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1 Conference in Bedlewo (Poland) 21th of july – 27th of july 2012 In memoriam of professors Henryk Kotlarski and Zygmunt Ratajczyk Nézondet ‘s p-destinies for theories with two and three quantifiers Denis RICHARD, professeur émérite à l’Université d’Auvergne The five missing numbers

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2 The notion of p-destiny is due to Francis Nézondet In his thesis (1997). Annie Château (2000) gave the best written survey of this thesis in 2000 and also wrote the first programm in CAML deciding Th 2 (N,S, ⊥ ). A group including YIN Jilei (Fudan DaXue), GUILLAUME Marcel(Université Blaise Pascal) and DR (Université d’Auvergne) attacked in 2005 the problem of constructing the essential exhaustive tranverse destiny of Th 3 (N,S, ⊥ ) which would provide an algorithm of decision and an axiomatisation of this theory. Therefore, there are 5 remaining cases. In each case it has to be construct a number or to prove it does not exist.

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3 A p-destiny is a tree of height p of possibilities in a theory with p quantifiers and a language L of finitely many relations. Suppose p=2 and N is the set of nodes of the tree and take the predicates < and P for primeness. We intend to obtain a description of all possible (up to isomorphism) situations which can exist with two variables. 0. What are p-destinies ? 0.1 The (trivial) 2-destiny Th 2 (N,S,P).

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4 Working with the tree of root 0 and denoting by a circle the prime numbers, we get the exhaustive 2-destiny of 0 inTh 2 (N,<,P) Which, up to isomorphism, produce the following essential 2-destiny of 0 in Th (N,<,P)

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5 Of course, the same is possible for any integer, giving the exhaustive simplified 2-destiny of Th 2 (N,<,P) :

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6 Just keeping one representative for each class of isomorphic trees, we can construct the so-call exhaustive essential 2-destiny ( also called essential tranverse 2- destiny) of Th 2 (N,<,P)

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7 With the essential tranverse 2-destiny, we have An algorithm of decision for Th 2 (N,<,P) : for Q 1 x 1 Q 2 x 2 F(x 1, x 2 ) The variable x 1 takes its value(s) satisfying F(x 1, x 2 ) on the roots of the trees (all values if Q 1 is universal and at least one value if Q 1 is existential). The same for x 2 but at the depth 2. The same essential tranverse 2-destiny provides an (trivial) axiomatisation (more than 22 axioms) of Th 2 (N,<,P ) ∃ x( ∃ y(y>x) ∧ (x>y))) ∃ x( ∃ y(P(y) ∧ (x

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8 0.2. Some definitions about p-destinies Def. 0.2.1 (destiny). A p-destiny is a regular tree (X,P,r) of height p with a structuration map c : X is the set of nodes of the tree P is the fatherhood relation R is the root of the tree c maps a L-structure denoted c(n) to a node n, with domain the branche [r,n] so that if n 1 and n 2 are on a same branch, n 1 being an ancestor of n 2 then c(n 1 ) is a substructure of c(n 2 ) Def. 0.2.2 (M-complete tree of height p). We call M-complete tree of height p with root x the regular tree (LIST(M,p,x), P, (x)), where: (LIST(M,p,x) is the set of lists of elements of M with lenght at most p reaching the root x the root (x) is the list with a unique element x P is the binary relation defined on the set of lists by x 1 P x 2 iff x 1 is the list x 2 without its first element.

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9 Def. 0.2.3 (induced canonical structuration) Let M be a L-structure and x M. We define a structuration map c over (LIST(M,p,x),P,(x)) by the following condition: For every predicate R of L with arity k, for all families x 1 = (y 1,…), x 2 = (y 2,…), …,(x k = (y k,…) of lists with respective first elements y 1,y 2,…,y k belonging to the support of a same branch having z as a leaf of the tree (LIST(M,p,x),P,(x)), we have the following equivalence: c(z) satisfies R(x 1, x 2, …,x k ) iff M satisfies R(y 1, y 2, …,y k ) Def. 0.2.4 (exhaustive p-destiny) Let M be a L-structure p an integer and x M. We define the exhaustive p-destiny of x as being the p- destiny : (LIST(M,p,x),P,(x),c) where (LIST(M,p,x),P,(x)) is the M-complete tree with height p and root x, and c is the structuration canonical map induced on this tree by M.

