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Displaying 61 – 80 of 150

On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia (2014)

Colloquium Mathematicae

For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j})$ . Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.

On the number of representations of integers by quadratic forms in twelve variables.

Lomadze, G. (1998)

Georgian Mathematical Journal

On the number of representations of integers by some quadratic forms in ten variables.

Lomadze, G. (1995)

Georgian Mathematical Journal

On the number of representations of n by ax² + bxy + cy²

Zhi-Hong Sun, Kenneth S. Williams (2006)

Acta Arithmetica

On the number of representations of n by ax²+by(y-1)/2, ax²+by(3y-1)/2 and ax(x-1)/2+by(3y-1)/2

Zhi-Hong Sun (2011)

Acta Arithmetica

On the number of representations of positive integers by a direct sum of binary quadratic forms with discriminant $- 23$ .

Lomadze, G. (1997)

Georgian Mathematical Journal

On the number of representations of positive integers by quadratic forms as the basis of the space $S_{4} (Γ_{0} (47), 1)$ .

Tekcan, Ahmet, Bizim, Osman (2004)

International Journal of Mathematics and Mathematical Sciences

On the number of representations of positive integers by the quadratic form $x_{1}^{2} + \dots + x_{8}^{2} + 4 x_{9}^{2}$ .

Lomadze, G. (2001)

Georgian Mathematical Journal

On the Pythagoras number of a real irreducible algebroid curve.

Jesus J. Ortega (1991)

Mathematische Annalen

On the quartic character of quadratic units

Zhi-Hong Sun (2013)

Acta Arithmetica

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d = 2^{r} d ₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ${(b + \sqrt (b ² + 4^{α}) / 2)}^{(p - 1) / 4)} (m o d p)$ for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ${(2 a + \sqrt 4 a ² + 1)}^{(p - 1) / 4} (m o d p)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p - 1) / 4} (m o d p)$ and the criterion for $p | U_{(p - 1) / 8}$ (if p ≡ 1 (mod 8)), where Uₙ is the Lucas sequence given by U₀ = 0, U₁ = 1 and $U_{n + 1} = b U ₙ + U_{n - 1} (n \geq 1)$ , and b ≢...