On systems of composite Lehmer numbers with prime indices
J. Wójcik (1996)
Colloquium Mathematicae
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J. Wójcik (1996)
Colloquium Mathematicae
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Maohua Le (1993)
Colloquium Mathematicae
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Paulo Ribenboim, Wayne McDaniel (1998)
Colloquium Mathematicae
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William D. Banks, Florian Luca, Igor E. Shparlinski (2007)
Revista Matemática Complutense
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We study the behavior of the arithmetic functions defined by F(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1) where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.
Maohua Le (1991)
Colloquium Mathematicae
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Christoph Schwarzweller (2014)
Formalized Mathematics
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In this article we introduce Proth numbers and prove two theorems on such numbers being prime [3]. We also give revised versions of Pocklington’s theorem and of the Legendre symbol. Finally, we prove Pepin’s theorem and that the fifth Fermat number is not prime.
Moujie Deng, G. Cohen (2000)
Colloquium Mathematicae
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Let a, b, c be relatively prime positive integers such that . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of in positive integers is x=y=z=2. If n=1, then, equivalently, the equation , for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.
J. Browkin, A. Schinzel (1995)
Colloquium Mathematicae
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W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers (k = 1, 2,...) is of the form n - φ(n).
Maohua Le (1996)
Colloquium Mathematicae
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