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## An extension of Lucas’ theorem (2001)

Venue: | PROC. AMER. MATH. SOC |

Citations: | 24 - 14 self |

### Citations

811 |
A Classical Introduction to Modern Number Theory, 2nd ed
- Ireland, Rosen
- 1990
(Show Context)
Citation Context ...·· . 2 Suppose that uk =0forsomek ∈ Z + . Then ∆ ̸= 0,α ̸= β and α k = β k . Since the field Q( √ ∆) contains the root α/β ̸= ±1 of unity, by Propositions 13.1.5 and 13.1.6 of K. Ireland and M. Rosen =-=[3]-=- there exists a positive integer D such that ∆ = −D 2 and α/β ∈{±i}, or∆=−3D 2 and α/β ∈{±ω, ±ω 2 } where ω =(−1+ √ −3)/2. In the former case, (A + Di)/(A − Di) ∈{±i}; hence A 2 = D 2 and 2B =(A 2 − ∆... |

214 | History of the Theory - Dickson - 2005 |

101 |
History of the theory of numbers. Vol
- Dickson
- 1966
(Show Context)
Citation Context ... v 2 n − 2B n for n ∈ N. For a, b ∈ Z let (a, b) denote the greatest common divisor of a and b. Aniceresult of E. Lucas asserts that if (A, B) =1,then(um,un) =|u(m,n)| for m, n ∈ N (cf. L. E. Dickson =-=[1]-=-). In the case A2 = B = 1, by induction on n ∈ N we find that un =0if3| n, and { 1 if A = −1 &3| n − 1, or A =1&n ≡ 1, 2(mod6); un = −1 if A = −1 &3| n +1, or A =1&n ≡−1, −2 (mod6). Received by the ed... |

40 |
The power of a prime that divides a generalized binomial coefficient
- Knuth, Wilf
- 1989
(Show Context)
Citation Context ...1)/(q − 1) for j =0, 1, 2, ···. For generalized binomial coefficients formed from an arbitrary sequence of positive integers, the reader is referred to the elegant paper of D. E. Knuth and H. S. Wilf =-=[5]-=-. Let d>1andq>0 be integers with d | uq. If (A, B) =1andd∤uk for k =1,··· ,q− 1, then for any n ∈ N we have d | un ⇐⇒ d divides (un,uq) =|u(n,q)| ⇐⇒ q =(n, q) ⇐⇒ q | n; this property is usually called... |

36 |
Existence of primitive divisors of Lucas and
- Bilu, Voutier
(Show Context)
Citation Context ...g upon some ideas of A. Schinzel [6], C. L. Stewart [7] proved in 1977 that if A is prime to B and α/β is not a root of unity, then un has a primitive prime divisor for each n>e452267 ; P. M. Voutier =-=[9]-=- conjectured in 1995 that the lower bound e452267 can be replaced by 30. For m ∈ Z we use Zm to denote the ring of rationals in the form a/b with a ∈ Z, b ∈ Z + and (b, m) =1. Whenr ∈ Zm, byx ≡ r (mod... |

20 |
Primitive divisors of the expression A n − B n in algebraic number
- Schinzel
- 1974
(Show Context)
Citation Context ...u (n,q)) =1. When p is an odd prime not dividing B, p∗ exists because p | u ∆ p−( p ) as is well known where (−) denotes the Legendre symbol. On the other hand, drawing upon some ideas of A. Schinzel =-=[6]-=-, C. L. Stewart [7] proved in 1977 that if A is prime to B and α/β is not a root of unity, then un has a primitive prime divisor for each n>e452267 ; P. M. Voutier [9] conjectured in 1995 that the low... |

16 | Primitive divisors of Lucas and Lehmer sequences - Voutier - 1998 |

12 |
Congruence properties of ordinary and q-binomial coefficients
- Fray
- 1967
(Show Context)
Citation Context ... nonnegative integers with s, t < p. Inthecase A = a +1 andB = a where a ∈ Z and |a| > 1, as uq+1 =(a q+1 − 1)/(a − 1) = auq +1 ≡ 1(mod uq) forq ∈ Z + , our Theorem implies Theorem 3.11 of R. D. Fray =-=[2]-=-. Theorem 3 of B. Wilson [10] follows from our Theorem in the special case A = 1, B = −1 ands ≥ t. Wilson used a result of Kummer concerning the highest power of a prime dividing a binomial coefficien... |

11 |
Reduction of unknowns in Diophantine representations
- Sun
- 1992
(Show Context)
Citation Context ...t k, q ∈ Z +.Then (7) If uq ̸= 0,then = Aunvn + v 2 n − 2B n 2 u1 = un+1vn − B n ≡−B n (mod d). ukq+l ≡ u k q+1 ul (mod uq) for l =0, 1, 2, ··· . ukq (8) kuq Proof. Let l ∈ N. By Lemma 2 of Z.-W. Sun =-=[8]-=-, k∑ ( k ukq+l = r ≡ u k−1 q+1 +(k − 1)Auq 2 r=0 ) c k−r u r qul+r (mod uq). where c = −Buq−1 = uq+1 − Auq. Clearly ukq+l ≡ u k q+1 ul (mod uq). In the case uq ̸= 0, k∑ ukq 1 = kuq k r=1 For any prime... |

6 |
Fibonacci triangles modulo p, Fibonacci Quart. 36
- Wilson
- 1998
(Show Context)
Citation Context ..., t < p. Inthecase A = a +1 andB = a where a ∈ Z and |a| > 1, as uq+1 =(a q+1 − 1)/(a − 1) = auq +1 ≡ 1(mod uq) forq ∈ Z + , our Theorem implies Theorem 3.11 of R. D. Fray [2]. Theorem 3 of B. Wilson =-=[10]-=- follows from our Theorem in the special case A = 1, B = −1 ands ≥ t. Wilson used a result of Kummer concerning the highest power of a prime dividing a binomial coefficient; see Knuth and Wilf [5] for... |

5 |
Some congruences for generalized binomial coefficients
- Kimball, Webb
- 1995
(Show Context)
Citation Context ... 1 (mod 2), k − 13474 HONG HU AND ZHI-WEI SUN Remark 2. In light of Lemma 1, by induction, if n ∈ N and [n] ̸= 0,then[ n k ] ∈ Z for all k ∈ N. This was also realized by W. A. Kimball and W. A. Webb =-=[4]-=-. In 1989 Knuth and Wilf [5] proved that generalized binomial coefficients, formed from a regularly divisible sequence of positive integers, are always integral. Lemma 2. Let q be a positive integer. ... |

4 |
Primitive divisors of Lucas and Lehmer sequences, in: Transcendence Theory
- Stewart
- 1977
(Show Context)
Citation Context ... is an odd prime not dividing B, p∗ exists because p | u ∆ p−( p ) as is well known where (−) denotes the Legendre symbol. On the other hand, drawing upon some ideas of A. Schinzel [6], C. L. Stewart =-=[7]-=- proved in 1977 that if A is prime to B and α/β is not a root of unity, then un has a primitive prime divisor for each n>e452267 ; P. M. Voutier [9] conjectured in 1995 that the lower bound e452267 ca... |