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10 Working with the tree of root 0 and denoting by a circle the prime numbers, we get the exhaustive 2-destiny of 0 in (N,<,P)

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11 Def. 0.2.5 (stalk isomorphism). An isomorphism of stalk between two sub trees (resp. two destinies) is a bijection preserving the tree structure (i.e. the fatherhood relation) and being un isomorphism for the structuration map. A stalk (pétiole in french) is a subtree of a destiny equipped with the structuration induced by the structuration of the whole destiny Def. 0.2.6 (superimposition (or simplification) of rank k ) Given a p-destiny (A,P,r,c), we say (A’,P,’r,’c’) is a superimposition of rank k ( 0

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12 simplification of rank 2 for destinies associated to the congruence modulo 3 ( a cross means there no equality mod 3 )

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13 simplification of rank 2 for destinies associated to the congruence modulo 3 ( a cross means there no equality mod 3 )

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14. Def. 0.2.7 (simplication of rank k and general simplification) Given a family of superimpositions over the set of p- destinies of a fixed language Sj, Sj,…, Sp-1, each with a rank corresponding to its index, we call simplification at the k rank, the map Sj o Sj … o Sp-1. We call general simplification a simplification at first rank. Def. 0.2.8 (essential destiny). We call an essential p-destiny or an essential tranverse p-destiny or a tranverse p-destiny any range by a general simplication of a p-destiny

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15 Def. 0.2.8 (essential destiny). An essential p-destiny is the range of a general (of rank p) simplification of a p- destiny Transverse essential 2-destiny for the congruence modulo3

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16 0.3 some reasons of becoming attached to p-destinies This the best possibility I know for effectively writing a programm deciding a theory with fixed number of quantifiers and finitely many relations (FRT), Destinies are good tools to test the truth in FRT to computers Destinies provide an axiomatization for FRT, Transform problems into the research of finitely many elements determined by conditions, In the case of arithmetical theories, construction of destinies turns into finding integers realizing some conditions or does not exist. Destinies lead to the number-theoritical key-questions of a theory, and focus on the actual expressive power of a FRT Use the powerful technic of FRAISSE-EHRENFURT back-and- - forth in the steps of simplification Could be useful in data basis

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17 Alan WOODS (1981) and DR (1982) gave independent proofs of undecidability of Th(N, S, ⊥ ) and Th(N, =, S, ⊥ ). Interlude : Why to choose relations S and ⊥ ?? Many interesting problems in number theory emerge from the thesis of Alan Woods. One is now known as Erdös-Woods conjecture: The following are equivalent: (i) z=xy is (S, ⊥ )- definable in N (ii) z = x + y is (S, ⊥ )- definable in N (iii) x y is (S, ⊥ )- definable in N (iv) x = y is (S, ⊥ )- definable in N ( v) (Erdös-Woods Conjecture) There is some k >0 such that every natural number x is determined uniquely by the sequence S 0, S 1, …, S k of sets of (distinct) prime numbers defined by S i = {p / p divides x + i} ( vi) For any fixed prime p, the map n→p n is (S, ⊥ ) - definable in N.

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18 Importance of EWC. Let us call Supp(a) the set of prime (distinct) divisors of a. Consider the following conjectures of Number Theory : Oesterlé-Masser’s conjecture. (Also called a-b-c conjecture) For all (a,b) Î(N * ) 2, there exists an effectively computable constant C such that : Supp((a + b)ab) > C[ (a+ b)/ gcd (a,b) ] 1-ε(a) with ε(a) tends to 0 when a tends to infinity. Hall’s conjecture. Suppose x 3 ≠ y 2, for an effectively computable constant C, we have │ x 3 − y 2 │> [C. Max (x 3, y 2 )] 1/ 6 Hall and Schinzel ‘s conjecture Suppose x m ≠ y n, for an effectively computable constant C, we have │ x m − y n │> [Max (x 3, y 2 )] C M. Langevin results in [1988] the three previous conjectures are false in case EWC fails

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19 Moreover every conjecture which is false if Hall’s conjecture is false, becomes false in turn if EWC is. This is the case for Lang-Waldschmidt‘s conjecture (which provides lower bounds of linear forms of logarithms) and for Vojta ‘s conjecture about abelian varieties. Obviously a positive solution to one of this conjecture would give a positive solution to EWC. More precisely Hall and Schinzel‘s conjecture implies that if a is sufficiently large, then a is determined by S o,S 1, …, S 20.

