scholarly journals CONGRUENCES OF THE FIBONACCI NUMBERS MODULO A PRIME

2021 ◽  
pp. 15-21
Author(s):  
V.M. Zyuz'kov ◽  
◽  
Keyword(s):  
Fibonacci Numbers ◽  
Integer Coefficients ◽  
Wolfram Mathematica ◽  
Modulo P ◽  
Pisano Period ◽  
Mod P

Congruences of the form F(expr1) ≡ εF(expr2) (mod p) by prime modulo p are proved, whenever expr1 is a polynomial with respect to p. The value of ε equals 1 or –1 and expr2 does not contain p. An example of such a theorem is as follows: given a polynomial A(p) with integer coefficients ak, ak–1, …, a2, a1, a0 and with respect to p of form 5t ± 2; then, F(A(p)) ≡ F(ak + ak–1 + … + a2 + a1 + a0) (mod p). In particular, we consider the case when the coefficients of the polynomial expr1 form the Pisano period modulo p. To search for existing сongruences, experiments were performed in the Wolfram Mathematica system.

A note on some relations among special sums of reciprocals modulo p

2008 ◽  
Vol 58 (1) ◽  
Author(s):  
Ladislav Skula
Keyword(s):  
Linear Relation ◽  
Fibonacci Numbers ◽  
Linear Relations ◽  
Fermat Quotient ◽  
Modulo P ◽  
Mod P

AbstractIn this note the sums s(k, N) of reciprocals $$\sum\limits_{\tfrac{{kp}}{N} < x < \tfrac{{(k + 1)p}}{N}} {\tfrac{1}{x}(mod p)} $$ are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers.

On Cyclotomic Numbers of Order Sixteen

1954 ◽  
Vol 6 ◽  
pp. 449-454 ◽  
Author(s):  
Emma Lehmer
Keyword(s):  
Linear Combination ◽  
Primitive Root ◽  
Number Of Solutions ◽  
Integer Coefficients ◽  
Cyclotomic Numbers ◽  
Image Position ◽  
Mod P

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of (mod p),then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where,anda ≡ b ≡ 1 (mod 4),while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

Cubic polynomials with the same residues (mod p)

1968 ◽  
Vol 64 (3) ◽  
pp. 655-658 ◽  
Author(s):  
K. McCann ◽  
K. S. Williams
Keyword(s):  
Recent Work ◽  
Linear Transformation ◽  
Integer Coefficients ◽  
Cubic Polynomial ◽  
Cubic Polynomials ◽  
Mod P

Some recent work by the authors (1) on the distribution of the residues of a cubic polynomial modulo an odd prime p led to the conjecture that, in general, two cubic polynomials with integer coefficients possessing the same residues modulo p (not necessarily occurring to the same multiplicity) are equivalent, that is are related by a linear transformation modulo p. The purpose of the present paper is to prove this conjecture. We establish the following theorem.

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.

scholarly journals ON SUMS OF FIBONACCI NUMBERS MODULO p

2010 ◽  
Vol 83 (3) ◽  
pp. 413-419
Author(s):  
VICTOR C. GARCÍA ◽  
FLORIAN LUCA ◽  
V. JANITZIO MEJÍA HUGUET
Keyword(s):  
Residue Class ◽  
Fibonacci Numbers ◽  
Modulo P

AbstractHere, we show that for most primes p, every residue class modulo p can be represented as a sum of 32 Fibonacci numbers.

scholarly journals Primitive zeros of quadratic forms mod $p^2$

2015 ◽  
Vol 46 (3) ◽  
pp. 349-364
Author(s):  
Ali H. Hakami
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Integer Coefficients ◽  
Mod P

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,\ldots,x_n )$ be a quadratic form with integer coefficients, $p$ be an odd prime and $\left\| \bf{x} \right\| = \max _i \left| {x_i } \right|.$ A solution of the congruence $Q({\mathbf{x}}) \equiv {\mathbf{0}}\;(\bmod\; p^2 )$ is said to be a primitive solution if $p\nmid x_i $ for some $i$. In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the type $ \mathcal{B} = \{ {\mathbf{x}} \in \mathbb{Z}^n : |x_i| \le M_i ,\;1 \leqslant i \leqslant n\} $ where for $1 \le i \le l$ we have $M_i \le p$, while for $i>l$ we have $M_i>p$. In particular, we show that if $n \ge 4$, $n$ even, $l \le \frac n2-2$, and $Q$ is nonsingular $\pmod p$, then there exists a primitive solution with $x_i = 0$, $1 \le i \le l$, and $|x_i| \le 2^{\frac {4n+3}{n-l}} p^{\frac n{n-l}} +1$, for $l<i \le n$.

CONGRUENCES CONCERNING LUCAS SEQUENCES

2014 ◽  
Vol 10 (03) ◽  
pp. 793-815 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Lucas Sequences ◽  
Modulo P ◽  
Mod P

Let p be a prime greater than 3. In this paper, by using expansions and congruences for Lucas sequences and the theory of cubic residues and cubic congruences, we establish some congruences for [Formula: see text] and [Formula: see text] modulo p, where [x] is the greatest integer not exceeding x, and m is a rational p-adic integer with m ≢ 0 ( mod p).

scholarly journals COMMON FACTORS OF RESULTANTS MODULO p

2009 ◽  
Vol 79 (2) ◽  
pp. 299-302 ◽  
Author(s):  
DOMINGO GOMEZ ◽  
JAIME GUTIERREZ ◽  
ÁLVAR IBEAS ◽  
DAVID SEVILLA
Keyword(s):  
Common Factors ◽  
Greatest Common Divisor ◽  
Integer Coefficients ◽  
Modulo P ◽  
Open Question

AbstractWe show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

Sign in / Sign up

Export Citation Format

Share Document