Displaying similar documents to “A generalization of Scholz’s reciprocity law”

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 · 509203 (k = 1, 2,...) is of the form n - φ(n).

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 ≥ 1 ) . It is well known that F p - ( 5 / p ) ≡ 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 ∑ k ≡ r ( m o d 10 ) 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...

Kloosterman sums for prime powers in -adic fields

Stanley J. Gurak (2009)

Journal de Théorie des Nombres de Bordeaux

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Let K be a field of degree n over Q p , the field of rational p -adic numbers, say with residue degree f , ramification index e and differential exponent d . Let O be the ring of integers of K and P its unique prime ideal. The trace and norm maps for K / Q p are denoted T r and N , respectively. Fix q = p r , a power of a prime p , and let η be a numerical character defined modulo q and of order o ( η ) . The character η extends to the ring of p -adic integers ℤ p in the natural way; namely η ( u ) = η ( u ˜ ) , where u ˜ denotes the residue...

On the trace of the ring of integers of an abelian number field

Kurt Girstmair (1992)

Acta Arithmetica

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Let K, L be algebraic number fields with K ⊆ L, and O K , O L their respective rings of integers. We consider the trace map T = T L / K : L → K and the O K -ideal T ( O L ) ⊆ O K . By I(L/K) we denote the group indexof T ( O L ) in O K (i.e., the norm of T ( O L ) over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of T ( O L ) (Theorem 1)....

On the Carlitz problem on the number of solutions to some special equations over finite fields

Ioulia N. Baoulina (2011)

Journal de Théorie des Nombres de Bordeaux

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We consider an equation of the type a 1 x 1 2 + ⋯ + a n x n 2 = b x 1 ⋯ x n over the finite field 𝔽 q = 𝔽 p s . Carlitz obtained formulas for the number of solutions to this equation when n = 3 and when n = 4 and q ≡ 3 ( mod 4 ) . In our earlier papers, we found formulas for the number of solutions when d = gcd ( n - 2 , ( q - 1 ) / 2 ) = 1 or 2 or 4 ; and when d > 1 and - 1 is a power of p modulo  2 d . In this paper, we obtain formulas for the number of solutions when d = 2 t , t ≥ 3 , p ≡ 3 or 5 ( mod 8 ) or p ≡ 9 ( mod 16 ) . For general case, we derive lower bounds for the number of solutions.

Gross’ conjecture for extensions ramified over four points of ℙ 1

Po-Yi Huang (2006)

Journal de Théorie des Nombres de Bordeaux

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In this paper, under a mild hypothesis, we prove a conjecture of Gross for the Stickelberger element of the maximal abelian extension over the rational function field unramified outside a set of four degree-one places.