## Some results on pseudosquares (1996)

Venue: | Math. Comp |

Citations: | 16 - 5 self |

### BibTeX

@ARTICLE{Lukes96someresults,

author = {R. F. Lukes and C. D. Patterson and H. C. Williams},

title = {Some results on pseudosquares},

journal = {Math. Comp},

year = {1996},

volume = {65},

pages = {361--372}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L271. We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp √ e p/2, an inequality that must hold under the extended Riemann Hypothesis. 1.

### Citations

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Citation Context ...olynomial-time (in log N) algorithm for determining the prime character of N. At the moment the best unconditional results on the primality testing problem are those of Adleman, Pomerance, and Rumely =-=[1]-=-, who show that the problem can be solved for a given N by a deterministic algorithm of Received by the editor August 23, 1993 and, in revised form, April 8, 1994. 1991 Mathematics Subject Classificat... |

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Citation Context ... a most useful test of primality if it could be shown that Mp grows very quickly as a function of p. At the moment this seems very far from being achieved. We can, however, appeal to a result of Bach =-=[3]-=-: Theorem 2.3. Let G be a nontrivial subgroup of Z/(m) ∗ such that n ∈ G for all 0 <n<x. Then if the ERH holds, we must have x<2(log m) 2 . Consider the subgroup G of Z/(Mp) ∗ whichismadeupofall ksuch... |

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Citation Context ...ed by NSERC of Canada grant #A7649. 361 c○1996 American Mathematical Societys362 R. F. LUKES, C. D. PATTERSON, AND H. C. WILLIAMS time complexity O((log N) c log log log N ), and of Adleman and Huang =-=[2]-=-, who show that there is a probabilistic polynomial-time algorithm for solving this problem. However, both of these algorithms are very complicated and difficult to implement, whereas the (conditional... |

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Citation Context ...reasoning as used earlier that (pk/N)=−1forsome odd pk ≤ p when N is a prime. Thus, if B =1,andN<Lp, the conditions (3) and (4) must hold. Define negative pseudosquares (see Lehmer, Lehmer and Shanks =-=[18]-=-) Np for odd primes p by: (i) Np ≡−1(mod8), (ii) (−Np/pi) = 1 for all odd primes pi such that 2 <pi≤p, (iii) Np is the least positive integer satisfying (i) and (ii). i=1sSOME RESULTS ON PSEUDOSQUARES... |

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Citation Context ... of these algorithms are very complicated and difficult to implement, whereas the (conditional) algorithms provided in Section 2 are very simple. We should also mention here that Bach and Huelsbergen =-=[4]-=- have used the table of pseudosquares in [22] to support their heuristic argument that G(n), the smallest value of x such that the primes ≤ x generate the multiplicative group modulo n,is asymptotical... |

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Citation Context ...nevertheless not a perfect square. Hall [10] has shown that the values of Lp must be unbounded as p increases. Kraitchik [11, pp. 41–46] seems to have been the first to consider these numbers, and in =-=[11]-=- provides a table of them up to L47. Since then various authors, most notably D. H. Lehmer, who gave the pseudosquares their name (see Lehmer [16]), have extended Kraitchik’s list up to L223 (Stephens... |

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Citation Context ...ares should grow quickly. Incidentally, this problem of perfect power recognition, which can be of importance in primality testing and factoring, has been discussed more recently by Bach and Sorenson =-=[5]-=-. Also, as shown in Section 2, if pseudosquare growth is sufficiently rapid, then there exists a deterministic polynomial-time (in log N) algorithm for determining the prime character of N. At the mom... |

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Citation Context ... all primes pi such that 2 ≤ pi ≤ p, (4) p (N−1)/2 j ≡−1(modN)for some prime pj such that 2 ≤ pj ≤ p, then N is a prime or the power of a prime. A randomized version of this test was given by Lehmann =-=[14]-=-. Of course, this would be a most useful test of primality if it could be shown that Mp grows very quickly as a function of p. At the moment this seems very far from being achieved. We can, however, a... |

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Citation Context ... is the least positive nonsquare integer satisfying (i) and (ii). Thus, the pseudosquare Lp behaves locally like a perfect square modulo all primes ≤ p, but is nevertheless not a perfect square. Hall =-=[10]-=- has shown that the values of Lp must be unbounded as p increases. Kraitchik [11, pp. 41–46] seems to have been the first to consider these numbers, and in [11] provides a table of them up to L47. Sin... |

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Citation Context ...such that 2 <pi≤p, (iii) Np is the least positive integer satisfying (i) and (ii). i=1sSOME RESULTS ON PSEUDOSQUARES 365 Recently, the table of Np given in [22] has been extended by Bronson and Buell =-=[9]-=- from N211 to N227. By quadratic reciprocity it follows that (pi/Lp)=(pi/Np)=1 for all primes pi such that 2 ≤ pi ≤ p; thus it is easy to see that (2.2) Mp =min{Lp,Np}. It is possible, then, for the t... |

