Matijasevi˘c’sTheorem: Diophantine descriptions of recursively enumerable sets Bachelor’s thesis S.R. Groen ∗ First supervisor: prof. dr. J. Top Second supervisor: dr. A.E. Sterk 2017 Abstract In 1970, Yuri Matijasevi˘cfinished the proof that all recursively enumerable sets are Dio- phantine, rendering Hilbert’s tenth problem unsolvable. He did so by showing that exponential Diophantine sets are Diophantine, which complemented earlier work done by Martin Davis, Hilary Putnam and Julia Robinson. In this thesis, we analyze, explore and apply this result. We reconstruct ap known way to create a Diophantine description of exponentiation: using the unit group of Z[ d]. This provides a mechanism with which we can create a Diophantine de- scription of any recursively enumerable set. We apply this to find Diophantine descriptions of some specific sets of integers. We also study the complexity of such Diophantine descriptions. Furthermore, we try to create a new method of creating a Diophantine description of expo- p3 nentiation,p using the unit group of Z[ d], whose structure is similar to that of the unit group of Z[ d]. It turns out that such a similar method does not work, as the desired divisibility sequences don’t exist. Keywords: Hilbert’s tenth problem, Diophantine sets, Matijasevi˘c’stheorem, number rings, al- gebraic number theory. ∗Faculty of mathematics and natural sciences, Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands, e-mail: [email protected] 1 Contents 1 Introduction 3 1.1 Hilbert’s tenth problem.................................3 1.2 Diophantine sets.....................................3 1.3 Recursively enumerable sets...............................4 1.4 Matijasevi˘c’sTheorem..................................4 1.5 The DPR-theorem....................................5 1.6 From exponential Diophantine to Diophantine.....................5 1.7 The aim of this thesis..................................6 p 2 A Diophantinep description of exponentiation using Z[ d] 7 2.1 Z[ d] and its unit group.................................7 2.2 The Pell equation.....................................8 2.3 Cyclicity..........................................9 2.4 A suitable choice for d .................................. 10 2.5 Behavior of xn(a) and yn(a) ............................... 11 2.6 Finding the solution number using divisibility properties............... 14 2.7 A Diophantine description of xn(a) and yn(a) ..................... 20 2.8 A Diophantine description of exponentiation...................... 21 3 Expanding the language of Diophantine descriptions 23 3.1 Diophantine descriptions of important functions.................... 23 3.2 The Bounded Universal Quantifier Theorem...................... 24 3.3 The Sequence Number Theorem............................ 24 3.4 Putnam’s trick...................................... 25 4 Application 26 4.1 The set of primes..................................... 26 4.1.1 The straightforward definition.......................... 26 4.1.2 Wilson’s Theorem................................ 27 4.2 The divisor number function.............................. 28 4.3 The divisor sum function................................ 28 4.4 Euler’s φ function..................................... 29 4.5 Gödel’s incompleteness theorems............................ 29 5 The complexity of Diophantine descriptions 31 5.1 Degree........................................... 31 5.2 Dimension......................................... 32 p 3 6 A Diophantinep description of exponentiation using Z[ d] 33 3 6.1 Z[ d] and its unit group................................. 33 6.2 Application of Dirichlet’s theorem........................... 34 6.3 A suitable choice for d .................................. 35 6.4 The three-dimensional Pell equation.......................... 36 6.5 Behavior of xn(a), yn(a) and zn(a) ........................... 38 6.6 Finding the solution number using divisibility properties............... 41 6.7 Are xn(a), yn(a) and zn(a) Diophantine?....................... 43 6.8 Comparison to two-dimensional case.......................... 44 7 Conclusion and outlook 45 2 1 Introduction 1.1 Hilbert’s tenth problem In 1900, David Hilbert posed 23 then unsolved problems in mathematics that he encouraged mathematicians to solve in the twentieth century. Some of these problems, such as the Riemann hypothesis, are still unsolved. Hilbert’s tenth problem plays an important role in this thesis. Through work by Martin Davis, Julia Robinson, Hilary Putnam and Yuri Matijasevi˘c,this problem has been shown to be unsolvable. The problem was posed in 1900 as follows. Hilbert’s tenth problem: Devise an algorithm that, given any polynomial equation with in- teger coefficients as input, gives as output whether this polynomial has any roots over the integers. [Dav73] In 1970, Matijasevi˘ccompleted the proof that such an algorithm does not exist and that Hilbert’s tenth problem is thus impossible to solve. [Mat70] In this thesis, we will explore and apply the method’s used in the proof of Matijasevi˘c’s Theorem A key notion in the theory applicable to this problem is the notion of Diophantine sets. This will be a vital concept throughout this thesis. 