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20 1) An algorithm of decision for Th 2 (N,S, ⊥ ) The root x can be or not coprime with x cannot be in a relation of succession with x; for instance: For a leaf k, there are relations of k to k and of k to x, : so that we have a vector with 3 componenets as shown below ; the arrow from left to right indicates x=S(k) and from right to left that k=S(x).

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21 Construction of the exhaustive essential destiny of Th 2 (N,<,P) Destinies of 0,1, 2 and 3 are straightforward:

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22 When n>3, the essential 2-destiny of n is as follows:

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23 Now, it is easy to produce the essential exhaustive transverse destiny of Th 2 (N,S, ⊥ ) which provides an algorithm of decision together with an axiomatisation of this theory:

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24 2) Results and questions on Th 3 (N,S, ⊥ ) As usual, le case with 3 is much more difficult than with 2 We denote Supp(n) the set of prime divisors of n. Below for 3-destiny, n is the root of the tree, k is a son of x and u a son of k Then, there are 5 basis cases associated to u which are : u = 0 is never coprime with an integer 2.2) The nodes k and n are distant (S is not involved) u = 1 is coprime with any integer u is not coprime to itself and coprime with k and n u is not coprime to itself and coprime with k but not with n u is not coprime to itself and coprime with n but not with k

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25 Therefore, there are at most 5 branches (n,k,u) for u far off from k and n for given k and n. The number of branches depends on four relative positions of Supp(k) and Supp(n) as follows: Supp(k) ⊂ Supp(n) and Supp(k) ≠ Supp(n) Supp(n) ⊂ Supp(k) and Supp(n) ≠ Supp(k) Supp(n) = Supp(k) Supp(n) \ Supp(k) ≠ ⍉ and Supp(k) \Supp(k) ≠ ⍉ In this situation 2.2, the 3-destinies are easy to construct (cz14)

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26 2.3) The node u is close to k or to n when k and n are distant (S is involved ) Because u { k-1, k+1, n-1, n+1} and k-1, k+1 have no successor link with n (similarly n-1, n+1 have no successor relation with k), it remains to discuss the coprimeness relationship between k-1, k+1 (resp. n-1, n+1) with n (resp. k). For each case, there are 2 possibilities so that the number of possible cases is 16, giving the following set of configurations of branches (n,k,u) for u close to n or to k

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28 Counting. For each (n,k) with n far off k, there are 5 possibilities for the branches (n,k,u) when u is distant from k and from n, and there are 16 possibilities for the branches (n,k,u) when u is close to k or to n, so that we have 80 possible stalks (pétiole in french) for n distant from k. But each of these branches can or cannot appear in the destiny of a given n, so that, a priori, there are 2 80 trees in the essential tranverse destiny of Th 3 (N,S, ⊥ ) Fortunately, we can enormously reduce the number of possible trees.

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29 2.4) Branches (n,k) when k is close to n (S is involved) We begin by n>3 (so that n-2 ≠0 and n-2 ≠1). The discussion is rather technical depending on the parity of n, and on the combinatorial of Supp(n) and Supp(n-2) introducing the cases n=4, n=6, all numbers of the form 2(2 α -1) and 2(2 α +1) This is the first time, we observe Th 3 (N,S, ⊥ ) depends of exponents into the primary decomposition of integers and the following cases appear After cleaning (n=4, …) the key-cases are the following : (n-3 ⊥ n) which appears iff n = 0 (mod 3) ; (n+1 ⊥ n-2) which appears iff n = 2 (mod 3); (n - 3 ⊥ n) and (n+1 ⊥ n-2) which appears iff n = 1 (mod 3), So that congruences modulo 3 appear in the construction

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30 Stalk (n,n-2) : we know exactly when the cases a, b, c, d and e appear or not (case k=n-2). There are 9 possible cases (excepting n=4 and n=5

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31 Stalk (n,n-1) : There are only 2 possible cases : (n is even and (n-2 ⊥ n) (case a) or (n+1 ⊥ n-1)(case b) (n is odd and (n-2 ⊥ n) (case a) or (n+1 ⊥ n-1) (case b)

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32 Stalk (n,n) : This is the unique possible case

