@ARTICLE{HILB10_8.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Prime Representing Polynomial with 10 Unknowns -- Introduction. Part II},
  SECTION1 = {Theta Notation},
  SECTION2 = {More on Solutions to Pell's Equation},
  SECTION3 = {Prime Diophantine Representation}
  DAY = {27},
  MONTH = {December},
  YEAR = {2022},
  ADDRESS1 = {Institute of Computer Science\\University of Bia{\l}ystok\\ 
        Konstantego Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
  SUMMARY = {
In our previous work \cite{HILB10_6.ABS} we prove that
the set of prime numbers is diophantine
using the 26-variable polynomial proposed in \cite{AMM76}.
In this paper, we focus on the reduction of the number
of variables to 10 and it is the smallest variables number known today
\cite{YuriMatiyasevich81,sun2021results}.
We formalize the first step in this direction
by proving Theorem 1 \cite{YuriMatiyasevich81}
formulated as follows:
Let $k \in \mathbb{N}$. Then $k$ is prime if and only if there exists
$f,i,j,m,u\in \mathbb{N}^+$, $r,s,t\in\mathbb{N}$ unknowns such that
\begin{eqnarray}
&DFI\mbox{\texttt{ is square }}\wedge
      (M^2\!-\!1)S^2\!+\!1\mbox{\texttt{ is square }}\wedge &\nonumber\\
      &((MU)^2 -1)T^2\!+\!1\mbox{\texttt{ is square }}\wedge&\nonumber\\
      &(4f^2 -1)(r-mSTU)^2 + 4u^2S^2T^2 < 8fuST(r-mSTU)&\nonumber\\
&FL \mid (H-C)Z + F(f+1)Q + F(k+1)((W^2-1)Su-W^2u^2 +1)&\label{HILB10_8}.
\end{eqnarray}
where auxiliary variables $A-I,L,M,S-W,Q\in \mathbb{Z}$ 
are simply abbreviations defined as follows
$W = 100fk(k\!+\!1) $,
$U = 100u^3W^3\!+\!1$,
$M = 100mUW\!+\!1 $,
$S = (M\!-\!1)s\!+\!k\!+\!1 $,
$T = (MU\!-\!1)t\!+\!W\!-\!k\!+\!1 $,
$Q = 2MW\!-\!W^2\!-\!1 $,
$L = (k\!+\!1)Q $,
$A = M(U\!+\!1)$,
$B = W\!+\!1$,
$C = r\!+\!W\!+\!1$,
$D= (A^2\!-\!1)C^2\!+\!1 $,
$E= 2iC^2LD$,
$F= (A^2\!-\!1)E^2\!+\!1 $,
$G = A\!+\!F(F\!-\!A)$,
$H = B\!+\!2(j\!-\!1)C$,
$I = (G^2\!-\!1)H^2\!+\!1 $.
It is easily see that \eqref{HILB10_8} uses 8 unknowns explicitly along
with five implicit one for each diophantine relationship: \texttt{is square}, inequality, and divisibility. 
Together with $k$ this gives a total of 14 variables.}}

@article{YuriMatiyasevich81,
      title={Primes are nonnegative values of a polynomial in 10 variables},
      author={Yuri Matiyasevich},
      year={1981},
    JOURNAL = {Journal of Soviet Mathematics},
      VOLUME = {15},
    PAGES = {33--44},
    DOI = {10.1007/BF01404106}
}

@article{sun2021results,
      title={Further results on {H}ilbert's {T}enth {P}roblem},
      author={Zhi-Wei Sun},
      year={2021},
    JOURNAL = {Science China Mathematics},
      VOLUME = {64},
    PAGES = {281--306},
    DOI = {10.1007/s11425-020-1813-5}
}
