On integers not of the form n - φ (n)

J. Browkin; A. Schinzel

Displaying similar documents to “On integers not of the form n - φ (n)”

The diophantine equation x² + 19 = yⁿ

J. H. E. Cohn (1992)

Acta Arithmetica

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Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun (1992)

Acta Arithmetica

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Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n + 1} = F ₙ + F_{n - 1} (n \geq 1)$ . It is well known that $F_{p - (5 / p)} \equiv 0 (m o d p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p ² | F_{p - (5 / p)}$ is always impossible; up to now this is still open. In this paper the sum $\sum_{k \equiv r (m o d 10)} (\binom{n}{k})$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p - (5 / p)} / p$ and a criterion for the relation $p | F_{(p - 1) / 4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative...

Square Lehmer numbers

Wayne McDaniel (1993)

Colloquium Mathematicae

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A note on primes p with $σ (p^{m}) = z^{n}$

Maohua Le (1991)

Colloquium Mathematicae

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Modular forms and class number congruences

Antone Costa (1992)

Acta Arithmetica

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Primes, permutations and primitive roots.

Lewittes, Joseph, Kolyvagin, Victor (2010)

The New York Journal of Mathematics [electronic only]

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Simple proofs of some generalizations of the Wilson’s theorem

Jan Górowski, Adam Łomnicki (2014)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.

Weyl almost periodic selections of supports of measure-valued functions.

Danilov, L.I. (2006)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

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On the parity of generalized partition functions, III

Fethi Ben Saïd, Jean-Louis Nicolas, Ahlem Zekraoui (2010)

Journal de Théorie des Nombres de Bordeaux

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Improving on some results of J.-L. Nicolas [], the elements of the set $𝒜 = 𝒜 (1 + z + z^{3} + z^{4} + z^{5})$ , for which the partition function $p (𝒜, n)$ (i.e. the number of partitions of $n$ with parts in $𝒜$ ) is even for all $n \geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

Lagrange’s Four-Square Theorem

Yasushige Watase (2014)

Formalized Mathematics

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This article provides a formalized proof of the so-called “the four-square theorem”, namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in [14], [1] or [23]. This theorem is item #19 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.