Prime Representing Polynomial

Abstract

Summary The main purpose of formalization is to prove that the set of prime numbers is diophantine, i.e., is representable by a polynomial formula. We formalize this problem, using the Mizar system [1], [2], in two independent ways, proving the existence of a polynomial without formulating it explicitly as well as with its indication. First, we reuse nearly all the techniques invented to prove the MRDP-theorem [11]. Applying a trick with Mizar schemes that go beyond first-order logic we give a short sophisticated proof for the existence of such a polynomial but without formulating it explicitly. Then we formulate the polynomial proposed in [6] that has 26 variables in the Mizar language as follows (w·z+h+j−q)2+((g·k+g+k)·(h+j)+h−z)2+(2 · k3·(2·k+2)·(n + 1)2+1−f2)2+ (p + q + z + 2 · n − e)2 + (e3 · (e + 2) · (a + 1)2 + 1 − o2)2 + (x2 − (a2 −′ 1) · y2 − 1)2 + (16 · (a2 − 1) · r2 · y2 · y2 + 1 − u2)2 + (((a + u2 · (u2 − a))2 − 1) · (n + 4 · d · y)2 + 1 − (x + c · u)2)2 + (m2 − (a2 −′ 1) · l2 − 1)2 + (k + i · (a − 1) − l)2 + (n + l + v − y)2 + (p + l · (a − n − 1) + b · (2 · a · (n + 1) − (n + 1)2 − 1) − m)2 + (q + y · (a − p − 1) + s · (2 · a · (p + 1) − (p + 1)2 − 1) − x)2 + (z + p · l · (a − p) + t · (2 · a · p − p2 − 1) − p · m)2 and we prove that that for any positive integer k so that k + 1 is prime it is necessary and sufficient that there exist other natural variables a-z for which the polynomial equals zero. 26 variables is not the best known result in relation to the set of prime numbers, since any diophantine equation over ℕ can be reduced to one in 13 unknowns [8] or even less [5], [13]. The best currently known result for all prime numbers, where the polynomial is explicitly constructed is 10 [7] or even 7 in the case of Fermat as well as Mersenne prime number [4]. We are currently focusing our formalization efforts in this direction.