scholarly journals CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

On the number of representations of a positive integer as a sum of two binary quadratic forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1395-1420 ◽  
Author(s):  
Şaban Alaca ◽  
Lerna Pehlivan ◽  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Definite Integral ◽  
Binary Quadratic Forms ◽  
Finite Linear Combination ◽  
Positive Integers ◽  
Finite Linear

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

THE CUBIC CONGRUENCE x3 + Ax2 + Bx + C ≡ 0 (mod p) AND BINARY QUADRATIC FORMS II

2001 ◽  
Vol 64 (2) ◽  
pp. 273-274 ◽  
Author(s):  
BLAIR K. SPEARMAN ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
Mod P

It is shown that the splitting modulo a prime p of a given monic, integral, irreducible cubic with non-square discriminant is equivalent to p being represented by forms in a certain subgroup of index 3 in the form class group of discriminant equal to the discriminant of the field defined by the cubic.

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

1998 ◽  
Vol 5 (1) ◽  
pp. 55-70
Author(s):  
N. Kachakhidze
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Direct Sums

Abstract The systems of bases are constructed for the spaces of cusp forms Sk (Γ0(3), χ) (k≥6), Sk (Γ0(7), χ) (k≥3) and Sk (Γ0(11), χ) (k≥3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .

scholarly journals CONCENTRATION OF POINTS ON MODULAR QUADRATIC FORMS

2011 ◽  
Vol 07 (07) ◽  
pp. 1835-1839 ◽  
Author(s):  
ANA ZUMALACÁRREGUI
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Irreducible Polynomial ◽  
Number Of Solutions ◽  
Modulo P ◽  
Absolutely Irreducible Polynomial ◽  
Mod P

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

RATIONAL REPRESENTATIONS OF PRIMES BY BINARY QUADRATIC FORMS

2007 ◽  
Vol 03 (04) ◽  
pp. 513-528
Author(s):  
RONALD EVANS ◽  
MARK VAN VEEN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Binary Quadratic Forms ◽  
Rational Solutions ◽  
Algebraic Integers ◽  
Squarefree Integer

Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2 + qv2 has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that [Formula: see text], where [Formula: see text] is the set of norms of algebraic integers in [Formula: see text]. Let k(q) denote the smallest positive integer k such that [Formula: see text] for all q-admissible primes p. It is shown that k(q) has subexponential but suprapolynomial growth in q, as q → ∞.

REPRESENTATION OF PRIMES BY THE PRINCIPAL FORM OF NEGATIVE DISCRIMINANT $\Delta$ WHEN $h(\Delta)$ IS 4

1994 ◽  
Vol 25 (4) ◽  
pp. 321-334
Author(s):  
KENNETH S. WILLIAMS ◽  
D. LIU
Keyword(s):  
Cyclic Group ◽  
Quadratic Forms ◽  
Class Group ◽  
Positive Definite ◽  
Negative Integer ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
Quartic Polynomial ◽  
Mod P

Let $\Delta$ be a negative integer which is congruent to 0 or 1 (mod 4). Let $H(\Delta)$ denote the form class group of classes of positive-definite, primitive integral binary quadratic forms $ax^2 +bxy +cy^2$ of discriminant $\Delta$. If $H(\Delta)$ is a cyclic group of order 4, an explicit quartic polynomial $\rho \Delta(x)$ of the form $x^4-bx^2 +d$ with integral coefficients is determined such that for an odd prime $p$ not dividing $\Delta$, $p$ is represented by the principal form of discriminant $\Delta$ if and only if the congruence $\rho \Delta(x) \equiv 0$ (mod $p$) has four solutions.

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

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