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Citation Context ...n any apparent residue of N is in fact a quadratic residue of N, and any apparent nonresidue of N is a nonresidue of N. In response to a nonrigorous method of primality testing advocated by Kraitchik =-=[12]-=- (for a discussion of this test see Lehmer [15], Beeger [6], Kraitchik [13]), Hall [10] produced a mathematically correct version of Kraitchik’s idea. This is provided in Theorem 2.1. If all the facto... |

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Citation Context ...f N, and any apparent nonresidue of N is a nonresidue of N. In response to a nonrigorous method of primality testing advocated by Kraitchik [12] (for a discussion of this test see Lehmer [15], Beeger =-=[6]-=-, Kraitchik [13]), Hall [10] produced a mathematically correct version of Kraitchik’s idea. This is provided in Theorem 2.1. If all the factors (not necessarily prime factors ) of N are below Lp and i... |

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Citation Context ...so a quadratic residue of N and every bibj is a quadratic residue of N,thenNis a prime or a power of a prime. Notice that this is a primality test that involves the pseudosquares. Furthermore, Beeger =-=[7]-=- actually used this test to prove that a certain 13 digit factor of 12 45 +1 is a prime. The main difficulty in utilizing Hall’s test is the problem of determining whether a given integer m is a quadr... |

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Citation Context ...p seem to grow very rapidly with respect to p. The growth rate of pseudosquares is of great importance in two problems in computational number theory: square recognition and primality testing. Cobham =-=[8]-=- has shown that if a number is not a perfect square, then under the Extended Riemann Hypothesis (ERH) it must fail to be a square modulo a small prime p. Thus, we expect that the pseudosquares should ... |

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Citation Context ...parent nonresidue of N is a nonresidue of N. In response to a nonrigorous method of primality testing advocated by Kraitchik [12] (for a discussion of this test see Lehmer [15], Beeger [6], Kraitchik =-=[13]-=-), Hall [10] produced a mathematically correct version of Kraitchik’s idea. This is provided in Theorem 2.1. If all the factors (not necessarily prime factors ) of N are below Lp and if {−1,2, −3,...,... |

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Citation Context ... all primes pi such that 2 ≤ pi ≤ p, (4) p (N−1)/2 j ≡−1(modN)for some prime pj such that 2 ≤ pj ≤ p, then N is a prime or the power of a prime. A randomized version of this test was given by Lehmann =-=[14]-=-. Of course, this would be a most useful test of primality if it could be shown that Mp grows very quickly as a function of p. At the moment this seems very far from being achieved. We can, however, a... |

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Citation Context ...tic residue of N, and any apparent nonresidue of N is a nonresidue of N. In response to a nonrigorous method of primality testing advocated by Kraitchik [12] (for a discussion of this test see Lehmer =-=[15]-=-, Beeger [6], Kraitchik [13]), Hall [10] produced a mathematically correct version of Kraitchik’s idea. This is provided in Theorem 2.1. If all the factors (not necessarily prime factors ) of N are be... |

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Citation Context ... In what follows we give a brief description of this new sieve and its capabilities. The Manitoba Scalable Sieve Unit (MSSU) employs Very Large Scale Integration (VLSI) circuits designed by Patterson =-=[19, 20]-=- at the University of Calgary. These sieve chips were manufactured using mature (2µm) CMOS gate array technology with a circuit complexity equivalent to 10,000 logic gates. Each device can accommodate... |

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Citation Context ...ld K = Q( √ D). Also, denote by L(1,χ) the value of the Dirichlet L-function L(s, χ)ats=1,where ∞� L(s, χ)= χ(n)n −s , n=1 χ(n) istheKroneckersymbol(d/n), and d is the discriminant of K. After Shanks =-=[21]-=- we define the Upper Littlewood Index (ULI) of K to be the value of L(1,χ)/(2e γ log log |d|). If the Riemann Hypothesis on L(s, χ) above holds, we must have L(1,χ)<{1+o(1)}2e γ log log |d|; thus, we ... |

1 |
An open architecture number sieve, Number Theory and Cryptography
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(Show Context)
Citation Context ...ble of them up to L47. Since then various authors, most notably D. H. Lehmer, who gave the pseudosquares their name (see Lehmer [16]), have extended Kraitchik’s list up to L223 (Stephens and Williams =-=[22]-=-). Notice that the values of Lp seem to grow very rapidly with respect to p. The growth rate of pseudosquares is of great importance in two problems in computational number theory: square recognition ... |

1 | Primality testing on a computer, ArsCombin.5(1978), 127–185 - Williams |

1 | Primality testing on a computer, Ars Combin. 5 - Williams - 1978 |