Let p be an odd prime and $w \lt p$w<p be a positive integer. The authors continue to investigate the binary sequence $\lpar f_u\rpar $(fu) over $\lcub 0\comma \; \, 1\rcub ${0,1} defined from ...polynomial quotients by $\lpar u^w - u^{wp}\rpar /p$(uw−uwp)/p modulo p. The $\lpar f_{u}\rpar$(fu) is generated in terms of $\lpar {-1}\rpar^{f_u}$(−1)fu which equals to the Legendre symbol of $\lpar u^{w} - u^{wp}\rpar /p \;\lpar {\rm mod}\;p\rpar$(uw−uwp)/p(modp) for u ≥ 0. In an earlier work, the linear complexity of $\lpar f_u\rpar $(fu) was determined for $w = p - 1$w=p−1 (i.e. the case of Fermat quotients) under the assumption of $2^{p - 1}\not{ \equiv }1\lpar \bmod \, p^2\rpar $2p−1⧸≡1(modp2). In this work, they develop a naive trick to calculate all possible values on the linear complexity of $\lpar f_u\rpar $(fu) for all $1 \le w \lt p - 1$1≤w<p−1 under the same assumption. They also state that the case of larger $w\lpar \ge p\rpar $w(≥p) can be reduced to that of $0 \le w \le p - 1$0≤w≤p−1. So far, the linear complexity is almost determined for all kinds of Legendre-polynomial quotients.
We show that the Diophantine system
f
(
z
)
=
f
(
x
)
f
(
y
)
=
f
(
u
)
f
(
v
)
has infinitely many nontrivial positive integer solutions for
f
(
X
)
=
X
2
-
1
, and infinitely many nontrivial ...rational solutions for
f
(
X
)
=
X
2
+
b
with nonzero integer
b
.
Presented is a square root algorithm in 𝔽q which generalises Atkins's square root algorithm see reference 6 for q ≡ 5 (mod 8) and Müller's algorithm see reference 7 for q ≡ 9 (mod 16). The presented ...algorithm precomputes a primitive 2s-th root of unity ξ where s is the largest positive integer satisfying 2s|q−1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favourably compared with the algorithms of Atkin, Müuller and Kong et al.
Perfect periodic cross-correlation is achievable in non-coherent pulse compression using Legendre sequences of any prime length. This paper briefly describes the (previously known) transmit and ...reference signals for the N = 4k − 1 case where N, the sequence length, is prime and k is a positive integer. It is then shown that a slight modification allows perfect periodic correlation with N = 4k − 3, doubling the number of available sequences.
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\mathbb F_p(t)$, when $p$ is prime and ...$r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb F_q(t^1/d)$.
The data in this article are as a result of a quest to uncover alternative research routes of deepening researchers’ understanding of integers apart from the traditional number theory approach. ...Hence, the article contains the statistical properties of the digits sum of the first 3000 squared positive integers. The data describes the various statistical tools applied to reveal different statistical and random nature of the digits sum of the first 3000 squared positive integers. Digits sum here implies the sum of all the digits that make up the individual integer.
In this paper we introduce several natural sequences related to polynomials of degree
having coefficients in {1,2, …,
} (
∈
) which factor completely over the integers. These sequences can be seen as ...generalizations of A006218. We provide precise methods for calculating the terms and investigate the asymptotic behavior of these sequences for
∈ {1, 2, 3}.
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