## Detecting Perfect Powers In Essentially Linear Time (1998)

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Venue: | Math. Comp |

Citations: | 42 - 12 self |

### BibTeX

@ARTICLE{Bernstein98detectingperfect,

author = {Daniel J. Bernstein},

title = {Detecting Perfect Powers In Essentially Linear Time},

journal = {Math. Comp},

year = {1998},

volume = {67},

pages = {1253--1283}

}

### Years of Citing Articles

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### Abstract

This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n) . 1.

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Citation Context ...s is almost exclusively M-time. The same is true in practice. See [15, section 4.3.3] for a discussion of fast multiplication algorithms. For the best known bounds on multiplication speed see [29] or =-=[1]-=-; note that it is possible to build a different algorithm, achieving the same bounds, out of the method of [24]. 4. Floating-point arithmetic A positive floating-point number is a positive integer div... |

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Citation Context ... for example, when n ≈ 32768—then F (n) will be noticeably larger than expected. I will get a lower bound on F (n) by changing u to 1, but I cannot get an upper bound in any analogous way. Notes. =-=See [28] for bo-=-unds on ϑ. See [3, section 2.7] for a general approach to obtaining bounds on functions such as ϑ, ϑ2, andℓ. 15. Analysis of F (n) Lemma 15.1 gives a lower bound for F (n), roughly lg lg n−lg l... |

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Citation Context ...heorem 1 by showing that F (n) is essentially linear in log n. Part VI surveys several practical improvements. Motivation. Before attempting to factor n with algorithms such as the number field sieve =-=[16]-=-, one should make sure that n is not a perfect power, or at least not a prime power. This is a practical reason to implement some power-testing method, though not necessarily a quick one. Speed is mor... |

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Citation Context ...(k) multiplications; this is useful in practice. See [15, section 4.6.3] and [11, section 1.2] for further discussion. For large k it is probably better to compute r k as exp(k log r) bythemethods of =-=[9], which take es-=-sentially linear time. PART II. ROOTS 7. Some overly specific inequalities Lemma 7.1. If κ>0and 0 <ɛ<1then (1 + ɛ/4κ) 2κ < 1+ɛ. Lemma 7.2. If κ ≥ 1 then 7/8 ≤ (1 − 1/8κ) κ . Lemma 7.3. ... |

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Citation Context ...omplete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; =-=(3) p-=-roves, using Loxton’s theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n) 1+o(1) . 1. Introduction An integer n>1isaperfect power if ... |

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Citation Context ...at most the number of products ≤ lg U of those primes, which is at most the number of products ≤ lg U of the first m prime numbers, which in turn can be estimated by analytic techniques. See [28] =-=and [10]. -=-20. Final F (n) analysis In this section, I use Lemma 19.4 to bound the function F (n) introduced in section 12. This upper bound is in (lg n) 1+o(1) . Lemma 20.1. Fix n ≥ exp exp 1000. Setu=loglog2... |

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Citation Context ...t new. It is stated in, e.g., [17, section 2.4]. But God is in the details: without good methods for computing n 1/k and for checking whether x k = n, Algorithm X is not so attractive. The authors of =-=[17] go on t-=-o say that one can “save time” by adding a battery of tests to Algorithm X. Variants of Algorithm X are also dismissed in [11, page 38] (“This is clearly quite inefficient”) and [4]. Observe t... |

41 |
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Citation Context ...lgorithms is almost exclusively M-time. The same is true in practice. See [15, section 4.3.3] for a discussion of fast multiplication algorithms. For the best known bounds on multiplication speed see =-=[29]-=- or [1]; note that it is possible to build a different algorithm, achieving the same bounds, out of the method of [24]. 4. Floating-point arithmetic A positive floating-point number is a positive inte... |

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Citation Context ...essarily a quick one. Speed is more important in other applications. According to [18] there is a theoretically interesting method of finding all small factors of n (to be presented in a successor to =-=[19]-=-) for which perfect-power classification can be a bottleneck. Received by the editor October 11, 1995 and, in revised form, April 10, 1997. 1991 Mathematics Subject Classification. Primary 11Y16; Seco... |