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33 Stalk (n,n+1) : It is similar to the case k=n-1 (n is even and (n-1 ⊥ n+1) (case a) or(n+2 ⊥ n)(case b) (n is odd and(n-1 ⊥ n+1) (case a) or (n+2 ⊥ n) (case b)

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34 Stalk (n,n+2) : It is similar to the case k=n-2

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35 (Pathological) cases of Stalk(n,0) : here Supp(n) Supp(k) Stalk(n,1) : 2 possibilities according the parity of n Stalk(n,2) : Case a appears iff n=2 α ; case b appears iff n is odd; Case c appears iff n=0(mod3); case d and e appear iff n is odd

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36 2.3 Counting all possibilities Counting the cases when k is close to n and the destinies of 0,1,2, we find 14 cases Since 80 configurations was a majoration of the cases for k distant from n, we have at most 14 × 2 80 possible destinies. What is huge But in fact one can reduce to 646 cases allowing an exhaustive inspection by a computer : Annie château did it and proved the number of 3-destinies is at most 72.

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37 2.4 – Five open number theory questions necessary to construct an essential transverse destiny of Th 3 (N, S, ⊥ ). In fact, surprisingly, S and ⊥ do express properties of the exponents of the prime factors of integers. It turns out that two sets are the keys of our construction: X = {nN / ∃ kN (Supp(k) Supp(n) ∧ (k-1 ⊥ n) ∧ ( k+1 ⊥ n)} Y ={nN / ∃ kN (Supp(k) Supp(n) ∧ (k-1 ⊥ n) ∧ ( k+1 ⊥ n)} This construction depends on the intersections of { 2 n +1 / n N } and { 2 n -1 / n N } with X and Y This means an algorithm of decision through an essential transverse 3- destiny will need to solve some questions in number theory

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38 Prop 1. (Characterization of Y) Y is the set of natural integers having two distinct prime factors p and q such that Ord(p,q) (the order of the element q within the group (Z/pZ) * ) is even Prop 2. (Characterization of X∩(2N+1)). A positive odd integer belongs to X iff there is one of its prime divisors p, and two integers b and c with disjoint supports included in Supp(n) \ {p} such that v 2 (Ord(b,p) > v 2 (Ord(c,p) The work around these sets consists to find, for every form of n we discuss previously, a witness in X, or to prove there is no such integer a witness outside of Y, or to prove there is no such integer a witness in Y\X, or to prove there is no such integer. More precisely, on the 9 questions sufficient to achieve the 3- destiny, 5 remains open:

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39 Sketch of the proofs Finite groups theory Quadratic reciprocity law Aurifeuille’s decomposition of numbers of the form exp(2, ij2 α ) + 1. The construction of the essential 3-destiny of Th3(N, S, ⊥ ) depends on the answer (by yes or no) to nine questions

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40 1) Is there a = ( 2 n - 1) ∧ │Supp (a)│= 2 ∧ (a in Y) Answer :YES with a = 2 227 - 1 or a = 2 269 - 1 2) Is there a = ( 2 n - 1) ∧ │Supp (a)│= 2 ∧ (a in Y) Answer : YES with a = 2(2 269 - 1) 3) Is there a = ( 2 n - 1) ∧ │Supp (a)│ 3 ∧ (a in Y) OPEN 4) Is there a =( 2 n + 1 ∧ n even ∧ │Supp (a)│= 2 ∧ (a in Y) OPEN 5) = Is there a = ( 2 n + 1) ∧ n even ∧ │Supp (a)│ 3 ∧ (a in Y) OPEN

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41 6) = Is there a = ( 2 n - 1) ∧ n odd ∧ │Supp (a)│ 3 ∧ (a in Y\X) Answer: YES 2 25 -1 = 31.601.1801 7) = Is there a = ( 2 n - 1) ∧ n even ∧ │Supp (a)│ 3 ∧ (a in X) Answer : NO 8) = Is there a = ( 2 n + 1) ∧ n odd ∧ │Supp (a)│ 3 ∧ (a in X) Answer : OPEN 9) = Is there a = ( 2 n + 1) ∧ n even ∧ │Supp (a)│ 3 ∧ (a in Y\X) Answer : OPEN

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42 Conclusion. To – day, we know 67 trees of this 3-destiny (unpublished) which is made of at most 72 trees

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