Lehmer's semi-symmetric cyclotomic sums

Andrew J.S. Lazarus

Displaying similar documents to “Lehmer's semi-symmetric cyclotomic sums”

The diophantine equation x² + 19 = yⁿ

J. H. E. Cohn (1992)

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On the theory of cubic residues and nonresidues

Zhi-Hong Sun (1998)

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Primes, permutations and primitive roots.

Lewittes, Joseph, Kolyvagin, Victor (2010)

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Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun (1992)

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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...

On integers not of the form n - φ (n)

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 $2^{k} \cdot 509203$ (k = 1, 2,...) is of the form n - φ(n).

The Diophantine equation $x^{4} - y^{4} = z^{2}$ in three quadratic fields.

Szabó, Sándor (2004)

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Square Lehmer numbers

Wayne McDaniel (1993)

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Modular forms and class number congruences

Antone Costa (1992)

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A note on a conjecture of Jeśmanowicz

Moujie Deng, G. Cohen (2000)

Colloquium Mathematicae

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Let a, b, c be relatively prime positive integers such that $a^{2} + b^{2} = c^{2}$ . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ${(a n)}^{x} + {(b n)}^{y} = {(c n)}^{z}$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${(u^{2} - v^{2})}^{x} + {(2 u v)}^{y} = {(u^{2} + v^{2})}^{z}$ , 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.