17 |
private communication
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Citation Context ...ower, or at least not a prime power. This is a practical reason to implement some power-testing method, though not necessarily a quick one. Speed is more important in other applications. According to =-=[18]-=- there is a theoretically interesting method of finding all small factors of n (to be presented in a successor to [19]) for which perfect-power classification can be a bottleneck. Received by the edit... |

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Citation Context ... multiplication algorithms. For the best known bounds on multiplication speed see [29] or [1]; note that it is possible to build a different algorithm, achieving the same bounds, out of the method of =-=[24]-=-. 4. Floating-point arithmetic A positive floating-point number is a positive integer divided by a power of 2. The computer can store a pair (a, n), with a an integer and n a positive integer, to repr... |

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Citation Context ...lgorithms is almost exclusively M-time. The same is true in practice. See [15, section 4.3.3] for a discussion of fast multiplication algorithms. For the best known bounds on multiplication speed see =-=[29-=-] or [1]; note that it is possible to build a dierent algorithm, achieving the same bounds, out of the method of [24]. 4. Floating-point arithmetic A positivesoating-point number is a positive integer... |

13 | The complexity of multiple-precision arithmetic
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Citation Context ...rithms themselves are short and straightforward. The basic outline of my method is well known, as is its approximate run time. For Newton’s method with increasing precision see [9] (which popularize=-=d [8]) or-=- [7, section 6.4]. For the specific case of inversion see also [15, Algorithm 4.3.3–R] or [1, page 282]. For a controlled number of steps of binary search as preparation for Newton’s method see [4... |

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Citation Context ...g enough work into the subroutines, I have made Algorithm X quite fast—so fast, in fact, that typical modifications will slow it down. I use the Sieve of Eratosthenes to enumerate the primes p<f. Se=-=e [27]-=- for faster methods. Note that the best order of operations in Algorithm X depends on the distribution of inputs; for example, if the input source is very likely to produce 37th powers, then p = 37 sh... |

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Citation Context ...⌈lg 8k⌉)µ(2g + ⌈lg 8k⌉) <P(k)(4g + ⌈lg 2g⌉⌈lg 8k⌉)µ(2g + ⌈lg 8k⌉)sDETECTING PERFECT POWERS IN ESSENTIALLY LINEAR TIME 1263 Notes. My use of increasing precision is at the heart o=-=f my improvement over [4]-=-. A 50-digit number that starts with 9876 is not an 11th power; the remaining 46 digits are irrelevant. In general, Algorithm C does not inspect many more bits of n than are necessary to distinguish n... |

5 |
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Citation Context ...3–1283 S 0025-5718(98)00952-1 DETECTING PERFECT POWERS IN ESSENTIALLY LINEAR TIME DANIEL J. BERNSTEIN Abstract. This paper (1) gives complete details of an algorithm to compute approximate kth roots=-=; (2) u-=-ses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton’s theorem on multiple linear forms in logari... |

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(Show Context)
Citation Context ...t new. It is stated in, e.g., [17, section 2.4]. But God is in the details: without good methods for computing n 1=k and for checking whether x k = n, Algorithm X is not so attractive. The authors of =-=[17] go on to -=-say that one can \save time" by adding a battery of tests to Algorithm X. Variants of Algorithm X are also dismissed in [11, page 38] (\This is clearly quite inecient") and [4]. Observe that... |

1 | Masser (editors), Transcendence theory: advances and applications - Baker, William - 1977 |

1 |
Some problems involving powers of integers, Acta Arithmetica 46
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Citation Context ...ohn Loxton, Michael Parks, Igor Shparlinski, Jon Sorenson, and the referee for their helpful comments. Lenstra pointed out that the method of Part III had a p-adic analogue; Shparlinski suggested that=-=[20]-=-wouldhelpintheF(n) analysis; Loxton supplied the corrected proof of [20, Theorem 1] shown in section 19. PART I. ARITHMETIC 3. Integer arithmetic I represent a positive integer n inside a computer as ... |

1 | der Poorten, Multiplicative dependence in number fields, Acta Arithmetica 42 - Loxton, van - 1983 |

1 | der Poorten, Multiplicative dependence in number Acta Arithmetica 42 - Loxton, van - 1983 |