Displaying similar documents to “A sharp form of an embedding into multiple exponential spaces”

An arithmetic function arising from Carmichael’s conjecture

Florian Luca, Paul Pollack (2011)

Journal de Théorie des Nombres de Bordeaux

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Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n , the equation φ ( n ) = φ ( m ) has a solution m ≠ n . This suggests defining F ( n ) as the number of solutions m to the equation φ ( n ) = φ ( m ) . (So Carmichael’s conjecture asserts that F ( n ) ≥ 2 always.) Results on F are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of F contains every natural number k ≥ 2 . Also, the maximal order of F has been investigated by Erdős and Pomerance....

On Robin’s criterion for the Riemann hypothesis

YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Solé (2007)

Journal de Théorie des Nombres de Bordeaux

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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ ( n ) : = ∑ d | n d < e γ n log log n is satisfied for n ≥ 5041 , where γ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power > 1 . As consequence we obtain that RH holds true iff every natural number divisible by...

Palindromic powers.

Hernández, Santos Hernández, Luca, Florian (2006)

Revista Colombiana de Matemáticas

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Perfect powers in the summatory function of the power tower

Florian Luca, Diego Marques (2010)

Journal de Théorie des Nombres de Bordeaux

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Let ( a n ) n ≥ 1 be the sequence given by a 1 = 1 and a n = n a n - 1 for n ≥ 2 . In this paper, we show that the only solution of the equation a 1 + ⋯ + a n = m l is in positive integers l > 1 , m and n is m = n = 1 .