1.2 Diophantine sets n Definition 1.1. A set S ⊂ Z is Diophantine if there exists an m ≥ n and a p 2 Z[X1;X2; ··· ;Xm] for which the following holds: n m−n S = fs 2 Z j 9t 2 Z s.t. p(s; t) = 0g A Diophantine set S ⊂ Zn is thus the projection of the set of zeros in Zm of the polynomial p 2 Z[X1;X2; ··· ;Xm] onto the first n coordinates. An equation of the form p(X1;X2; ··· Xm) = 0 is also called a Diophantine equation, and a Diophantine description of S. We will call p the polynomial corresponding to S. In the following examples, the set of numbers X1 for which the polynomial p has integer roots is Diophantine: • The even numbers, with the corresponding polynomial p = X1 − 2X2. 2 • The squares (of integers), with the corresponding polynomial p = X1 − X2 . 2 2 2 2 • The non-negative integers, with the corresponding polynomial p = X1 − X2 − X3 − X4 − X5 . Here we use Lagrange’s result that every non-negative integer is the sum of four squares, and obviously negative integers can’t have that property). 2 2 2 • The Pythagorean hypotenuse integers, with the corresponding polynomial p = X1 −X2 −X3 Another example of a Diophantine set is the set of composite numbers. On the first hand, p = X1 − X2X3 = 0 might seem like a suitable Diophantine description of this set, but it allows for one (or two) of the factors of X1 (which are X2 and X3) to be equal to 1. We therefore also need that both are larger than 1, in order to find the positive composite numbers. That will result in either of the following equivalent Diophantine descriptions of the set of composite numbers: 2 2 2 2 2 2 2 2 2 2 2 p = (X1 − X2X3) + X2 − X4 − X5 − X6 − X7 − 2 + X3 − X8 − X9 − X10 − X11 − 2 = 0 2 2 2 2 2 2 2 2 p = X1 − X2 + X4 + X5 + X6 + 2 X3 + X7 + X8 + X9 + 2 = 0 If we would also want to find the negative composite numbers, this would be equivalent to also allowing X3 to be smaller than −1 instead of greater than 1. We would then have the following polynomial: 2 2 2 2 2 2 p = (X1 − X2X3) + X2 − X4 − X5 − X6 − X7 − 2 + 2 2 2 2 2 2 2 2 2 X3 − X8 − X9 − X10 − X11 − 2 X3 + X12 + X13 + X14 + X15 + 2 = 0 We will later see that the complement of this last set, which is set of primes, is also Diophantine. However, this is far from trivial and may feel counterintuitive at this moment. 3 As the example of the set of composite numbers shows, using the four squares theorem so many times is quite a hassle. We therefore assume from now on that every variable can only be nonnegative. This is no loss of generality, as we can always introduce a minus sign to let a number be negative. We conclude that, for any set S, the following three are equivalent: 1. There exists a Diophantine description of S in the integers 2. There exists a Diophantine description of S in the non-negative integers 3. There exists a Diophantine description of S in the positive integers This is because we can always introduce a minus sign or use Lagrange’s four square theorem. It is straightforward to prove that the set of Diophantine sets is closed under union: if we have S1 and S2, with corresponding polynomials p1 and p2, The set S1[S2 has corresponding polynomial p1 ·p2. This polynomial is zero if and only if at least one of the polynomials p1 and p2 is zero, which means we are dealing with an element of S1 or S2. Similarly, the set S1 \ S2 has corresponding 2 2 polynomial p1 + p2. We have already applied this technique to our Diophantine description of composite numbers. Not all Diophantine sets have the property that their complement is also Diophantine, but this is also not trivially seen. We can now see what the algorithm Hilbert asked for should do precisely: it should be able to decide within a finite amount of time, given any polynomial as input, whether the corresponding Diophantine set is empty or non-empty. 1.3 Recursively enumerable sets Another important notion will be the notion of recursively enumerable sets. Definition 1.2. A set S is recursively enumerable if the Turing machine has an algorithm that enumerates S. Equivalently: an algorithm exists that halts precisely when its input is an element of S. Examples of such sets are the following: • Any finite set: S = fs1; s2; ··· ; sng = fs j s = s1 _ s = s2 ···_ s = sng • The positive numbers: S = fs j s > 0g • The even numbers: S = fs j 9y s = 2yg • The set of powers of 2: S = fs j 9y s = 2yg • The set of prime numbers: S = fs j :(9y)1<y<s (9z)1<z<s s.t. s = yzg It is again straightforward to see that unions and intersections of recursively enumerable sets are also recursively enumerable: the set of recursively enumerable sets is closed under union and intersection.