arXiv subjects
Publications and source records attributed to Yingpu Deng.
In this article, we focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group. As an application, we give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over a class of imaginary quadratic fields where $p$ is a prime element.
We describe an explicit generalized Lucasian test to determine the primality of numbers $h\cdot2^n\pm1$ when $h\nequiv0\pmod{17}$. This test is by means of fixed seeds which depend only on $h$. In particular when $h=16^m-1$ with $m$ odd, our paper gives a primality test with some fixed seeds depending only on $h$. Comparing the results of W. Bosma(1993) and P. Berrizbeitia and T. G. Berry(2004), our result adds new values of $h$ along with this line. Octic and bioctic reciprocity are used to deduce our result.
We describe a primality test for number $M=(2p)^{2^n}+1$ with odd prime $p$ and positive integer $n$. And we also give the special primality criteria for all odd primes $p$ not exceeding 19. All these primality tests run in polynomial time in log$_{2}(M)$. A certain special $2p$-th reciprocity law is used to deduce our result.
We propose an algorithm determining the primality of numbers $M=Ap^n+w_n$ where $w_n^{p-1}\equiv1\pmod{p^n}$ and $A<p^n$ and give example when $p=7$. $p$ th reciprocity law is involved. The algorithm runs in polynomial time in $\log_2(M)$ for fixed $p$ and $A$.
It is well known that in an exact covering system in $\mathbb{Z}$, the biggest modulus must be repeated. Very recently, Kim gave an analogous result for certain quadratic fields, and Kim also conjectured that it must hold in any algebraic number field. In this paper, we prove Kim's conjecture. In other words, we prove that exact covering systems in any algebraic number field must have repeated moduli.
Define $ψ_m$ to be the smallest strong pseudoprime to the first $m$ prime bases. The exact value of $ψ_m$ is known for $1\le m \le 8$. Z. Zhang have found a 19-decimal-digit number $Q_{11}=3825\,12305\,65464\,13051$ which is a strong pseudoprime to the first 11 prime bases and he conjectured that $$ψ_9=ψ_{10}=ψ_{11}=Q_{11}.$$ We prove the conjecture by algorithms.