 |
bounds for the Chebyshev functions θ(x)andψ(x
- Sharper
- 1976
(Show Context)
Citation Context ...cannot be a power of 2, since n is odd. Hence we find P2(x) ≤ p1p2 ···pk. The prime number theorem tells us that P2(x) ≤ � p≤x p = e ϑ(x) = e x(1+o(1)) . Furthermore, a result announced by Schoenfeld =-=[20]-=- provides the more explicit upper bound e 1.000081x . We can obtain a lower bound on P2(x) if we assume the ERH: Theorem 3. If the Riemann hypothesis holds for Dedekind zeta functions, then there is a... |