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LEAST TOTIENT IN A RESIDUE CLASS
 BULL. LONDON MATH. SOC. 39 (2007) 425–432
, 2007
"... For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a (mod m). Here, as usual ϕ(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler ..."
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Cited by 4 (2 self)
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For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a (mod m). Here, as usual ϕ(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler function in reduced residue classes modulo m. Some discussion and results are also given for classes with gcd(a, m)>1, in which case such n do not always exist, and also on the related problem for ‘cototients’.
Values of the Euler Function in Various Sequences
 MONATSH. MATH. 146, 1–19
, 2005
"... Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ p 1 holds with some prime p, as wel ..."
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Cited by 4 (3 self)
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Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ p 1 holds with some prime p, as well as those positive integers n such that the equation ’ðnÞ f ðmÞ holds with some integer m, where f is a fixed polynomial with integer coefficients and degree deg f> 1.
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
Compositions with the Euler and Carmichael Functions
"... Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1 ..."
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Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1
Abstract
, 2008
"... We present an algorithm to invert the Euler function ϕ(m). The algorithm, for a given integer n ≥ 1, in polynomial time “on average”, finds the set Ψ(n) of all solutions m to the equation ϕ(m) = n. In fact, in the worst case the set Ψ(n) is exponentially large and cannot be constructed by a polynom ..."
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We present an algorithm to invert the Euler function ϕ(m). The algorithm, for a given integer n ≥ 1, in polynomial time “on average”, finds the set Ψ(n) of all solutions m to the equation ϕ(m) = n. In fact, in the worst case the set Ψ(n) is exponentially large and cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under some widely accepted number theoretic conjecture, that the Partition Problem, an NPcomplete problem, can be reduced, in polynomial time, to the problem of deciding whether ϕ(m) = n has a solution, for polynomially (in the input size of the Partition problem) many values of n. In fact, the following problem is NPcomplete: Given a set of positive integers S, decide whether there is an n ∈ S satisfying ϕ(m) = n, for some integer m. Finally, we establish close links between the problem of inverting the Euler function and the integer factorisation problem. 1
Article electronically published on January 23, 2006 COMPLEXITY OF INVERTING THE EULER FUNCTION
"... Abstract. Given an integer n, how hard is it to find the set of all integers m such that ϕ(m) =n, whereϕ is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of n, in polynomial time “on average ” (that is, (log n) O(1)), finds the set of al ..."
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Abstract. Given an integer n, how hard is it to find the set of all integers m such that ϕ(m) =n, whereϕ is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of n, in polynomial time “on average ” (that is, (log n) O(1)), finds the set of all such solutions m. In fact, in the worst case this set of solutions is exponential in log n, and so cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that the Partition Problem, anNPcomplete problem, can be reduced in polynomial (in the input size) time to the problem of deciding whether ϕ(m) =n has a solution, for polynomially (in the input size of the Partition Problem)manyvaluesofn (where the prime factorizations of these n are given). What this means is that the problem of deciding whether there even exists a solution m to ϕ(m)=n, let alone finding any or all such solutions, is very likely to be intractable. Finally, we establish close links between the problem of inverting the Euler function and the integer factorization problem. 1.