We show that for each positive integer a there exist only finitely many prime numbers p such that a appears an odd number of times in the period of continued fraction of p or 2p. We also prove that ...if p is a prime number and D=p or 2p is such that the length of the period of continued fraction expansion of D is divisible by 4, then 1 appears as a partial quotient in the continued fraction of D. Furthermore, we give an upper bound for the period length of continued fraction expansion of D, where D is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas MU.
Economics and Finance undergraduate students from four cohorts played LUPI, a game rewarding the person submitting the lowest unique positive integer, for a small bonus in an exam. Some months later, ...they played this game again with financial incentives and took a cognitive reflection test (CRT). We find that submitted responses to different configurations of LUPI are correlated with short-term (i.e., exam grade) and medium-term (i.e., final grade and GPA) academic performance, as well as the score in the CRT.
Let $ k, l, m_1 $ and $ m_2 $ be positive integers and let both $ p $ and $ q $ be odd primes such that $ p^k = 2^{m_1}-a^{m_2} $ and $ q^l = 2^{m_1}+a^{m_2} $ where $ a $ is a positive integer with ...$ a\equiv {\pm 3}\pmod 8 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that Je$ \acute{s} $manowicz' a conjecture holds for the following set of primitive Pythagorean numbers:
<disp-formula> <tex-math id="FE1"> \begin{document}$ \frac{q^{2l}-p^{2k}}{2}, p^kq^l, \frac{q^{2l}+p^{2k}}{2}. $\end{document} </tex-math></disp-formula>
We also prove that Je$ \acute{s} $manowicz' conjecture holds for non-primitive Pythagorean numbers:
<disp-formula> <tex-math id="FE2"> \begin{document}$ n\frac{q^{2l}-p^{2k}}{2}, np^kq^l, n\frac{q^{2l}+p^{2k}}{2}, $\end{document} </tex-math></disp-formula>
for any positive integer $ n $ if for $ a = a_1a_2 $ with $ a_1\equiv 1 \pmod 8 $ not a square and $ \gcd(a_1, a_2) = 1 $, then there exists a prime divisor $ P $ of $ a_2 $ such that $ \left(\frac{a_1}{P}\right) = -1 $ and $ 2|m_1, a\equiv 5 \pmod 8 $ or $ 2\not|m_2, a\equiv 3\pmod 8 $.
The Traveling Salesman Problem (TSP) is one of the typical NP-hard problems. Efficient algorithms for the TSP have been the focus on academic circles at all times. This article proposes a discrete ...invasive weed optimization (DIWO) to solve TSP. Firstly, weeds individuals encode positive integer, on the basis that the normal distribution of the IWO does not change, and then calculate the fitness value of the weeds individuals. Secondly, the 3-Opt local search operator is used. Finally, an improved complete 2-Opt (I2Opt) is selected as a second local search operator for solve TSP. A benchmarks problem selected from TSPLIB is used to test the algorithm, and the results show that the DIWO algorithm proposed in this article can achieve to results closed to the theoretical optimal values within a reasonable period of time, and has strong robustness.
Let $ k, l, m_1, m_2 $ be positive integers and let both $ p $ and $ q $ be odd primes such that $ p^k = 2^{m_1}-a^{m_2} $ and $ q^l = 2^{m_1}+a^{m_2} $ where $ a $ is odd prime with $ a\equiv 5\pmod ...8 $ and $ a\not\equiv 1\pmod 5 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that the exponential Diophantine equation $ \left(\frac{q^{2l}-p^{2k}}{2}n\right)^x+(p^kq^ln)^y = \left(\frac{q^{2l}+p^{2k}}{2}n\right)^z $ has only the positive integer solution $ (x, y, z) = (2, 2, 2) $.
Let $ n $ be a positive integer with $ n > 1 $ and let $ a, b $ be fixed coprime positive integers with $ \min\{a, b\} > 2 $. In this paper, using the Baker method, we proved that, for any $ n $, if ...$ a > \max\{15064b, b^{3/2}\} $, then the equation $ (an)^x+(bn)^y = ((a+b)n)^z $ has no positive integer solutions $ (x, y, z) $ with $ x > z > y $. Further, let $ A, B $ be coprime positive integers with $ \min\{A, B\} > 1 $ and $ 2|B $. Combining the above conclusion with some existing results, we deduced that, for any $ n $, if $ (a, b) = (A^2, B^2), A > \max\{123B, B^{3/2}\} $ and $ B\equiv 2\pmod 4 $, then this equation has only the positive integer solution $ (x, y, z) = (1, 1, 1). $ Thus, we proved that the conjecture proposed by Yuan and Han is true for this case.
Let
G
be a simple graph with order
n
and adjacency matrix
A
(
G
)
. The characteristic polynomial of
G
is defined by
ϕ
(
G
;
λ
)
=
det
(
λ
I
-
A
(
G
)
)
=
∑
i
=
0
n
a
i
(
G
)
λ
n
-
i
, where
a
i
(
G
...)
is called the
i
-th adjacency coefficient of
G
. Denote by
B
n
,
m
the collection of all connected bipartite graphs having
n
vertices and
m
edges. A bipartite graph
G
is referred as 4-Sachs optimal if
a
4
(
G
)
=
min
{
a
4
(
H
)
∣
H
∈
B
n
,
m
}
.
For any given integer pair (
n
,
m
), in this paper we investigate the 4-Sachs optimal bipartite graphs. Firstly, we show that each 4-Sachs optimal bipartite graph is a difference graph. Then we deduce some structural properties on 4-Sachs optimal bipartite graphs. Especially, we determine the unique 4-Sachs optimal bipartite (
n
,
m
)-graphs for
n
≥
5
and
n
-
1
≤
m
≤
2
(
n
-
2
)
. Finally, we provide a method to construct a class of cospectral difference graphs, which disprove a conjecture posed by Andelić et al. (J Czech Math 70:1125–1138, 2020).
Uniform constant composition (UCC) codes are introduced, which are p-ary constant composition codes with block length mp where each p-ary symbol appears exactly m times in each codeword, being m a ...positive integer and p a prime. UCC codes are derived from a new class of p-ary repeated-root cyclic codes and two constructions are presented. Both constructions are at least asymptotically optimal with increasing p when compared with known best bounds for constant composition codes.
In 1911, Dubois determined all positive integers that are represented by sums of k nonvanishing squares for any k≥4. In this article, we extend the Dubouis' results to real quadratic fields Q(m) and ...we will show that, for each positive integer k≥5, there exists a bound C(m,k) such that every totally positive integer in the real quadratic field Q(m) whose norm exceeds C(m,k) can be expressed as a sum of k nonvanishing integral squares in Q(m).
We study a simple model of similarity-based global cumulative imitation in symmetric games with large and ordered strategy sets and a salient winning player. We show that the learning model explains ...behavior well in both field and laboratory data from one such “winner-takes-all” game: the lowest unique positive integer game in which the player that chose the lowest number not chosen by anyone else wins a fixed prize. We corroborate this finding in three other winner-takes-all games and discuss under what conditions the model may be applicable beyond this class of games. Theoretically, we show that global cumulative imitation without similarity weighting results in a version of the replicator dynamic in winner-takes-all games.
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