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On the density of primes in arithmetic progression having a prescribed primitive root
, 1999
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Residue classes free of values of Euler’s function
 In: Gy}ory K (ed) Proc Number Theory in Progress, pp 805–812. Berlin: W de Gruyter
, 1999
"... Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains at least one multiple of 4, then it contains ..."
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Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains at least one multiple of 4, then it contains
Residue Classes Having Tardy Totients
, 2008
"... We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/log mk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also s ..."
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We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/log mk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function λ(n) and the least such n satisfies n ≪ m 13. 1
ON THE STABILITY OF CERTAIN LUCAS SEQUENCES MODULO 2*
, 1995
"... Let {i/.  J G N} be the twoterm recurrence sequence defined by u0 = 0, ux\, and ut = aut_x +hui_2 for all /> 2, where a and b are fixed integers. Let m be an integer and consider the corresponding sequence {^.}, where u. GZ/mZ is obtained via the natural projection Z>Z//wZ. ..."
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Let {i/.  J G N} be the twoterm recurrence sequence defined by u0 = 0, ux\, and ut = aut_x +hui_2 for all /> 2, where a and b are fixed integers. Let m be an integer and consider the corresponding sequence {^.}, where u. GZ/mZ is obtained via the natural projection Z>Z//wZ.
Modulus Period Distribution
, 1987
"...  2) denote the sequence of Fibonacci numbers. For an integer m> 1, recall that (Fn) is uniformly distributed modulo m if all residues modulo m occur with the same frequency in any period (see [2], [4]). This happens precisely when m = 5 k with k> 0, in which case (Fn) has (shortest) period of lengt ..."
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 2) denote the sequence of Fibonacci numbers. For an integer m> 1, recall that (Fn) is uniformly distributed modulo m if all residues modulo m occur with the same frequency in any period (see [2], [4]). This happens precisely when m = 5 k with k> 0, in which case (Fn) has (shortest) period of length 4 • 5 k, and each residue occurs four times (see [1]5 [3])., In this paper we study moduli with more complex distributions. For any r, 0 < r < m, denote by v(p) the number of times r occurs as a residue in one (shortest) period of Fn (mod m). If m is a power of 5, then v(p) = 4 for all p. However, if m = 11, then the period of Fn (mod 11) is 0, 1 * 1> 2, 3, 5, 85 2, 10, 1, so that v(p) takes on four different values. Definition: For an integer m> 1, (Fn) is almost uniformly distributed modulo m [notation: (Fn) AUD (mod m)] if V(P) assumes exactly two values for 0 < r < m. In this paper we describe four infinite sequences of AUD moduli, along with describing the function v precisely for these moduli. Our proof makes use of a recent result of Velez [2], which we state here for the reader's convenience. Lemma: For any integer s> 0, the sequence