## The Lucas–Pratt primality tree

Venue: | Math. Comp |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Bayless_thelucas–pratt,

author = {Jonathan Bayless},

title = {The Lucas–Pratt primality tree},

journal = {Math. Comp},

year = {},

pages = {495--502}

}

### OpenURL

### Abstract

Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1.

### Citations

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(Show Context)
Citation Context ...r than C log p for a set of primes p with relative asymptotic density 1. 1. Introduction Over a quarter of a century before Agrawal, Kayal, and Saxena showed PRIMES is in P in [1], V. Pratt showed in =-=[11]-=- that PRIMES is in NP by utilizing the following theorem of Lucas: Theorem 1. Suppose p > 1 is an odd integer and (1.1) � a p−1 2 ≡ −1 mod p, a p−1 2q �≡ −1 mod p for every odd prime q | p − 1. Then p... |

41 |
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(Show Context)
Citation Context ...o prove Theorem 4. Lemma 1. For any value of x ≥ ee and any prime p < x, the following holds: � ≤ r≤ x 1 r log 2 r≡1 mod p x r 30 log log x p log x . p Proof. We use the Brun-Titchmarsh inequality of =-=[9]-=-; namely, for coprime integers k and ℓ, the number of primes q ≤ x with q ≡ ℓ mod k, denoted π(x; k, ℓ), satisfies π(x; k, ℓ) ≤ 2x ϕ(k) log x k for x > k. For prime p ≥ 2 and (ℓ, p) = 1, we can use th... |

17 |
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(Show Context)
Citation Context ...ntegers n ≤ x with p 2 | FK(n) for some prime p > y is at most x y K(C2 log log x) 2K . Clearly, the bound in the proposition still holds if we restrict the count to prime n. Using the arguments from =-=[4]-=- and [7] together with a short computation, we may take C2 = 27. We need one last proposition before moving on to the proof of the theorems. Proposition 3. There exists an absolute constant C3 > 0 suc... |

8 |
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(Show Context)
Citation Context ...al to the squarefree product of the primes appearing in the Lucas-Pratt tree for p. As an example, F (9461) = 9461 · 5 · 11 · 43 · 3 · 7 = 469880565. We first prove a theorem analogous to a result in =-=[7]-=- and suggested in the same paper. The proof of this theorem follows the same plan as in [7], but certain extra tools are brought into play. Theorem 4. For each ɛ > 0, the set of primes p for which ϕ(F... |

8 |
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(Show Context)
Citation Context ...least log 2 p − 2, as this is a lower bound for the number of multiplications necessary to evaluate the first condition of Theorem 1. A relevant result of Pomerance is the following theorem (found in =-=[10]-=-): Theorem 3. For every prime p there is a proof that it is prime which requires for its verification ( 5 2 + o(1)) log 2 p multiplications modulo p. Thus, in principle, for each prime p there exists ... |

5 | On values taken by the largest prime factor of shifted primes
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(Show Context)
Citation Context ...imited attempts to use the result of this paper to answer this question have met with little success, but it would be interesting to see if a similar lower bound on the bit complexity is possible. In =-=[2]-=-, the related question of the height of the Lucas-Pratt tree is considered by Banks and Shparlinski, who prove a lower bound on the height for almost all primes p. Throughout, p, q, and r will always ... |

3 |
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(Show Context)
Citation Context ...≡ −1 mod p for every odd prime q | p − 1. Then p is prime and a is a primitive root of p. Conversely, if p is an odd prime, then every primitive root a of p satisfies conditions (1.1). In 1877, Lucas =-=[8]-=- stated a result essentially equivalent to Theorem 1; it is based on his work from a year earlier. The actual statement presented here can be found in [3, Theorem 4.1.8]. To achieve this Lucas-Pratt c... |