Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Abstract

Summary In our previous work [7] we prove that the set of prime numbers is diophantine using the 26-variable polynomial proposed in [4]. In this paper, we focus on the reduction of the number of variables to 10 and it is the smallest variables number known today [5], [10]. Using the Mizar [3], [2] system, we formalize the first step in this direction by proving Theorem 1 [5] formulated as follows: Let k ∈ ℕ. Then k is prime if and only if there exists f, i, j, m, u ∈ ℕ+, r, s, t ∈ ℕ unknowns such that D F I   is   square       ∧   ( M 2 - 1 ) S 2 + 1     is     square     ∧ ( ( M U ) 2 - 1 ) T 2 + 1     is     square ∧ ( 4 f 2 - 1 ) ( r - m S T U ) 2 + 4 u 2 S 2 T 2 < 8 f u S T ( r - m S T U ) F L | ( H - C ) Z + F ( f + 1 ) Q + F ( k + 1 ) ( ( W 2 - 1 ) S u - W 2 u 2 + 1 ) \matrix{ {DFI\,is\,square\,\,\,{\Lambda}\,\left( {{M^2} - 1} \right){S^2} + 1\,\,is\,\,square\,\,{\Lambda}} \hfill \cr {\left( {{{\left( {MU} \right)}^2} - 1} \right){T^2} + 1\,\,is\,\,square{\Lambda}} \hfill \cr {\left( {4{f^2} - 1} \right){{\left( {r - mSTU} \right)}^2} + 4{u^2}{S^2}{T^2} < 8fuST\left( {r - mSTU} \right)} \hfill \cr {FL|\left( {H - C} \right)Z + F\left( {f + 1} \right)Q + F\left( {k + 1} \right)\left( {\left( {{W^2} - 1} \right)Su - {W^2}{u^2} + 1} \right)} \hfill \cr } where auxiliary variables A − I, L, M, S − W, Q ∈ ℤ are simply abbreviations defined as follows W = 100fk(k + 1), U = 100u 3 W 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 2 LD, 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 (0.1) uses 8 unknowns explicitly along with five implicit one for each diophantine relationship: is square, inequality, and divisibility. Together with k this gives a total of 14 variables. This work has been partially presented in [8].