The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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.